power comparison of non-parametric tests: small-sample properties
TRANSCRIPT
Journal of Applied Statistics, Vol. 24, No. 5, 1997, 603± 632
Power comparison of non-parametric tests:Small-sample properties from Monte Carloexperiments*
HISASHI TANIZAKI, Kobe University, Japan
SUMMARY Non-parametric tests that deal with two samples include scores tests (such as
the Wilcoxon rank sum test, normal scores test, logistic scores test, Cauchy scores test,
etc.) and Fisher’s randomization test. B ecause the non-parametric tests generally require
a large amount of computational work, there are few studies on small-sample properties,
although asymptotic properties with regard to various aspects were studied in the past. In
this paper, the non-parametric tests are compared with the t-test through Monte Carlo
experiments. Also, we consider testing structural changes as an application in economics.
1 Introduction
There are many kinds of non-parametric test (distribution-free tests), such as
scores tests, Fisher’ s test, etc. However, almost all the studies in the past have
related to asymptotic properties. In this paper, we examine small-sample properties
of non-parametric two-sample tests by Monte Carlo experiments.
One of the features of non-parametric tests is that we do not have to impose any
assumption on the underlying distribution. With no restriction on the distribution,
it can be expected that non-parametric tests are less powerful than the conventional
parametric tests, such as the t-test. However, Hodges and Lehman (1956) and
ChernoŒand Savage (1958) showed that the Wilcoxon rank sum test is as powerful
as the t-test under the location shift alternatives and, moreover, that the Wilcoxon
test is sometimes much more powerful than the t-test. In particular, the remarkable
* This paper is an extention of Diebold et al. (1992) , where the Wilcoxon test, Fisher test and t-test
were compared with respect to sample power. In this paper, more non-parametric tests are examined.
Correspondence: H. Tanizaki, Faculty of Economics, Kobe University, Rokkodai, Nadaku, Kobe 657,
Japan.
0266-476 3/97/050603-3 0 $7.00 � 1997 Carfax Publishing Ltd
604 H. Tanizaki
fact about the Wilcoxon test is that it is about 95% as powerful as the usual t-test
for normal data. ChernoŒ and Savage (1958) proved that Pitman’ s asymptotic
relative e� ciency of the normal scores test relative to the t-test is greater than unity
under the location shift alternatives.1 This implies that the power of the normal
scores test is always greater than that of the t-test. According to Mehta and Patel
(1992), the normal scores test is less powerful than the Wilcoxon test if the tails of
the underlying distributions are diŒuse.
Fisher’ s test statistic is the diŒerence between two sample means, so it is
asymptotically equivalent to the t-test (Bradley, 1968).2 The scores tests are similar
to the Fisher test, except that the test statistic of the scores test is the sum of
scores, while the Fisher test statistic is the diŒerence between two sample means.
Both test statistics are discretely distributed and we have to obtain all the possible
combinations for the tests.
It is quite di� cult to obtain all the possible combinations, and the computational
time is also quite long. Mehta and Patel (1983, 1986a) and Mehta et al. (1984,
1985, 1988) carried out a program on the Fisher permutation test (a generalization
of the Fisher two-sample test treated in this paper, i.e. independence test by r 3 c
contingency table) using a network algorithm.3
In this paper, we consider small-sample properties of two-sample non-parametric
tests (i.e. the scores tests and the Fisher test) by comparing with the t-test, which
is the usual parametric test. Finally, the test of structural change is examined as an
application in economics.
2 Overview of non-parametric tests
It is well known for testing two-sample means that the t-test gives us a uniform
powerful test under the normality assumption, but not under non-normality. We
consider a distribution-free test in this paper, which is also called a non-parametric
test. The normal scores test, Wilcoxon (1945) rank sum test and Fisher (1935)
test are famous non-parametric tests, which are similar tests. We have two sample
groups. We test if two samples are generated from the same distribution. Let x1 ,
x2 , . . . , xn1 be mutually independently distributed as F(x), and let y1 , . . . , yn2
be mutually independently distributed as G( y). F(x) and G( y) are continuous
distribution functions. Under the assumptions, we consider the null hypothesis of
no diŒerence between two sample means. The null hypothesis H0 is represented by
H0 : F(x) 5 G(x)
The scores tests and the Fisher test are usually applied under the alternative of
location shift.4 One possible alternative hypothesis H1 is given by
H 1 : F(x) 5 G(x 2 l ), l > 0
where a shift in the location parameter l is tested.
Let n1 be the sample size of group 1 and let n2 be that of group 2. We consider
randomly taking n1 samples out of n1 + n2 samples, and mixing the two groups.
Then, we have n1+ n2Cn1 combinations. Each event of n1+ n2C n1 combinations occurs
with equal probability 1 /n1+ n2Cn1 . For the scores tests and the Fisher test, all the
possible combinations are compared with the original two samples.
Power of non-parametric tests 605
2.1 Scores tests
For the scores tests, the two samples {x i}n1i 5 1 and {y j}
n2j 5 1 are converted into the data
ranked by size. Let {Rx i}n1i 5 1 and {Ry j}
n2j 5 1 be the ranked samples that correspond
to {x i}n1i 5 1 and {y j}
n2j 5 1 . The scores test statistic s0 is represented by
s0 5 Rn1
i 5 1
a(Rx i) (1)
where a(´) is a function to be speci® ed.
For all the possible combinations of taking n1 samples out of n1 + n2 samples
(i.e. n1+ n2C n1 combinations), we compute the sum of the scores. Let the scores sum
be sm , m 5 1, 2, . . . , n1+ n2Cn1 . (Note that at least one of sm , m 5 1, 2, . . . , n1+ n2Cn1 , is
equal to s0 .) sm occurs with equal probability (i.e. 1 /n1+ n2C n1 ) for all the combina-
tions. Comparing s0 and sm , the following probabilities can be computed. We have
Prob(s< s0 ) 5Number of combinations less than s0 out of sm , m 5 1, 2, . . . , n1+ n2 Cn1
n1+ n2 C n1 (Number of all the possible combinations)
Prob(s 5 s0 ) 5Number of combinations equal to s0 out of sm , m 5 1, 2, . . . , n1+ n2 Cn1
n1+ n2 C n1 (Number of all the possible combinations)
Prob(s> s0 ) 5Number of combinations greater than s0 out of sm , m 5 1, 2, . . . , n1+ n2 Cn1
n1+ n2 Cn1 (Number of all the possible combinations)
where s is taken as a random variable generated from the scores test statistic.
If Prob(s< s0 ) is small enough, then s0 is located at the right tail of the
distribution, which implies that F(x)< G(x) for all x. Similarly, if Prob(s> s0 ) is
small enough, then s0 is located at the left tail of the distribution, which implies
that F(x)> G(x) for all x. Therefore, in the case of the null hypothesis H0 :
F(x) 5 G(x) and the alternative H1 : F(x) ¹ G(x), the null hypothesis is rejected at
the 10% signi ® cance level when Prob(s< s0) < 0.05 or Prob(s> s0) < 0.05.
We can consider various scores tests by specifying the function of a(´). The
scores tests examined in this paper are the Wilcoxon rank sum test, the normal
scores test, the logistic scores test and the Cauchy scores test.
2.1.1 Wilcoxon rank sum test. One of the most famous non-parametric tests is the
Wilcoxon rank sum test. The Wilcoxon test statistic w0 is the scores test de® ned as
a(Rx i) 5 Rx i, which is given by
w0 5 Rn1
i 5 1
Rx i (2)
In the past, it was too di� cult to obtain the exact distribution of w, from a
computational point of view. Therefore, under the null hypothesis, we have tested
utilizing the fact that w has approximately normal distribution, with mean E(w)
and variance Var(w). We have
E(w) 5n1(n1 + n2 + 1)
2
Var(w) 5n1 n2(n1 + n2 + 1)
12
606 H. Tanizaki
Therefore, in the past, the following statistic was used for the Wilcoxon test
statistic:
aw0 5w0 2 E(w)
[Var(w)]1 /2 (3)
This is called the asymptotic Wilcoxon test statistic in this paper. aw is asymptotic-
ally distributed as a standard normal random variable. Mann and Whitney (1947)
demonstrated that the normal approximation is quite accurate when n1 and n2 are
larger than 7 (see Mood et al., 1974).
Hodges and Lehman (1956) showed that Pitman’ s asymptotic relative e� ciency
of the Wilcoxon test relative to the t-test is quite good. They obtained the result
that the asymptotic relative e� ciency is greater than 0.864 under the null hypothesis
of location shift. This result implies that the Wilcoxon test does not perform too
poorly compared with the t-test and, moreover, that the Wilcoxon test may be
much better than the t-test. In particular, they showed that the relative e� ciency
of the Wilcoxon test is 1.33 when the density function f(x) takes the form 5
f(x) 5x
2 exp( 2 x)
C (3)(4)
where C (3) is a gamma function with parameter 3. In general, for the distributions
with large tails, the Wilcoxon test is more powerful than the t-test.
All the past studies are concerned with asymptotic properties. In Section 3, we
examine the small-sample cases, i.e. n1 5 n2 5 5, 7, 9.
2.1.2 Normal scores test. The normal scores test statistic ns0 is
ns0 5 Rn1
i 5 1
U2 1( Rx i
n1 + n2 + 1) (5)
where U (´) is a standard normal distribution. The scores test that a(´) in equation
(1) is assumed to be
a(x) 5 U2 1( x
n1 + n2 + 1)is called the normal scores test.6
ChernoŒand Savage (1958) proved that the asymptotic relative e� ciency of the
normal scores test relative to the t-test is greater than or equal to unity, i.e. that
the normal scores test is equivalent to the t-test under the normality assumption
and that the power of the normal scores test is greater than that of the t-test
otherwise.
2.1.3 Logistic scores test. The logistic scores test statistic ls0 is given by
ls0 5 Rn1
i 5 1
F2 1( Rx i
n1 + n2 + 1) (6)
Power of non-parametric tests 607
where
F(x) 51
1 + e 2 x
which is a logistic distribution.
2.1.4 Cauchy scores test. The Cauchy scores test statistic cs0 is represented by
cs0 5 Rn1
i 5 1
F2 1( Rx i
n1 + n2 + 1) (7)
where F(x) 5 1 /2 + (1 / p ) tan 2 1x, which is a Cauchy distribution.
By specifying a functional form for a(´), various scores tests can be constructed.
In this paper, the four scores tests discussed above and the Fisher test in the
following section are compared.
2.2 Fisher’s two-sample test
While the Wilcoxon test statistic is the rank sum of the two samples, the Fisher
test statistic uses the diŒerence between two sample means, i.e. xÅ 2 yÅ for the two
samples {x i}n1i 5 1 and {yj }
n2j5 1 . Thus, the test statistic is given by
f0 5 xÅ 2 yÅ (8)
where xÅ 5 (1 /n1) R x i and yÅ 5 (1 /n2) R y i. For all the possible combinations (i.e.
n1+ n2Cn1 combinations taking n1 out of n1 + n2), we compute the diŒerence between
the sample means. Let fm , m 5 1, 2, . . . , n1+ n2Cn1 , be the diŒerence between the two
samples means for all the possible combinations. (Note that at least one out of
fm , m 5 1, 2, . . . , n1+ n2C n1 , is equal to f0 .) For all m , fm occurs with equal probability
(i.e. 1 /n1+ n2Cn1 ). Comparing f0 and fm , we can compute Prob(f< f0 ), Prob(f 5 f0)
and Prob(f> f0 ), where f is a random variable generated from the Fisher test
statistic.
Fisher’ s two-sample test is of the same type as the scores tests, in the sense of
the use of all the possible combinations, but the Fisher test uses more information
than do the scores tests, because the scores tests utilize the ranked data as the test
statistics, while the Fisher test uses the original data. It might be expected that the
Fisher test is more powerful than the scores tests. However, Bradley (1968) stated
that the Fisher test and the t-test are asymptotically equivalent, because they both
use the diŒerence between two sample means as the test statistic.7 Therefore, it
can be shown that the asymptotic relative e� ciency of the Fisher test is sometimes
better or worse, compared with the Wilcoxon test.
3 Power comparison (small-sample properties)
In Section 2, we have introduced the four scores tests and the Fisher test. In this
section, we examine the small-sample properties by Monte Carlo experiments.
Assuming a speci® c distribution for group 1 samples (i.e. {x i}n1i 5 1 ) and group
2 samples (i.e. {yj }n2j 5 1), and generating random draws, we compare the non-
parametric tests and t-test with respect to sample power. The t-test is compared
with the non-parametric tests introduced in this paper.
608 H. Tanizaki
The t-test statistic is represented as
t0 5xÅ 2 yÅ
s(1 /n1 + 1 /n2)1 /2
where the degree of freedom is n1 + n2 2 2. xÅ , yÅ and s are given by
xÅ 5 Ri
x i , yÅ 5 Rj
y j , s25
(n1 2 1)s2x + (n2 2 1)s
2y
n1 + n2 2 2
s2x 5
1
n1 2 1 R i
(x i 2 xÅ )2 , s2y 5
1
n2 2 1 R j
(yj 2 yÅ )2
In the case of Var(x) 5 Var(y), the t-test provides a uniformly very powerful test.
Otherwise, the test statistic t0 does not follow a t-distribution, so the t-test is
meaningless.
The null hypothesis is H0 : F(x) 5 G(x). We compare the sample powers for shifts
not only in the location parameter (i.e. l ) but also in the scale parameter (i.e. r ).
Therefore, the alternative hypothesis is given by
H1 : F(x) 5 G(x 2 l
r )We perform Monte Carlo experiments in the following cases: n1 5 n2 5 5, 7, 9,
l 5 0.0, 0.5, 1.0 , and r 5 1.0, 1.5, 2.0 for each of the signi® cance levels, i.e.
a 5 0.10, 0.05, 0.01. The underlying distributions of {x i} and {y j} are normal in
Table 1, uniform in Table 2, logistic in Table 3, Cauchy in Table 4 and v2(6) /2 in
Table 5.
First, we generate normal random draws as x i ~ N(0, 1) for i 5 1, . . . , n1, and
y i ~ N( l , r2) for j 5 1, . . . , n2. Theoretically, in the case of r 5 1, the t-test is more
powerful than any other test, because the underlying distribution is normal with
equal variance; in the case of r ¹ 1, the non-parametric tests generally are more
powerful than the t-test. (However, because the studies by Hodges and Lehman
(1956) and ChernoΠand Savage (1958) can be applied only to a shift in the
location parameter, it is not appropriate to conclude that the non-parametric tests
are more powerful than the t-test in the case of r ¹ 1.) The results are in Table 1.8
t, w, aw, ns, ls, cs and f represent the t-test, Wilcoxon test, asymptotic Wilcoxon
test, normal scores test, logistic scores test and Cauchy scores test respectively.
Even in the case of r 5 1, the Wilcoxon test exhibits greater power than the Fisher
test and t-test. ns, ls and cs are not as powerful as t. Also aw does not perform as
well as w, although aw approaches w as the sample size increases. The Fisher two-
sample test uses more information than does the Wilcoxon test, in that the Fisher
test utilizes the original data, while the Wilcoxon test utilizes the ranked data.
Therefore, it is expected that the small-sample Fisher test is more powerful than
the Wilcoxon test. However, the Monte Carlo experiment shows that f performs
between t and w but f is close to t.
In Table 2, the uniform random draws are generated as follows: x i ~ U( 2 5, 5)
for i 5 1, . . . , n1, and y j 5 l + r v j , where v j ~ U( 2 5, 5), for j 5 1, . . . , n2. For a large
shift in the location parameter (i.e. large l ), the Cauchy scores test is the most
powerful test when r 5 1, l increases, and n1 and n2 are large. w is the best test
for almost all the cases.
Power of non-parametric tests 609
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610 H. Tanizaki
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6
w0
.24
37
ëëë
0.2
02
2ëë
ë0
.17
67
ëëë
0.2
62
3ëë
ë0
.20
51
ëëë
0.1
72
3ëë
ë0
.28
56
ëëë
0.3
23
4ëë
ë0
.18
65
ëëë
aw
0.1
77
0*
*0
.13
84
***
0.1
16
8**
*0
.21
80
**
0.1
70
8*
0.1
42
5*
0.2
56
3*
**
0.1
94
3*
**
0.1
62
9*
ns
0.1
96
3ëë
0.1
56
4ë
0.1
27
80
.22
30
*0
.16
75
*0
.13
00
**
*0
.26
65
*0
.19
86
**
0.1
49
7**
*
ls0
.19
63
ëë0
.15
64
ë0
.12
78
0.2
23
3*
0.1
65
8*
*0
.12
82
**
*0
.26
55
*0
.19
32
***
0.1
39
2**
*
cs0
.18
19
*0
.14
30
*0
.11
61
**
*0
.21
61
**
0.1
55
1*
**
0.1
14
3**
*0
.25
32
**
*0
.16
97
***
0.1
05
5**
*
1.0
t0
.42
19
0.3
17
10
.25
42
0.5
52
30
.40
81
0.3
09
40
.65
59
0.4
89
50
.37
07
f0
.42
69
ë0
.32
18
ë0
.25
85
0.5
52
40
.41
04
0.3
10
60
.65
63
0.4
89
30
.37
12
w0
.49
82
ëë0
.38
27
ëëë
0.3
19
0ëë
ë0
.58
48
ëëë
0.4
40
6ëë
ë0
.34
45
ëëë
0.6
58
30
.49
46
ë0
.38
32
ëë
aw
0.3
96
9*
**
0.2
90
8*
**
0.2
24
7**
*0
.52
52
**
*0
.38
60
***
0.2
95
6**
0.6
20
6*
**
0.4
56
3*
**
0.3
51
0**
*
ns
0.4
28
8ë
0.3
14
30
.24
32
**
0.5
34
0*
**
0.3
80
9*
**
0.2
74
0**
*0
.63
93
**
*0
.46
19
***
0.3
31
2**
*
ls0
.42
88
ë0
.31
43
0.2
43
2**
0.5
34
5*
**
0.3
79
5*
**
0.2
69
6**
*0
.63
68
**
*0
.45
38
***
0.3
15
4**
*
cs0
.40
55
**
*0
.29
12
***
0.2
18
8**
*0
.52
06
**
*0
.35
69
***
0.2
38
2**
*0
.60
47
**
*0
.39
85
***
0.2
43
0**
*
Power of non-parametric tests 611
TA
BL
E1
Ð(C
on
tin
ued
)
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( c)
a5
0.0
1
0.0
t0
.00
92
0.0
09
80
.01
14
0.0
10
30
.01
03
0.0
11
60
.01
17
0.0
12
20
.01
33
f0
.01
17
ëë0
.01
21
ëë0
.01
42
ëë0
.00
97
0.0
10
80
.01
27
ë0
.01
13
0.0
12
30
.01
36
w0
.01
65
ëëë
0.0
17
1ëë
ë0
.01
88
ëëë
0.0
13
3ëë
0.0
15
8ëë
ë0
.01
76
ëëë
0.0
14
1ëë
0.0
13
9ë
0.0
17
0ëë
ë
aw
0.0
07
6*
0.0
08
3*
0.0
09
9*
0.0
09
70
.00
96
0.0
12
10
.01
11
0.0
11
30
.01
30
ns
0.0
16
5ëë
ë0
.01
71
ëëë
0.0
18
8ëë
ë0
.01
11
0.0
12
4ëë
0.0
13
6ë
0.0
12
60
.01
09
*0
.01
16
*
ls0
.01
65
ëëë
0.0
17
1ëë
ë0
.01
88
ëëë
0.0
10
50
.01
21
ë0
.01
30
ë0
.01
24
0.0
10
7*
0.0
11
0**
cs0
.01
65
ëëë
0.0
17
1ëë
ë0
.01
88
ëëë
0.0
11
00
.01
20
ë0
.01
32
ë0
.01
20
0.0
10
8*
0.0
09
8**
*
0.5
t0
.04
95
0.0
40
10
.03
59
0.0
70
60
.05
05
0.0
41
60
.09
53
0.0
67
10
.05
09
f0
.05
52
ëë0
.04
58
ëë0
.04
30
ëëë
0.0
69
70
.05
12
0.0
42
70
.09
46
0.0
66
80
.05
22
w0
.07
11
ëëë
0.0
59
3ëë
ë0
.05
14
ëëë
0.0
84
2ëë
ë0
.06
10
ëëë
0.0
53
6ëë
ë0
.10
82
ëëë
0.0
73
7ëë
0.0
60
3ëë
ë
aw
0.0
39
6*
**
0.0
32
2*
**
0.0
30
0**
*0
.06
20
**
*0
.04
50
**
0.0
36
0**
0.0
88
1*
*0
.06
06
**
0.0
48
7*
ns
0.0
71
1ëë
ë0
.05
93
ëëë
0.0
51
4ëë
ë0
.07
01
0.0
52
00
.04
18
0.0
94
70
.06
01
**
0.0
45
2**
ls0
.07
11
ëëë
0.0
59
3ëë
ë0
.05
14
ëëë
0.0
68
40
.04
89
0.0
39
6*
0.0
94
90
.05
86
***
0.0
42
9**
*
cs0
.07
11
ëëë
0.0
59
3ëë
ë0
.05
14
ëëë
0.0
68
50
.04
90
0.0
39
3*
0.0
90
5*
0.0
55
0*
**
0.0
37
2**
*
1.0
t0
.16
61
0.1
07
00
.08
29
0.2
60
40
.16
59
0.1
16
30
.35
73
0.2
25
70
.15
09
f0
.17
98
ëëë
0.1
21
0ëë
ë0
.09
40
ëëë
0.2
62
90
.16
71
0.1
22
6ë
0.3
56
90
.22
67
0.1
56
1ë
w0
.21
04
ëëë
0.1
47
0ëë
ë0
.11
20
ëëë
0.2
82
0ëë
ë0
.18
74
ëëë
0.1
32
9ëë
ë0
.37
31
ëëë
0.2
40
0ëë
ë0
.16
80
ëëë
aw
0.1
32
0*
**
0.0
89
0*
**
0.0
71
7*
0.2
31
5*
**
0.1
42
7*
**
0.0
97
7**
*0
.33
59
**
*0
.20
83
***
0.1
43
3**
ns
0.2
10
4ëë
ë0
.14
70
ëëë
0.1
12
0ëë
ë0
.25
29
**
0.1
60
5*
0.1
07
4**
0.3
44
9*
*0
.20
69
***
0.1
31
3**
*
ls0
.21
04
ëëë
0.1
47
0ëë
ë0
.11
20
ëëë
0.2
48
3*
*0
.15
40
***
0.1
01
7**
*0
.34
23
**
0.2
01
4*
**
0.1
23
3**
cs0
.21
04
ëëë
0.1
47
0ëë
ë0
.11
20
ëëë
0.2
47
0*
**
0.1
54
1*
**
0.1
02
0**
*0
.32
81
**
*0
.18
88
***
0.1
09
1**
*
612 H. Tanizaki
TA
BL
E2
.U
nif
orm
dis
trib
uti
on
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( a)
a5
0.1
0
0.0
t0
.09
85
0.1
00
40
.10
37
0.1
01
80
.10
02
0.1
01
50
.09
66
0.1
02
30
.10
46
f0
.10
38
ë0
.10
51
ë0
.10
89
ë0
.10
31
0.1
01
00
.10
29
0.0
97
90
.10
30
0.1
05
3
w0
.11
27
ëëë
0.1
17
1ëë
ë0
.12
34
ëëë
0.1
07
6ë
0.1
12
7ëë
ë0
.11
87
ëëë
0.1
09
7ëë
ë0
.11
87
ëëë
0.1
26
2ëë
ë
aw
0.1
12
7ëë
ë0
.11
71
ëëë
0.1
23
4ëë
ë0
.10
76
ë0
.11
27
ëëë
0.1
18
7ëë
ë0
.09
40
0.1
02
80
.11
06
ë
ns
0.1
04
2ë
0.1
04
3ë
0.1
04
40
.10
30
0.0
97
50
.09
47
**
0.1
01
4ë
0.0
94
9*
*0
.09
40
**
*
ls0
.10
35
ë0
.09
54
*0
.08
23
**
*0
.10
27
0.0
93
1*
*0
.08
86
**
*0
.10
09
ë0
.08
97
***
0.0
82
7**
*
cs0
.10
35
ë0
.09
54
*0
.08
23
**
*0
.10
30
0.0
80
5*
**
0.0
58
6**
*0
.10
19
ë0
.06
14
***
0.0
34
5**
*
0.5
t0
.14
63
0.1
37
60
.13
40
0.1
62
40
.14
56
0.1
36
60
.17
64
0.1
59
00
.14
85
f0
.15
35
ëë0
.14
35
ë0
.13
85
ë0
.16
46
0.1
47
90
.13
73
0.1
78
60
.16
05
0.1
49
7
w0
.16
60
ëëë
0.1
55
1ëë
ë0
.15
27
ëëë
0.1
71
2ëë
0.1
54
8ëë
0.1
50
2ëë
ë0
.19
34
ëëë
0.1
73
3ëë
ë0
.16
53
ëëë
aw
0.1
66
0ëë
ë0
.15
51
ëëë
0.1
52
7ëë
ë0
.17
12
ëë0
.15
48
ëë0
.15
02
ëëë
0.1
69
6*
0.1
50
9*
*0
.14
44
*
ns
0.1
59
4ëë
ë0
.13
93
0.1
30
70
.17
22
ëë0
.13
78
**
0.1
24
1**
*0
.19
19
ëëë
0.1
41
4*
**
0.1
24
0**
*
ls0
.16
18
ëëë
0.1
28
5*
*0
.10
39
**
*0
.17
47
ëëë
0.1
30
1*
**
0.1
13
0**
*0
.19
62
ëëë
0.1
35
9*
**
0.1
12
8**
*
cs0
.16
18
ëëë
0.1
28
5*
*0
.10
39
**
*0
.18
26
ëëë
0.1
14
6*
**
0.0
76
2**
*0
.21
30
ëëë
0.0
93
7*
**
0.0
48
7**
*
1.0
t0
.20
98
0.1
84
00
.16
76
0.2
42
50
.20
63
0.1
83
00
.27
92
0.2
30
20
.19
96
f0
.21
72
ë0
.19
16
ë0
.17
30
ë0
.24
71
0.2
08
80
.18
47
0.2
82
10
.23
19
0.2
00
8
w0
.23
67
ëëë
0.1
97
1ëë
ë0
.18
25
ëëë
0.2
55
5ëë
ë0
.20
97
0.1
89
4ë
0.3
04
5ëë
ë0
.23
78
ë0
.21
10
ëë
aw
0.2
36
7ëë
ë0
.19
71
ëëë
0.1
82
5ëë
ë0
.25
55
ëëë
0.2
09
70
.18
94
ë0
.26
83
**
0.2
12
2*
**
0.1
88
1*
*
ns
0.2
29
9ëë
ë0
.17
86
*0
.15
74
**
0.2
68
6ëë
ë0
.19
43
**
0.1
58
0**
*0
.31
35
ëëë
0.2
03
3*
*0
.16
64
**
*
ls0
.24
02
ëëë
0.1
68
6*
**
0.1
27
1**
*0
.27
40
ëëë
0.1
85
0*
**
0.1
45
2**
*0
.32
24
ëëë
0.1
95
3*
**
0.1
48
6**
*
cs0
.24
02
ëëë
0.1
68
6*
**
0.1
27
1**
*0
.29
45
ëëë
0.1
63
5*
**
0.0
98
3**
*0
.36
16
ëëë
0.1
45
3*
**
0.0
68
5**
*
Power of non-parametric tests 613
TA
BL
E2
Ð(C
on
tin
ued
)
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( b)
a5
0.0
5
0.0
t0
.05
13
0.0
54
90
.06
03
0.0
47
00
.05
06
0.0
53
50
.04
85
0.0
53
70
.05
65
f0
.05
40
ë0
.05
50
0.0
59
50
.04
70
0.0
50
30
.05
33
0.0
48
90
.05
37
0.0
56
8
w0
.07
59
ëëë
0.0
81
8ëë
ë0
.08
82
ëëë
0.0
62
3ëë
ë0
.06
86
ëëë
0.0
72
9ëë
ë0
.08
76
ëëë
0.0
62
8ëë
ë0
.07
06
ëëë
aw
0.0
48
6*
0.0
49
5*
*0
.04
91
**
*0
.04
52
0.0
52
50
.05
72
ë0
.04
69
0.0
52
50
.05
90
ë
ns
0.0
56
5ëë
0.0
57
2ë
0.0
54
3**
0.0
49
00
.05
01
0.0
46
6**
*0
.05
05
0.0
49
8*
0.0
48
7**
*
ls0
.05
65
ëë0
.05
72
ë0
.05
43
**
0.0
49
7ë
0.0
49
30
.04
55
**
*0
.04
98
0.0
46
7*
**
0.0
42
1**
*
cs0
.05
04
0.0
53
80
.05
10
**
*0
.04
95
ë0
.04
78
*0
.03
88
**
*0
.05
00
0.0
39
4*
**
0.0
25
0**
*
0.5
t0
.07
99
0.0
74
90
.07
43
0.0
88
10
.08
07
0.0
75
70
.09
39
0.0
88
00
.08
24
f0
.08
21
0.0
76
10
.07
39
0.0
88
30
.08
06
0.0
75
40
.09
40
0.0
88
20
.08
22
w0
.11
51
ëëë
0.1
09
4ëë
ë0
.11
10
ëëë
0.1
12
8ëë
ë0
.09
87
ëëë
0.0
94
0ëë
ë0
.10
70
ëëë
0.0
94
9ëë
0.0
95
2ëë
ë
aw
0.0
75
2*
0.0
69
1*
*0
.06
49
**
*0
.08
48
*0
.07
73
*0
.07
53
0.0
89
3*
0.0
80
8*
*0
.07
96
*
ns
0.0
88
2ëë
ë0
.07
88
ë0
.07
20
0.0
92
9ë
0.0
73
3*
*0
.06
38
**
*0
.10
36
ëëë
0.0
79
1*
**
0.0
67
3**
*
ls0
.08
82
ëëë
0.0
78
8ë
0.0
72
00
.09
38
ëë0
.07
26
**
0.0
62
5**
*0
.10
57
ëëë
0.0
73
6*
**
0.0
60
0**
*
cs0
.08
26
0.0
73
80
.06
73
**
0.0
95
4ëë
0.0
71
1*
**
0.0
53
2**
*0
.11
44
ëëë
0.0
60
9*
**
0.0
35
6**
*
1.0
t0
.11
94
0.1
03
00
.09
61
0.1
45
80
.11
58
0.1
05
30
.16
89
0.1
36
70
.11
82
f0
.12
15
0.1
03
30
.09
56
0.1
45
40
.11
59
0.1
04
40
.16
87
0.1
36
50
.11
74
w0
.16
67
ëëë
0.1
42
8ëë
ë0
.13
64
ëëë
0.1
76
2ëë
ë0
.13
63
ëëë
0.1
20
0ëë
ë0
.18
24
ëëë
0.1
40
00
.12
57
ëë
aw
0.1
15
6*
0.0
92
3*
**
0.0
80
7**
*0
.14
09
*0
.10
83
**
0.0
98
3**
0.1
59
6*
*0
.11
95
***
0.1
09
6**
ns
0.1
34
3ëë
ë0
.10
63
ë0
.09
11
*0
.15
48
ëë0
.10
37
***
0.0
83
6**
*0
.18
39
ëëë
0.1
18
1*
**
0.0
93
0**
*
ls0
.13
43
ëëë
0.1
06
3ë
0.0
91
1*
0.1
56
9ëë
ë0
.10
25
***
0.0
81
3**
*0
.19
03
ëëë
0.1
14
3*
**
0.0
83
0**
*
cs0
.12
73
ëë0
.09
98
*0
.08
46
**
*0
.16
43
ëëë
0.1
02
9*
**
0.0
68
9**
*0
.21
26
ëëë
0.0
96
7*
**
0.0
51
0**
*
614 H. Tanizaki
TA
BL
E2
Ð(C
on
tin
ued
)
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( c)
a5
0.0
1
0.0
t0
.01
38
0.0
14
80
.01
74
0.0
09
90
.01
28
0.0
15
30
.01
21
0.0
12
70
.01
59
f0
.01
32
0.0
16
3ë
0.0
17
90
.00
90
0.0
11
4*
0.0
13
8*
0.0
10
5*
0.0
12
00
.01
45
*
w0
.01
70
ëë0
.02
03
ëëë
0.0
22
6ëë
ë0
.01
21
ëë0
.01
49
ë0
.01
87
ëë0
.01
30
0.0
15
2ëë
0.0
19
3ëë
aw
0.0
09
0*
**
0.0
10
5*
**
0.0
13
3**
*0
.00
84
*0
.00
94
***
0.0
11
4**
*0
.01
00
*0
.01
21
0.0
14
5*
ns
0.0
17
0ëë
0.0
20
3ëë
ë0
.02
26
ëëë
0.0
09
40
.01
19
0.0
13
7*
0.0
10
5*
0.0
12
20
.01
19
**
*
ls0
.01
70
ëë0
.02
03
ëëë
0.0
22
6ëë
ë0
.00
89
*0
.01
14
*0
.01
30
*0
.01
05
*0
.01
12
*0
.01
05
**
*
cs0
.01
70
ëë0
.02
03
ëëë
0.0
22
6ëë
ë0
.00
88
*0
.01
16
*0
.01
31
*0
.01
15
0.0
11
2*
0.0
10
4**
*
0.5
t0
.02
02
0.0
22
20
.02
54
0.0
19
70
.02
01
0.0
21
80
.02
51
0.0
23
90
.02
33
f0
.02
05
0.0
22
20
.02
45
0.0
17
5*
0.0
18
2*
0.0
20
70
.02
29
*0
.02
21
*0
.02
20
w0
.02
64
ëëë
0.0
27
9ëë
ë0
.03
00
ëë0
.02
29
ëë0
.02
40
ëë0
.02
48
ëë0
.02
57
0.0
25
10
.02
61
ë
aw
0.0
13
9*
**
0.0
14
7*
**
0.0
17
9**
*0
.01
55
**
*0
.01
58
***
0.0
16
6**
*0
.02
08
**
0.0
19
8*
*0
.02
11
*
ns
0.0
26
4ëë
ë0
.02
79
ëëë
0.0
30
0ëë
0.0
18
40
.01
95
0.0
18
5**
0.0
22
7*
0.0
19
4*
*0
.01
70
**
*
ls0
.02
64
ëëë
0.0
27
9ëë
ë0
.03
00
ëë0
.01
82
*0
.01
90
0.0
17
9**
0.0
22
9*
0.0
19
1*
**
0.0
15
4**
*
cs0
.02
64
ëëë
0.0
27
9ëë
ë0
.03
00
ëë0
.01
78
*0
.01
93
0.0
18
0**
0.0
24
20
.01
87
***
0.0
14
3**
*
1.0
t0
.03
22
0.0
31
10
.03
20
0.0
35
70
.03
21
0.0
30
30
.04
65
0.0
38
70
.03
61
f0
.03
14
0.0
29
2*
0.0
31
80
.03
19
**
0.0
29
0*
0.0
29
10
.04
31
*0
.03
62
*0
.03
45
w0
.03
96
ëëë
0.0
37
4ëë
ë0
.03
87
ëëë
0.0
41
5ëë
ë0
.03
58
ëë0
.03
38
ëë0
.05
18
ëë0
.04
16
ë0
.03
80
ë
aw
0.0
22
1*
**
0.0
22
1*
**
0.0
22
4**
*0
.02
84
**
*0
.02
49
***
0.0
22
3**
*0
.04
16
**
0.0
34
0*
*0
.03
17
**
ns
0.0
39
6ëë
ë0
.03
74
ëëë
0.0
38
7ëë
ë0
.03
64
0.0
29
2*
0.0
25
9**
0.0
46
30
.03
38
**
0.0
26
0**
*
ls0
.03
96
ëëë
0.0
37
4ëë
ë0
.03
87
ëëë
0.0
36
30
.02
89
*0
.02
47
**
*0
.04
69
0.0
33
7*
*0
.02
45
**
*
cs0
.03
96
ëëë
0.0
37
4ëë
ë0
.03
87
ëëë
0.0
35
90
.02
89
*0
.02
50
**
*0
.05
30
ëëë
0.0
32
6*
**
0.0
22
5**
*
Power of non-parametric tests 615
TA
BL
E3
.L
og
isti
cd
istr
ibu
tio
n
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( a)
a5
0.1
0
0.0
t0
.10
07
0.1
01
50
.10
56
0.1
05
60
.10
21
0.1
03
10
.10
03
0.1
01
90
.10
48
f0
.10
36
0.1
05
7ë
0.1
09
2ë
0.1
04
60
.10
22
0.1
01
00
.09
92
0.1
00
70
.10
29
w0
.11
27
ëëë
0.1
15
3ëë
ë0
.11
76
ëëë
0.1
07
60
.10
71
ë0
.11
14
ëë0
.10
97
0.1
16
6ëë
ë0
.12
12
ëëë
aw
0.1
12
7ëë
ë0
.11
53
ëëë
0.1
17
6ëë
ë0
.10
76
0.1
07
1ë
0.1
11
4ëë
0.0
94
0**
0.1
00
10
.10
45
ns
0.1
04
2ë
0.1
05
9ë
0.1
04
60
.10
30
0.1
00
90
.09
60
**
0.1
01
40
.09
91
0.0
96
5*
*
ls0
.10
35
0.1
01
10
.09
28
**
*0
.10
27
0.0
98
8*
0.0
94
0*
*0
.10
09
0.0
98
5*
0.0
91
8**
*
cs0
.10
35
0.1
01
10
.09
28
**
*0
.10
30
0.0
92
5*
**
0.0
79
4**
*0
.10
19
0.0
86
5*
**
0.0
66
2**
*
0.5
t0
.20
14
0.1
80
60
.16
59
0.2
24
20
.19
40
0.1
76
60
.24
59
0.2
09
90
.18
95
f0
.20
54
0.1
83
00
.16
75
0.2
22
50
.19
17
0.1
74
50
.24
47
0.2
08
10
.18
74
w0
.21
32
ëë0
.19
13
ëë0
.18
10
ëëë
0.2
29
6ë
0.1
98
20
.18
72
ëë0
.26
86
ëëë
0.2
29
3ëë
ë0
.21
13
ëëë
aw
0.2
13
2ëë
0.1
91
3ëë
0.1
81
0ëë
ë0
.22
96
ë0
.19
82
ë0
.18
72
ëë0
.23
94
*0
.20
49
*0
.18
68
ns
0.2
00
30
.17
71
0.1
62
50
.22
42
0.1
87
1*
0.1
64
8**
*0
.24
47
0.2
02
7*
0.1
74
7**
*
ls0
.19
71
*0
.17
01
**
0.1
46
5**
*0
.22
24
0.1
82
0*
**
0.1
57
5**
*0
.24
33
0.1
99
6*
*0
.16
48
**
*
cs0
.19
71
*0
.17
01
**
0.1
46
5**
*0
.21
17
**
0.1
66
4*
**
0.1
32
1**
*0
.22
24
**
*0
.16
77
***
0.1
14
9**
*
1.0
t0
.34
32
0.2
79
00
.24
12
0.4
05
60
.32
65
0.2
78
10
.45
91
0.3
66
00
.30
70
f0
.34
96
ë0
.28
60
ë0
.24
45
0.4
05
40
.32
42
0.2
74
70
.45
74
0.3
64
30
.30
32
w0
.35
89
ëëë
0.3
00
4ëë
ë0
.26
23
ëëë
0.4
16
8ëë
0.3
36
9ëë
0.2
88
2ëë
0.4
90
70
.39
29
ëëë
0.3
30
1ëë
ë
aw
0.3
58
9ëë
ë0
.30
04
ëëë
0.2
62
3ëë
ë0
.41
68
ëë0
.33
69
0.2
88
2ëë
0.4
53
2*
0.3
56
1*
*0
.29
77
**
ns
0.3
43
60
.27
91
0.2
36
7*
0.4
01
40
.31
87
*0
.26
04
**
*0
.45
56
0.3
52
4*
*0
.28
46
**
*
ls0
.33
70
*0
.26
78
**
0.2
12
2**
*0
.39
79
*0
.31
02
***
0.2
49
9**
*0
.45
30
*0
.34
37
***
0.3
69
9**
*
cs0
.35
70
*0
.26
78
**
0.2
12
2**
*0
.36
92
**
*0
.27
61
***
0.2
01
2**
*0
.39
66
**
*0
.27
88
***
0.1
84
9**
*
616 H. Tanizaki
TA
BL
E3
Ð(C
on
tin
ued
)
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
(b)
a5
0.0
5
0.0
t0
.04
93
0.0
51
40
.05
40
0.0
47
30
.05
04
0.0
53
10
.05
08
0.0
51
30
.05
46
f0
.05
25
ë0
.05
38
ë0
.05
77
ë0
.04
76
0.0
51
60
.05
44
0.0
51
30
.05
14
0.0
55
6
w0
.07
59
ëëë
0.0
79
6ëë
ë0
.08
33
ëëë
0.0
62
3ëë
ë0
.06
64
ëëë
0.0
69
5ëë
ë0
.05
76
ëëë
0.0
60
4ëë
ë0
.06
44
ëëë
aw
0.0
48
60
.04
93
0.0
51
2*
0.0
45
20
.04
97
0.0
52
70
.04
68
*0
.05
04
0.0
54
3
ns
0.0
56
5ëë
ë0
.05
63
ëë0
.05
68
ë0
.04
90
0.0
51
10
.05
05
*0
.05
05
0.0
52
00
.05
02
*
ls0
.05
65
ëëë
0.0
56
3ëë
0.0
56
8ë
0.0
49
7ë
0.0
51
00
.05
01
*0
.04
95
0.0
50
80
.04
67
**
*
cs0
.05
04
0.0
50
60
.05
01
*0
.04
95
ë0
.04
87
0.0
45
2**
*0
.05
00
0.0
45
4*
*0
.03
90
**
*
0.5
t0
.10
91
0.0
96
50
.08
78
0.1
29
50
.10
74
0.0
96
50
.14
53
0.1
20
00
.10
83
f0
.11
31
ë0
.09
98
ë0
.09
29
ë0
.13
05
0.1
09
60
.09
76
0.1
46
10
.12
06
0.1
08
7
w0
.15
47
ëëë
0.1
37
7ëë
ë0
.13
18
ëëë
0.1
60
6ëë
ë0
.13
51
ëëë
0.1
21
9ëë
ë0
.16
07
ëëë
0.1
38
5ëë
ë0
.12
46
ëëë
aw
0.1
04
5*
0.0
89
5*
*0
.08
25
*0
.12
56
*0
.10
48
0.0
96
60
.13
79
**
0.1
17
30
.10
63
ns
0.1
20
2ëë
ë0
.10
13
ë0
.09
23
ë0
.12
91
0.1
04
90
.09
15
*0
.14
29
0.1
18
80
.09
80
**
*
ls0
.12
02
ëëë
0.1
01
3ë
0.0
92
3ë
0.1
28
90
.10
41
*0
.08
99
**
0.1
41
90
.11
52
ë0
.09
25
**
*
cs0
.11
14
0.0
93
90
.08
30
*0
.12
25
**
0.0
98
2*
*0
.08
06
**
*0
.13
41
**
*0
.09
98
***
0.0
70
8**
*
1.0
t0
.21
03
0.1
65
80
.14
13
0.2
65
00
.20
02
0.1
64
00
.31
35
0.2
32
60
.18
77
f0
.21
90
ëë0
.17
22
ë0
.14
66
ë0
.26
63
0.2
03
20
.16
75
0.3
15
40
.23
47
0.1
88
9
w0
.28
11
ëëë
0.3
22
4ëë
ë0
.19
65
ëëë
0.3
07
6ëë
ë0
.23
77
ëëë
0.2
01
2ëë
ë0
.34
07
ëëë
0.2
53
6ëë
ë0
.21
04
ëëë
aw
0.2
01
3*
*0
.15
50
**
0.1
31
0**
0.2
62
50
.19
66
0.1
63
90
.30
88
*0
.22
25
**
0.1
84
8
ns
0.2
23
6ëë
ë0
.17
16
ë0
.14
29
0.2
62
10
.19
30
*0
.15
30
**
*0
.31
29
0.2
23
5*
*0
.17
16
**
*
ls0
.22
36
ëëë
0.1
71
6ë
0.1
42
90
.26
19
0.1
92
6*
0.1
49
3**
*0
.30
71
*0
.21
79
***
0.1
60
8**
*
cs0
.20
78
0.1
58
05
*0
.12
81
**
*0
.24
93
**
*0
.17
87
***
0.1
30
2**
*0
.28
01
**
*0
.18
64
***
0.1
23
8**
*
Power of non-parametric tests 617
TA
BL
E3
Ð(C
on
tin
ued
)
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
(c)
a5
0.0
1
0.0
t0
.00
90
0.0
09
30
.00
98
0.0
07
90
.00
82
0.0
10
30
.01
04
0.0
10
40
.01
15
f0
.01
30
ëëë
0.0
14
9ëë
ë0
.01
57
ëëë
0.0
09
50
.01
02
0.0
12
9ëë
0.0
10
40
.01
13
0.0
12
9ë
w0
.01
70
ëëë
0.0
18
5ëë
ë0
.01
97
ëëë
0.0
12
1ëë
ë0
.01
30
ëëë
0.0
16
2ëë
ë0
.01
30
0.0
13
9ëë
ë0
.01
49
ëëë
aw
0.0
09
00
.00
92
0.0
11
3ë
0.0
08
40
.00
80
0.0
09
60
.01
00
0.0
11
20
.01
29
ë
ns
0.0
17
0ëë
ë0
.01
85
ëëë
0.0
19
7ëë
ë0
.00
94
ë0
.01
02
ëë0
.01
18
ë0
.01
05
0.0
11
30
.01
22
l s0
.01
70
ëëë
0.0
18
5ëë
ë0
.01
97
ëëë
0.0
08
9ë
0.0
09
7ë
0.0
11
10
.01
05
0.0
11
00
.01
13
cs0
.01
70
ëëë
0.0
18
5ëë
ë0
.01
97
ëëë
0.0
08
80
.00
97
ë0
.01
11
0.0
11
5ë
0.0
10
10
.00
94
*
0.5
t0
.02
63
0.0
21
70
.02
07
0.0
27
70
.02
49
0.0
22
30
.03
92
0.0
32
40
.02
91
f0
.03
17
ëëë
0.0
28
4ëë
ë0
.02
89
ëëë
0.0
29
9ë
0.0
28
5ëë
0.0
26
9ëë
ë0
.04
11
0.0
33
70
.03
18
ë
w0
.04
05
ëëë
0.0
35
4ëë
ë0
.03
45
ëëë
0.0
37
9ëë
ë0
.03
41
ëëë
0.0
32
3ëë
ë0
.04
61
ëëë
0.0
39
2ëë
ë0
.03
75
ëëë
aw
0.0
22
2*
*0
.02
09
0.0
20
90
.02
70
0.0
22
9*
0.0
21
60
.03
73
0.0
30
90
.02
92
ns
0.0
40
5ëë
ë0
.03
54
ëëë
0.0
34
5ëë
ë0
.03
23
ëë0
.02
71
ë0
.02
51
ë0
.03
87
0.0
32
40
.02
80
ls0
.04
05
ëëë
0.0
35
4ëë
ë0
.03
45
ëëë
0.0
30
9ë
0.0
26
5ë
0.0
24
20
.03
78
0.0
31
70
.02
70
*
cs0
.04
05
ëëë
0.0
35
4ëë
ë0
.03
45
ëëë
0.0
31
1ë
0.0
26
40
.02
38
ë0
.03
81
0.0
30
3*
0.0
23
1**
*
1.0
t0
.05
86
0.0
45
00
.03
84
0.0
85
70
.05
96
0.0
48
70
.11
02
0.0
76
90
.06
06
f0
.07
22
ëëë
0.0
54
9ëë
ë0
.04
81
ëëë
0.0
92
7ëë
0.0
65
1ëë
0.0
53
7ëë
0.1
15
8ë
0.0
82
0ë
0.0
64
6ë
w0
.08
79
ëëë
0.0
68
7ëë
ë0
.05
81
ëëë
0.1
06
8ëë
ë0
.07
42
ëëë
0.0
62
2ëë
ë0
.13
07
ëëë
0.0
94
8ëë
ë0
.07
37
ëëë
aw
0.0
51
6*
*0
.04
02
**
0.0
36
4*
0.0
77
1*
**
0.0
54
0*
*0
.04
41
**
0.1
08
40
.07
73
0.0
60
9
ns
0.0
87
9ëë
ë0
.06
87
ëëë
0.0
58
1ëë
ë0
.08
96
ë0
.06
11
0.0
48
60
.11
33
0.0
75
90
.05
66
*
ls0
.08
79
ëëë
0.0
68
7ëë
ë0
.05
81
ëëë
0.0
86
30
.05
93
0.0
46
60
.11
28
0.0
75
00
.05
48
**
cs0
.08
79
ëëë
0.0
68
7ëë
ë0
.05
81
ëëë
0.0
86
70
.05
93
0.0
46
3*
0.1
05
7*
0.0
69
5*
*0
.04
92
**
*
618 H. Tanizaki
TA
BL
E4
.C
au
ch
yd
istr
ibu
tio
n
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( a)
a5
0.1
0
0.0
t0
.08
53
0.0
86
30
.08
57
0.0
84
60
.08
48
0.0
85
90
.08
86
0.0
88
70
.08
90
f0
.10
27
ëëë
0.1
04
0ëë
ë0
.10
47
ëëë
0.0
99
9ëë
ë0
.09
78
ëëë
0.1
01
4ëë
ë0
.10
01
ëëë
0.1
00
2ëë
ë0
.10
12
ëëë
w0
.11
27
ëëë
0.1
12
6ëë
ë0
.11
58
ëëë
0.1
07
6ëë
ë0
.10
59
ëëë
0.1
08
9ëë
ë0
.10
97
ëëë
0.1
13
8ëë
ë0
.11
88
ëëë
aw
0.1
12
7ëë
ë0
.11
26
ëëë
0.1
15
8ëë
ë0
.10
76
ëëë
0.1
05
9ëë
ë0
.10
89
ëëë
0.0
94
0ë
0.0
96
8ëë
0.1
03
6ëë
ë
ns
0.1
04
2ëë
ë0
.10
44
ëëë
0.1
06
1ëë
ë0
.10
30
ëëë
0.1
00
0ëë
ë0
.10
16
ëëë
0.1
01
4ëë
ë0
.10
19
ëëë
0.1
01
3ëë
ë
ls0
.10
35
ëëë
0.1
01
8ëë
ë0
.10
01
ëëë
0.1
02
7ëë
ë0
.09
91
ëëë
0.0
99
4ëë
ë0
.10
09
ëëë
0.1
02
8ëë
ë0
.10
09
ëëë
cs0
.10
35
ëëë
0.1
01
8ëë
ë0
.10
01
ëëë
0.1
03
0ëë
ë0
.09
68
ëëë
0.0
92
4ëë
0.1
01
9ëë
ë0
.09
88
ëëë
0.0
90
3
0.5
t0
.14
67
0.1
32
30
.12
42
0.1
48
30
.13
35
0.1
26
50
.15
02
0.1
38
10
.13
05
f0
.17
77
ëëë
0.1
63
4ëë
ë0
.15
24
ëëë
0.1
76
1ëë
ë0
.15
83
ëëë
0.1
48
2ëë
ë0
.17
37
ëëë
0.1
60
1ëë
ë0
.14
96
ëëë
w0
.20
73
ëëë
0.1
86
4ëë
ë0
.17
62
ëëë
0.2
19
6ëë
ë0
.19
55
ëëë
0.1
79
5ëë
ë0
.25
88
ëëë
0.2
23
2ëë
ë0
.20
53
ëëë
aw
0.2
07
3ëë
ë0
.18
64
ëëë
0.1
76
2ëë
ë0
.21
96
ëëë
0.1
95
5ëë
ë0
.17
95
ëëë
0.2
27
8ëë
ë0
.19
75
ëëë
0.1
82
6ëë
ë
ns
0.1
92
3ëë
ë0
.17
09
ëëë
0.1
61
7ëë
ë0
.20
37
ëëë
0.1
78
3ëë
ë0
.16
36
ëëë
0.2
20
4ëë
ë0
.19
04
ëëë
0.1
73
8ëë
ë
ls0
.18
11
ëëë
0.1
60
7ëë
ë0
.14
81
ëëë
0.1
99
4ëë
ë0
.17
36
ëëë
0.1
59
9ëë
ë0
.21
45
ëëë
0.1
85
8ëë
ë0
.16
67
ëëë
cs0
.18
11
ëëë
0.1
60
7ëë
ë0
.14
81
ëëë
0.1
77
7ëë
ë0
.15
50
ëëë
0.1
38
2ëë
ë0
.17
59
ëëë
0.1
52
1ëë
ë0
.13
28
1.0
t0
.22
48
0.1
91
90
.17
00
0.2
32
60
.19
76
0.1
75
30
.23
13
0.2
00
20
.18
09
f0
.27
33
ëëë
0.2
33
8ëë
ë0
.20
94
ëëë
0.2
67
6ëë
ë0
.22
99
ëëë
0.2
04
9ëë
ë0
.26
77
ëëë
0.2
28
5ëë
ë0
.20
61
ëëë
w0
.33
23
ëëë
0.2
84
8ëë
ë0
.25
08
ëëë
0.3
79
5ëë
ë0
.31
49
ëëë
0.2
75
2ëë
ë0
.44
49
ëëë
0.3
70
1ëë
ë0
.32
01
ëëë
aw
0.3
32
3ëë
ë0
.28
48
ëëë
0.2
50
8ëë
ë0
.37
95
ëëë
0.3
14
9ëë
ë0
.27
52
ëëë
0.4
09
4ëë
ë0
.33
39
ëëë
0.2
86
6ëë
ë
ns
0.3
07
1ëë
ë0
.26
02
ëëë
0.2
27
0ëë
ë0
.33
72
ëëë
0.2
80
8ëë
ë0
.24
47
ëëë
0.3
75
8ëë
ë0
.30
80
ëëë
0.2
66
1ëë
ë
ls0
.27
55
ëëë
0.2
33
2ëë
ë0
.20
19
ëëë
0.3
30
5ëë
ë0
.27
41
ëëë
0.2
36
9ëë
ë0
.35
77
ëëë
0.2
93
5ëë
ë0
.25
69
ëëë
cs0
.27
55
ëëë
0.2
33
2ëë
ë0
.20
19
ëëë
0.2
65
5ëë
ë0
.22
30
ëëë
0.1
89
8ëë
ë0
.25
40
ëëë
0.2
14
6ëë
ë0
.18
38
Power of non-parametric tests 619
TA
BL
E4
Ð(C
on
tin
ued
)
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( b)
a5
0.0
5
0.0
t0
.02
74
0.0
26
40
.02
83
0.0
27
90
.02
85
0.0
30
10
.03
05
0.0
32
10
.03
26
f0
.05
36
ëëë
0.0
52
7ëë
ë0
.05
29
ëëë
0.0
48
8ëë
ë0
.05
15
ëëë
0.0
53
5ëë
ë0
.05
02
ëëë
0.0
51
3ëë
ë0
.05
26
ëëë
w0
.07
59
ëëë
0.0
77
7ëë
ë0
.07
94
ëëë
0.0
62
3ëë
ë0
.06
31
ëëë
0.0
67
6ëë
ë0
.05
76
ëëë
0.0
58
6ëë
ë0
.06
17
ëëë
aw
0.0
48
6ëë
ë0
.04
97
ëëë
0.0
49
8ëë
ë0
.04
52
ëëë
0.0
47
7ëë
ë0
.05
03
ëëë
0.0
46
9ëë
ë0
.04
89
ëëë
0.0
51
1ëë
ë
ns
0.0
56
5ëë
ë0
.05
64
ëëë
0.0
56
2ëë
ë0
.04
90
ëëë
0.0
49
2ëë
ë0
.05
16
ëëë
0.0
50
5ëë
ë0
.05
10
ëëë
0.0
52
7ëë
ë
ls0
.05
65
ëëë
0.0
56
4ëë
ë0
.05
62
ëëë
0.0
49
7ëë
ë0
.04
98
ëëë
0.0
81
8ëë
ë0
.04
95
ëëë
0.0
50
2ëë
ë0
.05
06
ëëë
cs0
.05
04
ëëë
0.0
49
9ëë
ë0
.04
81
ëëë
0.0
49
5ëë
ë0
.04
91
ëëë
0.0
47
9ëë
ë0
.05
00
ëëë
0.0
50
0ëë
ë0
.04
64
ëëë
0.5
t0
.06
05
0.0
53
10
.05
03
0.0
63
20
.05
44
0.0
50
30
.06
70
0.0
58
30
.05
48
f0
.10
38
ëëë
0.0
91
7ëë
ë0
.08
70
ëëë
0.1
06
1ëë
ë0
.09
19
ëëë
0.0
85
0ëë
ë0
.10
57
ëëë
0.0
92
7ëë
ë0
.08
65
ëëë
w0
.15
05
ëëë
0.1
36
2ëë
ë0
.12
83
ëëë
0.1
52
9ëë
ë0
.13
06
ëëë
0.1
18
7ëë
ë0
.15
20
ëëë
0.1
33
6ëë
ë0
.12
06
ëëë
aw
0.0
99
3ëë
ë0
.08
72
ëëë
0.0
79
9ëë
ë0
.12
15
ëëë
0.1
04
4ëë
ë0
.09
56
ëëë
0.1
29
2ëë
ë0
.11
43
ëëë
0.1
02
2ëë
ë
ns
0.1
11
2ëë
ë0
.09
82
ëëë
0.0
88
9ëë
ë0
.11
84
ëëë
0.1
01
2ëë
ë0
.08
84
ëëë
0.1
29
2ëë
ë0
.11
18
ëëë
0.0
98
9ëë
ë
ls0
.11
12
ëëë
0.0
98
2ëë
ë0
.08
89
ëëë
0.1
18
0ëë
ë0
.09
99
ëëë
0.0
87
0ëë
ë0
.12
60
ëëë
0.1
08
7ëë
ë0
.09
49
ëëë
cs0
.10
08
ëëë
0.0
88
8ëë
ë0
.07
92
ëëë
0.1
07
4ëë
ë0
.08
99
ëëë
0.0
76
4ëë
ë0
.10
97
ëëë
0.0
91
6ëë
ë0
.07
75
ëëë
1.0
t0
.11
59
0.0
94
40
.08
02
0.1
20
20
.09
38
0.0
80
80
.12
20
0.0
99
20
.08
52
f0
.17
99
ëëë
0.1
49
9ëë
ë0
.13
10
ëëë
0.1
81
3ëë
ë0
.14
77
ëëë
0.1
28
1ëë
ë0
.18
18
ëëë
0.1
49
9ëë
ë0
.13
04
ëëë
w0
.26
07
ëëë
0.2
16
5ëë
ë0
.19
00
ëëë
0.2
79
2ëë
ë0
.22
47
ëëë
0.1
90
8ëë
ë0
.30
27
ëëë
0.2
40
1ëë
ë0
.20
18
ëëë
aw
0.1
79
1ëë
ë0
.14
57
ëëë
0.1
28
1ëë
ë0
.23
33
ëëë
0.1
87
7ëë
ë0
.15
65
ëëë
0.2
70
8ëë
ë0
.21
24
ëëë
0.1
77
3ëë
ë
ns
0.1
91
7ëë
ë0
.15
85
ëëë
0.1
36
7ëë
ë0
.21
37
ëëë
0.1
71
2ëë
ë0
.14
21
ëëë
0.2
49
6ëë
ë0
.19
76
ëëë
0.1
62
1ëë
ë
ls0
.19
17
ëëë
0.1
58
5ëë
ë0
.13
67
ëëë
0.2
08
4ëë
ë0
.16
99
ëëë
0.1
41
5ëë
ë0
.23
85
ëëë
0.1
87
7ëë
ë0
.15
24
ëëë
cs0
.16
95
ëëë
0.1
40
9ëë
ë0
.11
76
ëëë
0.1
81
7ëë
ë0
.14
59
ëëë
0.1
17
8ëë
ë0
.17
92
ëëë
0.1
45
0ëë
ë0
.11
68
ëëë
620 H. Tanizaki
TA
BL
E4
Ð(C
on
tin
ued
)
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( c)
a5
0.0
1
0.0
t0
.00
19
0.0
01
90
.00
22
0.0
02
00
.00
18
0.0
02
40
.00
31
0.0
03
00
.00
33
f0
.01
29
ëëë
0.0
14
1ëë
ë0
.01
48
ëëë
0.0
09
4ëë
ë0
.00
97
ëëë
0.0
11
1ëë
ë0
.01
06
ëëë
0.0
10
5ëë
ë0
.01
16
ëëë
w0
.01
70
ëëë
0.0
18
0ëë
ë0
.01
83
ëëë
0.0
12
1ëë
ë0
.01
25
ëëë
0.0
13
5ëë
ë0
.01
30
ëëë
0.0
13
4ëë
ë0
.01
43
ëëë
aw
0.0
09
0ëë
ë0
.00
89
ëëë
0.0
10
4ëë
ë0
.00
84
ëëë
0.0
07
7ëë
ë0
.00
86
ëëë
0.0
10
0ëë
ë0
.01
01
ëëë
0.0
11
7ëë
ë
ns
0.0
17
0ëë
0.0
18
0ëë
ë0
.01
83
ëëë
0.0
09
4ëë
ë0
.00
96
ëëë
0.0
10
1ëë
ë0
.01
05
ëëë
0.0
10
5ëë
ë0
.01
18
ëëë
ns
0.0
17
0ëë
ë0
.01
80
ëëë
0.0
18
3ëë
ë0
.00
89
ëëë
0.0
08
9ëë
ë0
.00
94
ëëë
0.0
10
5ëë
ë0
.01
10
ëëë
0.0
11
2ëë
ë
cs0
.01
70
ëëë
0.0
18
0ëë
ë0
.01
83
ëëë
0.0
08
8ëë
ë0
.00
88
ëëë
0.0
09
5ëë
ë0
.01
15
ëëë
0.0
10
6ëë
ë0
.00
98
ëëë
0.5
t0
.00
77
0.0
06
00
.00
54
0.0
07
90
.00
69
0.0
06
10
.01
00
0.0
08
90
.00
83
f0
.03
36
ëëë
0.0
28
9ëë
ë0
.02
69
ëëë
0.0
28
1ëë
ë0
.02
58
ëëë
0.0
23
5ëë
ë0
.03
41
ëëë
0.0
27
8ëë
ë0
.02
44
ëëë
w0
.04
18
ëëë
0.0
36
3ëë
ë0
.03
27
ëëë
0.0
38
7ëë
ë0
.03
37
ëëë
0.0
29
8ëë
ë0
.04
55
ëëë
0.0
38
5ëë
ë0
.03
38
ëëë
aw
0.0
24
0ëë
ë0
.02
12
ëëë
0.0
20
1ëë
ë0
.02
62
ëëë
0.0
23
9ëë
ë0
.02
06
ëëë
0.0
37
5ëë
ë0
.03
13
ëëë
0.0
26
6ëë
ë
ns
0.0
41
8ëë
ë0
.03
63
ëëë
0.0
32
7ëë
ë0
.03
09
ëëë
0.0
27
2ëë
ë0
.02
33
ëëë
0.0
36
0ëë
ë0
.03
11
ëëë
0.0
26
1ëë
ë
ls0
.04
18
ëëë
0.0
36
3ëë
ë0
.03
27
ëëë
0.0
29
5ëë
ë0
.03
87
ëëë
0.0
22
0ëë
ë0
.03
49
ëëë
0.0
30
5ëë
ë0
.02
51
ëëë
cs0
.04
18
ëëë
0.0
36
3ëë
ë0
.03
27
ëëë
0.0
29
2ëë
ë0
.02
56
ëëë
0.0
21
8ëë
ë0
.03
32
ëëë
0.0
27
7ëë
ë0
.02
28
ëëë
1.0
t0
.02
27
0.0
16
10
.01
20
0.0
28
10
.01
78
0.0
13
20
.03
03
0.0
21
00
.01
72
f0
.07
38
ëëë
0.0
57
4ëë
ë0
.04
76
ëëë
0.0
74
3ëë
ë0
.05
63
ëëë
0.0
47
5ëë
ë0
.07
58
ëëë
0.0
58
1ëë
ë0
.04
87
ëëë
w0
.08
63
ëëë
0.0
67
0ëë
ë0
.05
61
ëëë
0.1
00
1ëë
ë0
.07
30
ëëë
0.0
60
7ëë
ë0
.11
68
ëëë
0.0
85
7ëë
ë0
.07
15
ëëë
aw
0.0
54
8ëë
ë0
.04
30
ëëë
0.0
36
7ëë
ë0
.07
19
ëëë
0.0
52
4ëë
ë0
.04
27
ëëë
0.0
96
7ëë
ë0
.07
09
ëëë
0.0
57
9ëë
ë
ns
0.0
86
3ëë
ë0
.06
70
ëëë
0.0
56
1ëë
ë0
.07
90
ëëë
0.0
57
1ëë
ë0
.04
64
ëëë
0.0
91
1ëë
ë0
.06
63
ëëë
0.0
54
2ëë
ë
ls0
.08
63
ëëë
0.0
67
0ëë
ë0
.05
61
ëëë
0.0
72
7ëë
ë0
.05
40
ëëë
0.0
44
9ëë
ë0
.08
75
ëëë
0.0
63
1ëë
ë0
.05
21
ëëë
cs0
.08
63
ëëë
0.0
67
0ëë
ë0
.05
61
ëëë
0.0
72
2ëë
ë0
.05
30
ëëë
0.0
44
4ëë
ë0
.07
34
ëëë
0.0
53
9ëë
ë0
.04
38
ëëë
Power of non-parametric tests 621
TA
BL
E5
.v
2(6
)/2
dis
trib
uti
on
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( a)
a5
0.1
0
0.0
t0
.09
41
0.0
82
70
.07
71
0.1
02
00
.08
42
0.0
75
90
.10
06
0.0
87
10
.08
19
f0
.09
74
ë0
.08
58
ë0
.07
99
ë0
.10
14
0.0
82
00
.07
39
0.0
98
70
.08
64
0.0
79
8
w0
.10
44
ëëë
0.0
82
40
.07
90
0.1
04
30
.07
44
***
0.0
65
8**
*0
.10
85
ëë0
.07
32
***
0.0
64
9**
*
aw
0.1
04
4ëë
ë0
.08
24
0.0
79
00
.10
43
0.0
74
4*
**
0.0
65
8**
*0
.09
28
**
0.0
62
2*
**
0.0
55
3**
*
ns
0.0
97
9ë
0.0
73
8*
**
0.0
65
6**
*0
.09
94
0.0
62
9*
**
0.0
49
8**
*0
.09
73
*0
.05
88
***
0.0
46
4**
*
ls0
.09
69
0.0
69
2*
**
0.0
54
5**
*0
.09
94
0.0
60
0*
**
0.0
44
4**
*0
.09
74
*0
.05
48
***
0.0
41
2**
*
cs0
.09
69
0.0
69
2*
**
0.0
54
5**
*0
.10
08
0.0
50
2*
**
0.0
29
2**
*0
.09
89
0.0
40
8*
**
0.0
19
9**
*
0.5
t0
.20
89
0.1
50
80
.12
64
0.2
36
20
.17
75
0.1
45
40
.25
19
0.1
92
30
.15
57
f0
.21
78
ëë0
.15
32
0.1
28
10
.23
51
0.1
75
00
.14
04
*0
.25
06
0.1
88
80
.15
14
*
w0
.22
96
ëëë
0.1
53
20
.12
21
*0
.25
06
ëëë
0.1
54
0*
**
0.1
17
7**
*0
.28
69
ëëë
0.1
68
2*
**
0.1
25
6**
*
aw
0.2
29
6ëë
ë0
.15
32
0.1
22
1*
0.2
50
6ëë
ë0
.15
40
***
0.1
17
7**
*0
.25
63
ë0
.14
84
***
0.1
08
1**
*
ns
0.2
16
7ë
0.1
41
5*
*0
.10
81
**
*0
.24
82
ëë0
.14
04
***
0.0
95
9**
*0
.27
25
ëëë
0.1
43
5*
**
0.0
96
1**
*
ls0
.22
14
ëëë
0.1
34
9*
**
0.0
92
2**
*0
.24
97
ëëë
0.1
38
2*
**
0.0
90
1**
*0
.27
62
ëëë
0.1
39
4*
**
0.0
89
6**
*
cs0
.22
14
ëëë
0.1
34
9*
**
0.0
92
2**
*0
.24
80
ëë0
.12
37
***
0.0
64
0**
*0
.27
22
ëëë
0.1
11
8*
**
0.0
49
6**
*
1.0
t0
.36
30
0.2
57
80
.19
47
0.4
30
40
.31
64
0.2
45
80
.47
10
0.3
52
00
.27
09
f0
.37
32
ëë0
.26
34
ë0
.19
70
0.4
31
90
.31
44
0.2
42
90
.47
00
0.3
48
40
.26
73
w0
.39
65
ëëë
0.2
59
20
.18
89
*0
.46
52
ëëë
0.2
93
8*
**
0.2
04
0**
*0
.52
49
ëëë
0.3
34
4*
**
0.2
21
3**
*
aw
0.3
96
5ëë
ë0
.25
92
0.1
88
9*
0.4
65
2ëë
ë0
.29
38
***
0.2
04
0**
*0
.48
98
ëëë
0.3
00
3*
**
0.1
95
7**
*
ns
0.3
78
9ëë
ë0
.24
37
***
0.1
68
5**
*0
.46
09
ëëë
0.3
79
6*
**
0.1
75
4**
*0
.51
32
ëëë
0.3
05
7*
**
0.1
78
6**
*
ls0
.37
73
ëë0
.23
86
***
0.1
51
3**
*0
.45
98
ëëë
0.2
75
1*
**
0.1
65
4**
*0
.51
55
ëëë
0.3
00
7*
**
0.1
67
2**
*
cs0
.37
73
ëë0
.23
86
***
0.1
51
3**
*0
.44
39
ëë0
.25
68
***
0.1
28
7**
*0
.48
32
ëë0
.26
07
***
0.1
04
4**
*
622 H. Tanizaki
TA
BL
E5
Ð(C
on
tin
ued
)
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( b)
a5
0.0
5
0.0
t0
.04
53
0.0
34
70
.03
18
0.0
49
00
.03
75
0.0
32
50
.05
00
0.0
40
10
.03
59
f0
.04
83
ë0
.03
68
ë0
.03
47
0.0
50
30
.03
82
0.0
34
3ë
0.0
50
50
.04
10
0.0
36
2
w0
.07
24
ëëë
0.0
55
6ëë
ë0
.05
23
ëëë
0.0
63
9ëë
ë0
.04
33
ëëë
0.0
36
6ëë
0.0
57
4ëë
ë0
.03
84
0.0
33
7*
aw
0.0
46
00
.03
18
*0
.02
75
**
0.0
49
10
.03
21
**
0.0
28
4**
0.0
48
20
.03
22
***
0.0
27
8**
*
ns
0.0
53
3ëë
ë0
.03
71
ë0
.03
22
0.0
50
80
.03
02
***
0.0
23
2**
*0
.05
14
0.0
28
7*
**
0.0
21
7**
*
ls0
.05
33
ëëë
0.0
37
1ë
0.0
32
20
.05
04
0.0
29
7*
**
0.0
22
3**
*0
.05
07
0.0
26
5*
**
0.0
18
3**
*
cs0
.04
96
ëë0
.03
41
0.0
29
8*
0.0
51
00
.02
68
***
0.0
16
8**
*0
.04
98
0.0
22
8*
**
0.0
12
0**
*
0.5
t0
.10
78
0.0
75
00
.05
97
0.1
34
90
.08
58
0.0
66
50
.15
11
0.0
99
00
.07
89
f0
.11
38
ë0
.08
18
ëë0
.06
60
ëë0
.13
92
ë0
.08
98
ë0
.07
09
ë0
.15
32
0.1
01
40
.08
06
w0
.16
22
ëëë
0.1
08
3ëë
ë0
.08
95
ëëë
0.1
73
4ëë
ë0
.09
93
ëëë
0.0
71
9ëë
0.1
72
9ëë
ë0
.09
59
*0
.06
90
**
*
aw
0.1
13
0ë
0.0
68
5*
*0
.05
26
**
0.1
40
6ë
0.0
78
7*
*0
.05
74
**
*0
.14
82
0.0
82
7*
**
0.0
59
3**
*
ns
0.1
28
8ëë
ë0
.07
82
ë0
.05
93
0.1
44
4ëë
0.0
73
9*
**
0.0
48
2**
*0
.16
32
ëëë
0.0
79
5*
**
0.0
49
9**
*
ls0
.12
88
ëëë
0.0
78
20
.05
93
0.1
44
9ëë
0.0
73
9*
**
0.0
47
6**
*0
.16
33
ëëë
0.0
77
1*
**
0.0
45
5**
*
cs0
.11
91
ëëë
0.0
75
20
.05
35
**
0.1
46
1ëë
ë0
.06
96
***
0.0
39
4**
*0
.16
22
ëëë
0.0
68
3*
**
0.0
29
6**
*
1.0
t0
.22
50
0.1
36
00
.10
16
0.2
88
20
.18
50
0.1
26
40
.33
48
0.2
18
50
.15
00
f0
.23
69
ëë0
.14
83
ëëë
0.1
11
5ëë
ë0
.29
37
ë0
.19
16
ë0
.13
24
ë0
.33
75
0.2
22
00
.15
35
w0
.30
62
ëëë
0.1
88
5ëë
ë0
.14
18
ëëë
0.3
54
3ëë
ë0
.20
62
ëëë
0.1
33
1ëë
0.3
84
8ëë
ë0
.21
49
0.1
34
5**
*
aw
0.2
25
80
.13
22
*0
.09
14
**
*0
.30
11
ë0
.16
60
***
0.1
05
7**
*0
.34
83
ëë0
.18
58
***
0.1
17
2**
*
ns
0.2
53
0ëë
ë0
.14
75
ëëë
0.1
01
30
.31
33
ëëë
0.1
66
4*
**
0.0
94
4**
*0
.36
65
ëëë
0.1
88
8*
**
0.1
03
9**
*
ls0
.25
30
ëëë
0.1
47
5ëë
ë0
.10
13
0.3
15
1ëë
ë0
.16
76
***
0.0
93
2**
*0
.36
43
ëëë
0.1
85
3*
**
0.0
96
2**
*
cs0
.23
85
ëëë
0.1
37
90
.09
34
**
0.3
09
1ëë
ë0
.15
99
***
0.0
80
3**
*0
.35
42
ëëë
0.1
69
6*
**
0.0
72
8**
*
Power of non-parametric tests 623
TA
BL
E5
Ð(C
on
tin
ued
)
n1
5n
25
5n
15
n2
57
n1
5n
25
9
lr
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0r
51
.0r
51
.5r
52
.0
( c)
a5
0.0
1
0.0
t0
.00
84
0.0
06
60
.00
67
0.0
08
20
.00
54
0.0
05
10
.00
89
0.0
06
40
.00
57
f0
.01
15
ëëë
0.0
10
2ëë
ë0
.00
95
ëëë
0.0
09
40
.00
65
0.0
06
5ë
0.0
09
50
.00
84
ëë0
.00
70
ë
w0
.01
46
ëëë
0.0
12
5ëë
ë0
.01
17
ëëë
0.0
12
5ëë
ë0
.00
81
0.0
07
6ëë
ë0
.01
23
ëëë
0.0
07
4ë
0.0
06
5ë
lw0
.00
89
0.0
06
60
.00
61
0.0
08
20
.00
50
0.0
04
50
.00
96
0.0
06
20
.00
53
ns
0.0
14
6ëë
ë0
.01
25
ëëë
0.0
11
7ë
0.0
10
3ëë
0.0
06
00
.00
51
0.0
10
1ë
0.0
05
70
.00
44
*
ls0
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60
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ls0
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5ëë
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**
*
624 H. Tanizaki
Furthermore, in Tables 3 ± 5, we take three examples of the underlying distrib-
utions, such that the Wilcoxon test has a larger asymptotic relative e� ciency than
that of the t-test; the examples are the logistic distribution, Cauchy distribution
and v2(6) /2 distribution (i.e. equation (4)).
In Table 3, we generate random draws as follows: x i ~ exp( 2 x i) /[1 + exp( 2 x i)]2
for i 5 1, . . . , n1, and y i 5 l + r v j , where v j ~ exp( 2 v j ) /[1 + exp( 2 v j)]2 , for
j 5 1, . . . , n2. In the case of the logistic distribution, it is known that the asymptotic
relative e� ciency of the Wilcoxon test to the normal scores test is 1.047 (see
Kendall & Stuart, 1979). For large n1 and n2, w exhibits the best performance,
despite a . ns, ls and cs have the best small-sample powers when r ¹ 1.
When the underlying distribution is Cauchy, it is known that the asymptotic
relative e� ciency of the Wilcoxon test to the normal scores test is 1.413 (see
Kendall & Stuart, 1979). Thus, in Table 4, we consider the following random
draws: x i ~ 1 / p (1 + x2i ) for i 5 1, . . . , n1, and y j 5 l + r v j , where v j ~ 1 / p (1 + v
2j ), for
j 5 1, . . . , n2. All the non-parametric tests perform better than the t-test. Overall,
the order of greatest power is given by the Wilcoxon test, Fisher test and the t-test.
When the underlying distribution is given by equation (4), i.e. v2(6) /2, the
asymptotic relative e� ciency of the Wilcoxon test to the t-test is 1.33 (see Hodges
& Lehman, 1956). In Table 5, we generate random draws as follows: x i 5 u i /2 2 3,
where u i ~ v2(6), for i 5 1, . . . , n1, and y i 5 l + r (v j /2 2 3), where v j ~ v
2(6), for
j 5 1, . . . , n2.9 When a is small, the non-parametric tests perform better, especially
when r 5 1. For small l and large a , t and f perform quite well.
When the tails of distribution are large, the asymptotic property that the Wilcoxon
tests has more asymptotic relative e� ciency than the t-test holds in the case of
small samples from the Monte Carlo experiments in Tables 1 ± 5. Intuitively, it is
expected from an amount of information included in the test statistics that the
Fisher test performs better than the Wilcoxon test in the sample power. However,
judging from the results obtained in Tables 3 ± 5, the Wilcoxon test is more
powerful. The following three facts contribute to this:
(1) both the t-test and the Fisher test take the diŒerence between the two sample
means as the test statistic;
(2) both the Fisher test and the Wilcoxon test are non-parametric tests based
on all the possible combinations;
(3) for a distribution with fat tails, the Wilcoxon test exhibits more asymptotic
relative e� ciency than does the t-test.
From these three facts, we can consider that the Fisher test performs between the
t-test and the Wilcoxon test, with regard to the sample power. Therefore, the order
of the three sample powers is given by pà t < pà f < pà w for a distribution with fat tails,
and pà w < pà f < pà t otherwise.
The theorem proved by ChernoΠand Savage (1958), i.e. that the asymptotic
relative e� ciency of the normal scores test to the t-test is more than unity under
the alternative hypothesis of a shifting location parameter, holds in the case of
small samples. In the case of r 5 1 in Table 1, the t-test sometimes performs better
than the normal scores test. In Tables 2 ± 5, the normal scores test performs better
than the t-test for almost all the cases. However, the normal scores test is less
powerful than the Wilcoxon test, which is consistent with the results obtained by
Mehta and Patel (1992). Thus, the t-test and the normal scores test are similar,
but the normal scores test performs slightly better than the t-test.
Power of non-parametric tests 625
Generally, in the case of small samples, it might be concluded from Tables 1 ± 5
that we have the following inequality: pà f < pà ns < pà w .
4 Example: Testing structural changes
In a regression analysis, the disturbance term is usually assumed to be normal and
we perform testing of a hypothesis. However, sometimes, the normality assumption
is too strong. In this paper, `loosening’ the normality assumption, we test a
structural change without assuming any distribution for the disturbance term.
Consider the standard regression model
y t 5 x t b + u t , t 5 1, . . . , T
where y t , x t , b and u t respectively are a dependent variable at time t, a 1 3 k vector
of independent variables at time t, a k 3 1 unknown parameter vector to be
estimated, and the disturbance term at time t with mean zero and variance r2 . The
sample size is T. Let us de® ne X t 2 1 5 (xÂ1 xÂ2 . . . xÂt 2 1 ) Â and Y t 2 1 5 (y1 y2 . . . y t 2 1 ) Â . b t 2 1
denotes the ordinary least-squares (OLS) estimate of b using the data up to time
t 2 1, i.e. b t 2 1 5 (XÂt 2 1 X t 2 1 ) 2 1XÂt 2 1 Y t 2 1 . The recursive residual, i.e.
x t 5(y t 2 x t b t 2 1 )
[1 + xt(XÂt 2 1 X t 2 1 ) 2 1x Ât ]
1 /2, t 5 k + 1, . . . , T
can be computed by recursive OLS estimation, which is distributed with mean
zero and variance r2 The recursive residuals x t , t 5 k + 1, . . . , T, are mutually
independently distributed and normalized to mean zero and variance r2 .
Based on the recursive residual x t , we perform testing at the structural change.10
We can accept the structural change if the structure of the recursive residuals
changes in a period. Dividing the sample into two groups, we test if both
{ x t }n1t 5 k+ 1 and { x t}
Tt 5 n1+ 1 are generated from the same distribution, where
T 5 n1 + n2. The null hypothesis is represented by H0 : F( x ) 5 G( x ), while the
alternative is H1 : F( x ) ¹ G( x ). Let F(´) be the distribution of the ® rst n1 recursive
residuals, and let G(´) be that of the last n2 recursive residuals.
We take an example of a Japanese import function. Annual data from Annual
Report on National Accounts (Economic Planning Agency, Government of Japan)
are used. Let GDP t be the gross domestic product (1985 price, billions of Japanese
yen), let M t be the imports of goods and services (1985 price, billions of Japanese
yen), let P t be the terms of trade index, which is given by the imports of goods and
services implicit price de¯ ator (1985 5 1.00) divided by the gross domestic product
implicit price de¯ ator (1985 5 1.00).
The following two import functions are estimated and the recursive residuals are
computed.
log M t 5 b 0 + b 1 log GDP t + b 2 log P t (7)
log M t 5 b 0 + b 1 log GDP t + b 2 log P t + b 3 log M t 2 1 (8)
where b 0 , b 1 , b 2 and b 3 are the unknown parameters to be estimated.11
For each regression equation, we compute the recursive residuals from 1961 to
1991, divide the period into two groups, and test if the recursive residuals in the ® rst
period are the same as those in the last period. The results are in Tables 6 and 7. In
the tables, t, Fisher, Wilcoxon, Asy Wil, Normal, Logistic and Cauchy denote the
t-test, Fisher test, Wilcoxon test, asymptotic Wilcoxon test, normal scores test,
626 H. Tanizaki
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0.3
25
72
0.9
54
70
.32
46
21
.67
32
0.3
25
72
2.5
70
00
.34
48
0.5
06
50
.31
92
19
89
20
.16
91
0.4
33
42
0.0
08
00
.45
02
49
20
.46
82
20
.05
52
0.4
78
02
0.3
05
20
.43
71
20
.61
46
0.4
27
72
1.6
16
50
.37
52
0.5
91
00
.37
43
19
90
0.4
11
50
.65
82
0.0
22
20
.67
83
52
20
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44
0.7
51
50
.77
38
0.9
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90
.71
22
1.5
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10
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74
1.2
72
80
.62
96
19
91
0.3
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.64
34
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53
60
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0.6
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0.7
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80
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51
1.1
49
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56
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90
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54
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55
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00
.58
32
0.1
53
50
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55
0.2
43
60
.54
55
0.1
92
70
.54
55
Power of non-parametric tests 627
TA
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.T
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7.
0.5
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0.2
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0.4
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50
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6.0
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20
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0.4
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52
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01
50
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14
20
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96
0.2
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10
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95
0.5
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40
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60
0.9
15
80
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13
1.0
66
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.54
52
6.0
75
80
.99
89
19
69
20
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22
0.3
71
02
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08
40
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69
15
20
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63
20
.04
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0.4
83
92
0.0
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90
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79
0.0
10
10
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09
0.2
80
40
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5.4
87
80
.91
80
19
70
20
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0.4
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42
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01
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0.2
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20
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0.6
35
60
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1.0
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1.2
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90
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5.5
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12
0.8
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2.3
31
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8.1
89
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5.5
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50
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0.0
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23
90
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01
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49
3.5
79
10
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6.2
44
90
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60
11
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0.7
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34
0.9
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6
19
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2.1
08
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0.0
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27
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1.8
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5.7
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10
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2.0
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90
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4.9
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2.4
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0.0
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40
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80
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80
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21
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55
20
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74
19
75
1.8
93
70
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62
0.0
40
60
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56
30
10
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98
1.6
63
10
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19
5.0
65
50
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23
9.1
41
60
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46
29
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29
0.9
69
54
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89
0.9
94
9
19
76
1.7
21
50
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24
0.0
37
10
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22
31
60
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10
1.5
85
00
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35
4.9
12
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69
8.8
98
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99
28
.91
02
0.9
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54
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0.9
91
9
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77
1.6
45
80
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35
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33
30
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1.5
85
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4.9
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8.8
98
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99
28
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0.9
67
53
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0.9
88
8
19
78
1.8
21
80
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09
0.0
39
20
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15
35
60
.96
34
1.8
07
80
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47
5.3
85
30
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11
9.6
60
10
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29
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0.9
72
94
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0.9
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6
19
79
2.0
59
00
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60
0.0
44
10
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68
38
00
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2.0
76
30
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11
5.9
44
50
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10
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58
0.9
83
33
0.3
39
30
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5.1
67
40
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19
80
1.4
92
90
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72
0.0
33
30
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86
38
20
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1.5
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40
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91
4.2
49
10
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7.5
21
30
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23
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0.9
10
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0.9
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5
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1.1
95
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1.2
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3.5
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6.2
97
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22
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0.8
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0.9
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3
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0.8
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39
70
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40
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2.6
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4.8
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20
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0.8
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0.9
52
7
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0.0
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40
10
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0.3
91
70
.65
24
1.3
99
90
.69
89
2.7
60
50
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80
17
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64
0.8
49
91
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0.8
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3
19
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0.6
67
00
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0.0
16
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20
42
30
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0.6
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1.7
90
60
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3.3
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30
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18
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0.8
63
72
.15
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0.9
01
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0.3
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40
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42
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1.5
40
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16
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0.8
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12
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0.8
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7
19
86
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0.3
55
82
0.0
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30
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42
90
.28
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20
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25
0.2
83
32
1.4
78
00
.26
80
22
.63
40
0.2
67
32
4.7
30
30
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32
0.8
12
20
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23
19
87
20
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50
0.2
43
02
0.0
20
50
.23
93
44
10
.19
88
20
.84
02
0.2
00
42
1.8
68
70
.20
30
23
.25
97
0.2
07
02
5.2
45
90
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49
0.8
77
90
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08
19
88
20
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59
0.3
80
82
0.0
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60
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01
46
80
.33
94
20
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17
0.3
44
02
1.0
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30
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21
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0.3
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3.8
41
60
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66
1.0
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60
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19
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0.4
81
80
.68
33
0.0
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60
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98
49
90
.61
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0.3
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00
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0.4
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63
0.6
47
10
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0.2
80
50
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28
19
90
1.1
73
90
.87
53
0.0
45
10
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39
52
80
.83
61
1.1
27
20
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02
1.5
14
70
.81
67
2.4
92
90
.80
00
2.4
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20
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14
19
91
0.7
50
80
.77
08
0.0
35
20
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05
53
40
.67
99
0.5
28
10
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13
0.5
48
60
.65
34
0.8
83
50
.64
20
0.7
38
10
.61
17
19
92
0.3
46
90
.63
45
0.0
22
80
.51
52
54
50
.51
52
0.1
05
00
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18
0.0
75
30
.51
52
0.1
21
40
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52
0.0
95
50
.51
52
628 H. Tanizaki
FIG. 1. p-values of non-parametric tests, using import function of equation (7) (Table 6).
logistic scores test and Cauchy scores test. Moreover, each test statistic is given by t0 ,
f0 , w0 , aw0 , ns0 , ls0 and cs0. The probability which is less than the test statistic repre-
sents p-val. The results of the stepwise Chow test, which is usually used in testing the
structural change, are also shown in Tables 6 and 7. The stepwise Chow test statistic
is denoted by F0 and the p-value that corresponds to F0 is given by p-val.
From the import function of equation (7), Table 6 and Fig. 1 indicate that the
structural change occurs during the period 1973 ± 1982 for the t-test, Fisher test,
Wilcoxon test, asymptotic Wilcoxon test, normal scores test and logistic scores
test, during the period 1974 ± 1982 for the Cauchy scores test, and during the
period 1968 ± 1985 for the stepwise Chow test when the signi® cance level is 1%.
Also, when the signi® cance level is 5%, we have the break point during the period
1971 ± 1984 for the t-test and Fisher test, during the period 1971 ± 1983 for the
Wilcoxon test and asymptotic Wilcoxon test, during the period 1972 ± 1984 for the
normal scores test, during the period 1973 ± 1984 for the logistic scores test and
Cauchy scores test, and during the period 1967 ± 1985 for the stepwise Chow test.
It is concluded that the recursive residuals in the ® rst period are larger than those
in the last period.
From the import function of equation (8), Table 7 and Fig. 2 show that the
structural change occurs during the periods 1973, 1974 and 1979 for the t-test,
Fisher test, normal scores test, logistic scores test and Cauchy scores test, during
the periods 1974 and 1979 for the Wilcoxon test and asymptotic Wilcoxon test,
and during the period 1961 ± 1982 for the stepwise Chow test, where the signi® cance
Power of non-parametric tests 629
FIG. 2. p-values of non-parametric tests, using import function of equation (8) (Table 7).
level is 1%. However, except for the stepwise Chow test, we are unable to accept
the fact that the structural change occurred during the period 1961 ± 1992 when
the signi® cance level is 1%. For the stepwise Chow test the structural change is
detected during the periods 1961 ± 1976 and 1979.
Thus, according to the Chow test, which has often been used for the structural
change in previous studies, it is di� cult to know when the structural change takes
place. However, the non-parametric tests shown in this paper gives us the break
points more exactly.
5 Summarizing remarks
In previous studies, non-parametric test statistics have been approximated by a
normal random variable from both computational and programming points of
view. Recently, however, we have been able to perform the test exactly, thanks to
progress in computers. In this paper, we have compared the sample powers in
small samples, taking the scores tests (the Wilcoxon rank sum test, normal scores
test, logistic scores test and Cauchy scores test) and the Fisher test.
In the case where we compare the t-test, Wilcoxon test and Fisher test, the
following might be intuitively expected.
(1) When the underlying distribution is normal, the t-test gives us the most
powerful test.
630 H. Tanizaki
(2) In the situations where we cannot use the t-test, the Wilcoxon test and
Fisher test are more powerful than the t-test. Moreover, the Fisher test
performs better than the Wilcoxon test, because it utilizes more information
than the Wilcoxon test. Therefore, in the case where the underlying distribu-
tion is non-normal or where two samples are heteroskedastic, we might have
the following relationship among the sample powers: pà t < pà w < pà f.
However, the results of the Monte Carlo simulations are as follows. The Wilcoxon
test is as powerful as the t-test, even in the case where the two samples are
identically and normally distributed. Moreover, when the underlying distribution
is Cauchy, the Wilcoxon test performs much better than the t-test. In general, we
have pà t < pà f < pà w in the situation where we cannot use the t-test. The fact proved
by ChernoŒand Savage (1958), which is the theorem that, under the alternative
hypothesis of a shifting location parameter, the asymptotic relative e� ciency of the
normal scores test relative to the t-test is more than unity, holds even in the small-
sample case. However, the normal scores test is less powerful than the Wilcoxon
test. Therefore, in the small-sample case, it might be concluded from Tables 1 ± 5
that we have pà t < pà ns < pà w .
Finally, we take an example of testing structural change as an application to the
non-parametric tests. According to the Chow test, it is di� cult to know when the
structural change takes place. However, the non-parametric tests shown in this
paper gives us the break points more exactly.
Using the non-parametric tests, we can test the hypothesis, despite the functional
form of the distribution of the disturbances.
Acknowledgements
The author is grateful to Francis X. Diebold and Glen D. Rudebusch for valuable
suggestions.
Notes
1. Note that Pitman’s asymptotic relative e� ciency relative to the t-test is de® ned as follows. Let
N0 5 n10 + n20 be the sample size required to obtain the same power as the t-test for the sample
size N 5 n1 + n2. Then, the limit of N /N0 is called Pitman’ s asymptotic relative e� ciency (see,
for example, Kendall & Stuart, 1979), where n1 /N 5 n10 /N0 and n2 /N 5 n20 /N0. If we take
l 5 l  /N1 /2, then Pitman’ s asymptotic relative e� ciency does not depend on l  .
2. Bradley (1968) showed that the t-test does not depend on the functional form of the underlying
distribution for large n1, if there exists the fourth moment, which implies that the t-test is
asymptotically a non-parametric test.
3. StatXact is a computer software package on non-parametric inference, which computes the exact
probability using a non-parametric test. The network algorithm that Mehta and Patel (1983 , 1986a,
1986b) and Mehta et al. (1984, 1985 , 1988) developed is used for a permutation program.
4. For the non-parametric test in the case where we test if the functional form of the two distributions
is diŒerent, we have the runs test (Kendal & Stuart, 1979).
5. Let z be a chi-square random variable with six degrees of freedom. De® ne x 5 z /2. Then x is
distributed as f(x).
6. We can interpret the Wilcoxon test as the scores test assumed to be a uniform distribution for the
inverse function of a(´). In other words, the scores test de® ned as a(x) 5 x /(n1 + n2 + 1) is equivalent
to the Wilcoxon test.
7. HoeŒding (1952 ) showed that the Fisher test is asymptotically as powerful as t-test, even under
normality assumption.
Power of non-parametric tests 631
8. In Tables 1 ± 5, note the following:
(1) Perform m simulation runs, where m 5 10 000. Given the signi® cance levels a 5 0.10, 0.05,
0.01, each value in Table 1 is a ratio of rejection numbers to m simulation runs, which
represents the following probabilities: Prob(t< 2 t0 ) < a , Prob(w< w0 ) < a , Prob(aw< 2 aw0 )
< a , Prob(ns< ns0 ) < a , Prob(ls< ls0 ) < a , Prob(cs< cs0 ) < a and Prob(f< f0 ) < a , where t0 , w0 ,
aw0 , ns0 , ls0 , cs0 and f0 are the statistics from the original data for each simulation run. For the
t-distribution and the normal distribution, given the signi® cance level, we have the following
critical points (i.e. t0 and aw0 ):
a 0.10 0.05 0.01
t0 n1 5 n2 5 5 1.397 1.860 2.396
n1 5 n2 5 7 1.356 1.782 2.681
n1 5 n2 5 9 1.337 1.746 2.583
aw0 1.282 1.645 2.326
Let pà t , pà w , pà aw , pà ns , pà ls, pà cs and pà f be the ratios for each test, indicating the probabilities that reject
the null hypothesis under the alternative, i.e. the sample powers.
(2) The estimated variance of each value in Table 1 is given by var(pà k) 5 [pà k(1 2 pà k )] /m for k 5 t,
w, aw, ns, ls, cs, f and m 5 10 000. Therefore, the standard errors of the estimated sample
powers are at most 0.005 ( 5 [0.5(1 2 0.5) /10 000]1 /2), which is quite small.
(3) ë , ë ë , ë ë ë , *, ** and *** in Table 1 represent comparison with the t-test. We put the superscript
ë in the corresponding values when (pà k 2 pà t) /[Var(pà t)]1 /2, k 5 w, aw, ns, ls, cs, f, is greater than 1;
ë ë when it is greater than 2; and ë ë ë when it is greater than 3. The superscript * is put in the
corresponding values if (pà k 2 pà t ) / [Var(pà t )]1 /2 is less than 2 1; ** if it is less than 2 2; and *** if
it is less than 2 3. Therefore, the values with ë indicate a more powerful test than the t-test.
The values with * represent a less powerful test than the t-test. Moreover, the number of ë or
* shows the level of the sample power.
9. The reason why a v2(6) /2 random variable is subtracted by 3 is that the expectation of v
2(6) /2 is 3.
10. The reason why we use the recursive residual and not the conventional OLS residual is because
the OLS residuals have the following problem. Let et be the residuals obtained from the regression
equation y t 5 x t b + u t , t 5 1, . . . ,T, i.e. et 5 y t 2 xt b à . The residuals et , t 5 1, . . . , T, are not mutually
independently distributed. Clearly, we have E(es et) ¹ 0 for s ¹ t. Therefore, we utilize the recursive
residuals which are mutually independently distributed.
11. The estimation results by OLS are as follows. By equation (7), we have
log M t 5 2 5.324 81 + 1.258 13 log GDP t 2 0.110 73 log P t
(0.482 85) (0.040 37) (0.094 36)
with R25 0.983 93, RÅ
25 0.982 96, DW 5 0.4555 and SE 5 0.105 01, for the estimation period
1958 ± 1993. By equation (8), we have
log M t 5 2 1.072 50 + 0.397 26 log GDP t 2 0.157 35 log P t + 0.626 66 log M t 2 1
(0.831 59) (0.156 53) (0.067 56) (0.112 31)
with R25 0.992 53, RÅ
2-0.991 85, DW 5 2.0564 and SE 5 0.075 89, for the estimation period
1957 ± 1993. There, the values in parentheses are the standard errors. R2 , RÅ
2, DW and SE are the
coe� cient of multiple determination, the adjusted R2 , the Durbin± Watson statistic, and the standard
error of the disturbance respectively. For both equations (7) and (8), the recursive residuals are
obtained from 1961 ± 1993 (33 periods).
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