testing for and against an order restriction on multinomial parameters

Testing For and Against an Order Restriction on Multinomial ParametersAuthor(s): Tim RobertsonSource: Journal of the American Statistical Association, Vol. 73, No. 361 (Mar., 1978), pp. 197-202Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2286546 .

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Testing For and Against an Order Restriction on Multinomial Parameters

TIM ROBERTSON*

Likelihood-ratio statistics are considered for testing a simple null hypothesis on a collection of multinomial parameters against an order-restricted alternative and for testing an order restriction against all alternatives. For the former test, the asymptotic distribution of the test statistic under the null hypothesis is a version of the chi-bar- squared distribution. This extends the work of Chacko (1966). For the latter test, homogeneity is, asymptotically, the least favorable configuration and under this hypothesis the asymptotic distribution of the test statistic has tail probabilities which are weighted averages of standard chi-squared tail probabilities.

KEY WORDS: Chi-squared tests; Order restrictions; Likelihood ratio; Asymptotic distribution.

1. INTRODUCTION

Among the more widely used.statistical tools are the chi-squared tests for making inferences about a collection Pi, P2, ..., Pk of multinomial parameters (EZ=1 p, = 1). Many such collections encountered in practice exhibit a trend. For example, virtually all of the data in the examples and exercises of Chapter 6 of Derman, Gleser, and Olkin (1973) give evidence of a unimodal character of the underlying multinomial parameters (i.e., P1 ? P2 <... < pi and pi > p7+1 > ... > pk for some i). On the other hand, trends may be suggested by relationships between the variables associated with the parameters. In bioassay, the probability of a positive response is assumed to be a monotone function of the dosage level. In eco- nomics, regression functions are frequently assumed to increase at a decreasing rate. In fact, a particular parametric model may be accepted by an investigator largely because it has a "shape" which corresponds to his intuition about the underlying parameters. For example, the Poisson distribution was rejected as a model in cer- tain biology problems because the data were found to be "over dispersed." The alternative, riegative binomial distribution, was accepted at least partly because of its unimodal character (cf. Bliss and Fisher 1953). In other circumstances, a normal model might be chosen simply because it is felt that the underlying distribution is both symmetric and unimodal. In such instances, tests of fit for those models should use statistics which take those "shapes" into account.

It is the purpose of this article to consider two likelihood-ratio statistics. The first is for testing any simple

* Tim Robertson is Professor, Department of Statistics, The University of Iowa, Iowa City, IA 52242. This research was supported in part by the National Science Foundation under Grant MCS75- 23576. The author gratefully acknowledges the helpful comments and suggestions of the Editor, an Associate Editor, and two referees.

null hypothesis about P1, P2, ..., Ipk against an arbitrary order restriction. The asymptotic distribution of this statistic, under the null hypothesis, is a version of the chi-bar-squared distribution (cf. Theorem 1) discovered by Bartholomew (1959a, 1959b, 1961) and by Chacko (1959, 1963). This result generalizes that of Chacko (1966) who studied this statistic when the null and alternative hypotheses require that pi = P2 = ... = Pk = k-1 and Pl < P2 < . . . < Pk, respectively. The theory of inheritance provides examples of nonhomogeneous null hypotheses and, in Example 1, the methods presented here are illustrated using Mendel's classical data from pea-breeding experiments.

In addition, we consider a likelihood ratio statistic for a testing situation where the null hypothesis places an order restriction on the values of pj, P2, ... , Pk. Such a statistic might be used by our hypothetical investigator to test his intuition about the shape of the underlying distribution. The simple hypothesis Pl = P2 =...= Pk

= k-1 is asympotically least favorable, and under this hypothesis the asymptotic distribution of the test statistic has tail probabilities which are weighted averages of standard chi-squared tail probabilities (cf. Theorem 2). Van Eeden (1958) proposed a test statistic for testing an order restriction on a collection of normal means. This statistic is compared to the corresponding likelihood ratio statistic in Robertson and Wegman (1978).

A referee suggested a geometric way of thinking about the relationships between the problems considered in this article. Suppose k = 2. Chacko (1966) considered a test of the null hypothesis that (pI, P2) = (2, 2) against the alternative, Pl < P2, that (Pl, P2) lies on the line segment, t4, joining (2, 2) to (0, 1) (cf. Figure A). If we have prior information that Pi < P2, then Theorem 1 gives the asymptotic distribution of the likelihood ratio statistic for testing the null hypothesis that (pl, P2) is any particular point on t4 against the alternative that it lies on t4 but is not equal to the null hypothesized point. Theorem 2 gives the asymptotic distribution for the likelihood ratio statistic for testing the null hypothesis that

(pl, P2) lies on 4i against the alternative that it lies on - 4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

? Journal of the American Statistical Association March 1978, Volume 73, Number 361

Theory and Methods Section

197



198 Journal of the American Statistical Association, March 1978

Hypothesized Parameter Values.

P2

(0, 1)

<(1/2, 1/2)

Pi (1,0) 1

2. THEOREMS AND EXAMPLES

It will be convenient to think of k-tuples such as p = (pi, P2, *-., pk) and estimates of p as points in Euclidean k-space, Rk. Assume we have in mind an order restriction, 0, on the coordinates of p such as PI < P2

< ... < Pk. The restriction, 0, is then equivalent to re- quiring that the point p belongs to a closed convex cone, C, in Rk (a closed convex cone is closed in the topology and closed under addition and multiplication by non- negative reals). If pi is the relative frequency of the event having probability pi [i.e., p = p p2, ..., p) is the unrestricted maximum likelihood estimate of p], then the projection p = E(A IC) of p onto C is the maximum likelihood estimate of p which satisfies 0. Computation algo- rithms for p are given in Barlow et al. (1972).

Suppose we have a random sample of size n from our multinomial distribution and assume that q = (ql, q2, ..., qk) is given and satisfies 0. Define the hypothesis Hi: i = 0, 1, 2 by

Ho: p = q H1: p satisfies 0 and p , q

and H2 places no restriction on p other than Ek=, pi = 1 and p does not satisfy 0. Let To, = -2 ln X0l where

= flI (qi p,-) 'i is the likelihood ratio for testing Ho against H1. Consider an order restriction, induced by q and 0 on k-tuples 6 = (01, 02, ..., 6k) of real numbers which requires that 6i < 6j only when qi = qj and 0 required that pi < pj. For example, if k = 4, q = (, 6 1, ,) and 0 requires that pi < p2 ? p3 < p4 then the induced order restriction requires that 01 < 62 and 63 < 04. Let D be the cone of points in Rk which satisfies this induced restriction (C C D) and for any k-tuple 6 = (61, 02, ..., k) let E(6 ID) be the projection of 0 onto D. The proofs of the theorems in this article are given in the Appendix.

Let X - (X1, X2, ..., 27) where X1, X2, . .., Xk are independent standard normal random variables and for

= 1, 2, ..., k let Pq(f, k) denote the probability that E (X j D) takes on exactly f distinct values (recall that D

depends on both 0 and on q, the null hypothesized value of p).

Theorem 1: If Ho is true, then for any real number t k

lim P[To1 > t] = E Pq(gt, k)P[X2t_l > t] n - oo t=1

where X2t-i is a standard chi-squared variable with t - 1 degrees freedom (X20 = 0).

Two cases are of particular interest. If q, = k-1; i = 1, 2, ..., k, then D = C so that we obtain the expected generalization of Chacko's work to alternatives which are arbitrary partial orders. On the other extreme, if q has k distinct values, then D = Rk and E(X ID) = X. Since the Xi's have absolutely continuous distributions and they are independent, this implies that Pq(f, k) = 0; t= 1, 2, ..., k-1, and P,(k, k) = 1. Thus, the asymptotic distribution of To1 is the standard chi- squared with k - 1 degrees freedom.

The only difficulty in the application of Theorem 1 is in the computation of the weights, P0(f, k), on the various chi-squared distributions. However, these weights have been fairly thoroughly explored for the problem of testing homogeneity against an order restriction (cf. Barlow et al. 1972) and inequalities among the null hypothesized values actually facilitate their computation. We illustrate the required computation using data from Mendel's classical experiments on pea breeding.

Example 1: Mendel classified 556 peas which resulted from crosses of plants which issued from round yellow seeds and plants issuing from wrinkled green seeds. Let Pi be the probability that a pea resulting from such a mating is round and yellow; P2 be the probability that it is wrinkled and yellow; p3 be the probability that it is round and green; and p4 be the probability that it is wrinkled and green. The Mendelian theory of inheritance states that the probabilities should satisfy Ho: pi = 16,

P2 = 16 P = 16, P4 = 16 The experiments resulted in respective frequencies of 315, 101, 108, and 32. If we test Ho against all alternatives, the likelihood-ratio statistic has a value of .4754 and the probability that a X23 assumes a value this large or larger (i.e., the p-value) is approximately .9246.

Assume the simple model described in Section 5, Chapter V of Feller (1968) for each of the characteristics, shape (round or wrinkled) and color, and assume that these two characteristics are independent. The probabilities pi; i = 1, 2, 3, 4 can be expressed as follows:

PI = 0102, P2 = (l-l1)232,

p3 = i1 (1 -2), P4 = (1-$1) (1 -32)

where

#1 = a12 + 2a, (1 - al), /32 = a22 + 2a22(1 - a2)

ae1 is the gene relative frequency of round in the parent population and aB2 is the gene relative frequency of yellow in the parent population. Now, depending on the nature



Robertson: Testing Order Restrictions on Multinomial Parameters 199

of the parent population, any one of several alternatives might be of interest. The assumption that flu > 2 and 12 ? ' is equivalent to Pi > P2, Pl > P3, P2 > P4, and p3 > p4 (i.e., Pl > (P3) > p4). (The assumption that f3 > 2 is equivalent to assuming that a1 > 1 - V2/2

.29. The Mendelian theory is consistent with the assumption that a1 = a2 = 1.) If, in addition to assuming that /31 > 2 we assume that i1 = 132 then Pi ? p2 = P3 2 p4 would also be an appropriate alternative.

First, consider testing Ho against H1, where

HI: p, > () > p4 and not Ho P3

(recall that Ho: p = ( --L 93, 16)). In this case the relative frequencies pi; i = 1, 2, 3, 4 satisfy H1 so that p = p and the value of the likelihood ratio statistic To, is .4754. Now the hypothesized value of p has equality between P2 and p3, but the order restriction does not relate those values. Thus D = I4 and E(XID) = X. Since the Xi's have absolutely continuous distribution, Pq(l 4) = Pq(2, 4) = Pq(3, 4) -0 and Pq(4, 4) = 1. Thus our p-value is again P[x23 > .4754] = .9246.

Consider testing Ho against H1, where

Hi:pi > p2 = p3 > p4 andnotH0.

In this case the maximum likelihood estimates subject to H1 are 315/556, 209/1112, 209/1112, and 32/556, respectively, and the value of the likelihood ratio statistic is .2409. In order to compute the corresponding p-value, note that D- t0; 02 = 03} so that with probability one E(X D) has three distinct values. Thus our p-value is P[x22 > .2409] = .8865.

Finally, consider testing Ho against H1, where

H1: pi > P2 2 p3 > p4 and not Ho .

The value of the test statistic is .2409, Pq(l, 4) = Pq(2, 4) = 0, and Pq(3, 4) = Pq(4, 4) = 1. Our p-value is given by 2-l P[x22 > .2409] + 2-1. P[x23 > .2409] = .9287.

Now, returning to the general case, let T12 =-2 ln X12, where X12 is the likelihood ratio for testing HI against H2. The test based on T12 is not similar in that the asymptotic distribution of T12 under H1 depends on the particular p satisfying H1 under consideration. Thus, the significance level of the test which rejects for all values of T12 at least as large as t would be given by SupqGH, Pq[T12 > t], where Pq[T12 2 t] is the probability that T12 > t computed under the assumption that q is the vector of parameter values. Finding this supremum would seem to be very difficult, at least using the techniques developed in this article, due to the impossibility, in general, of finding a mapping from one discrete variable to another which changes the probabilities. For example, it is easy to see that if X has a Bernoulli distribution with parameter p #? 2 then there is no function f(.) such that f (X) has a Bernoulli distribution with parameter p = 2. We can, however, show that the hypothesis Pu=P2 =... = pk

= k'l is asymptotically least favorable for this test and in addition that under this hypothesis the asymptotic

distribution of T12 is the distribution discussed in Robertson and Wegman (1975) for testing an analogous hypothesis about a set of normal means.

Theorem 2: If q satisfies H1 and if p = q then for any real t,

lim P[T12 > t] = E Pq(t, k)P[X2 ke >t] n --) oo=1

Moreover, if Po (E) represents the probability of the event E computed under the assumption that Pl = P2

= Pk = k-h then

lim Pq[T12 > t] < lim Po[T12 ?t] nf-coo nfl-*o

for any real t.

In addition to providing the value of

sup lim Pq[T12 > t] q EHI n-too

Theorem 2 also gives a method for investigating the behavior of limn * Pq[T12 > t] for various values of q satisfying H1. For example, if there is a q satisfying Hi which has k distinct values then for this q, D = Rk and E(XID) = X. Thus for any t > 0,

lim Pq[T12 2 t] = 0 and inf lim Pq[T12 2 t] = 0 n gqEHI n-.oo

Note that lim n Pq[T12 2 t] depends on q only through the sets on which q is constant. Moreover, as we note in the proof of Theorem 2, this limit is, in a sense, a decreasing function of the number of distinct values of q. Specifically, if q and r both satisfy H1 and if ri = rj for all i and j such that qi = qj, then limn,o PrJT12 ? t] 2 limn,O PE[T12 ? t]. We illustrate the required com- putations using Mendel's data.

Example 1 (continued): Suppose we are interested in determining if Mendel's data is compatible with the hypothesis that Pl ? P2 ? P3 > P4. The value of T12 is .2345 and the asymptotic p-value is equal to

4

sup lim PE[T12 > .2345] = E Po(l, 4)P[X24-1 ? .2345]. qE H1 n -.oo t=1

The values P0(t, 4) are given in Table A.5 of Barlow et al. (1972). Thus the asymptotic p-value is (')P(X23 > .2345] + (11/24)P[x22 2 .2345) + (1)P[x21 > .2345] = .8114. Clearly, there are q's satisfying H1 which have four distinct values so that

inf lim Pq[T12 ? .2345] = 0 q EH1 n-oo

If the Mendelian theory of inheritance is true then the actual value of p is ( 9 3 3 1 ) . The limit, limn,w Pq[T12 2 t], is the same for any q such that ql > q2 = q3 > q4. In computing E (X D) we average X2 and X3 when 272 <273. Thus Pq(1, 4) = Pq(2, 4) 0 and Pq(3, 4)-=Pq(4, 4)-2 . The asymptotic p-value is 2-1P[X21 > .2345] + 21lP[X20 > .2345] = .3217. Asymp-




1. Asymptotic P-Values for Testing Pi ? P2 ? p3 2 p4

Configuration liimn c Pq[T12 2 .2345]

q = q2 =q3 =q4 .8114

ql = q2 = q3 > q4 .6182 q1 > q2 =q3 =q4

ql = q2 > q3 q q4 .5441

q1 > q2 > q3 =q4

q1>q2 =q3>q4 .3217 ql = q2 > q3 > q4

ql > q2>q3 >q4 0

totic p-values for all possible configurations are given in Table 1.

Finally, consider testing

Hi: pl > p2-P3 > P4

The value of T12 is again .2345. In this case the values of Po(1, 4) have not been tabled but can be computed using the work of Childs (1967). Our asymptotic p-value is given by: (13)P[x23 > .2345] + (2)P[x22 > .2345] + (t)P[X12 > .2345] = .8759. Asymptotic p-values for other configurations are given in Table 2.

2. Asymptotic P-Values for Testing Pi ? P2 = P3 ? P4

Configuration lim,,w Pq[T12 2 .2345]

q1 = q2= q3 = q4 .8759

q1 =q2=q3>q4 .7664 ql > q2 = q3 = q4

ql > q2 = q3 > q4 .6435

The maximum likelihood estimates which satisfied the order restriction for the data in the pea-breeding experi- ment were computed using the pool-adjacent-violator algorithm (cf. Barlow et al. 1972). This algorithm cannot be applied directly when the order restriction requires that the points be unimodal with mode at a particular

point. In this case the so-called minimum-lower-sets algorithm seems to be the most practical way of computing the restricted estimate.

Example 2: Frequently, in practice, a parametric model is fitted to one data set and its validity is sub- sequently checked on an independently chosen set of data. The traffic accident data on page 320 of Derman et al. (1973) were, using a table of random numbers, randomly allocated to two groups. The allocation was done in such a way that the probability that any data. point was assigned to Group 1 was 9/10 and to Group 2 was 1/10. As a result, 646 and 62 points were assigned to Groups 1 and 2, respectively. The results are given in Table 3. A negative binomial distribution, i.e.,

(a + x- 1\ px pX= V X )(1? p)a+x =

was fitted to the Group 1 data using the scoring technique presented in Bliss and Fisher (1953) with resulting values of a = 4.78 and p = .48. This distribution seems to fit the data very well. As a measure of this fit, if b- is the observed frequency and nx is the expected frequency for this negative binomial distribution then the value of 2 Yx bx ln (bx/nx) is 3.3 and P[X2lo > 3.3] = .97.

Now consider the problem of testing this negative binomial distribution as a null hypothesis distribution for the Group 2 data set. If we test against all alternatives, then the value of the likelihood ratio statistic (i.e., 2 x ax ln (ax/mx)) is 7.138 and P[X212 > 7.138] = .848. Bliss (1953, p. 177) remarks that "The curve defined by the P's (or ?'s) is unimodal, so that in fitting the negative binomial to an observed distribution any apparent bimodality (or multimodality) is attributed to random sampling." Now, this particular negative binomial distribution is unimodal with mode at 1 so consider testing against the alternative that Po < Pl > P2 > ... > Pl. Using the minimum-lower-sets algorithm, the maximum likelihood estimates are computed as follows. First find the maximum average value of px over sets of consecutive

3. Observations of Number X of Traffic Accidents Incurred During 1952-1955 by Each of 708 Bus Drivers

Expected frequencies Number X of traffic Group 1 Group 2 negative binomial accidents during Observed frequency frequencies frequencies (a =4.77, p = 2.30,

the period of Xaccidents b, a, N =62) a,=62

0 117 106 11 9.51 11 1 157 143 14 14.75 16.5 2 158 139 19 13.83 16.5 3 115 110 5 10.14 5 4 78 75 3 6.40 4 5 44 39 5 3.65 4 6 21 18 3 1.93 3 7 7 7 0 .96 .5 8 6 5 1 .46 .5 9 1 1 0 .21 .5

10 3 2 1 .10 .5 11 1 1 0 .04 0

12 or more 0 0 0 .03 0

Total 708 646 62



Robertson: Testing Order Restrictions on Multinomial Parameters 201

integers containing 1. In our problem, this set is f1, 2} and the corresponding maximum is 2-1(ps + P2) = .266 (.266-62 = 16.5). This is the value of our restricted estimate px at x 1, 2 (i.e., Pl = P2 = .266). The re- maining values may be computed using the pool adjacent violators algorithm, independently on the two sets {0 } and {3, 4, ...,11 } . (For example, p4 < p5, a violator, so they are both replaced by their average 2-1(74 + P5).) The value of the likelihood ratio statistic is 2 Ex ax ln (dx/m.) = 5.119. Moreover, since none of the hypothesized PxIs are equal, Pq(13, 13) = 1 and our p-value is P[x212 > 5.119] = .95.

APPENDIX: PROOFS

Let r1 > r2 > ... > rh be the distinct values among ql, q2, ..., qk and define the partition Si, S2, ..., Sh of {1, 2, ..., k} by Si = {j; qj = ri}. The projection E(. ID) may be found by independently computing its values for subscripts in each of the sets S1, S2, * * *, Sh.

The following lemmas are straightforward consequences of this fact and well-known properties of these projection operators (i.e., conditional expectations given o- lattices).

Lemma A.1: If U = (U1, U2, ..., Uk) and

V= (V1, V2, ...) Vk)

are points in Rk and if Vi = Vj whenever qi = qj, then

E(U-VID) E(LJID)-V.

If, in addition, V, > 0; i = 1, 2, ..., k, then

E(V.UID) = V.E(UID) .

Lemma A.2 is a consequence of the fact that q satisfies 0.

Lemma A.2: If U is any point in Rk such that

minUj> max Uj ; i=1,2...,h-1 jESi jeSi+l

then E(U[C) E(U ID)

Proof of Theorem 1: The statistic To, is equal to 2 5k=7 nj5{ln pi - ln qJ. Assuming Ho is true, using Taylor's Theorem with second-degree remainder term, and expanding In pi and In qi about pfi we can write

To, n' nEpi[ai-2(qi - pi)2- 0-2(pi-p 'i)I] i=

where ai and fi are random variables converging almost surely to qi. In fact, with probability one for sufficiently large n, ai is between qi and 'i and fi is between Pi and Pi. The almost sure convergence of a, and At follow from well-known properties of Pi and pi. The first-order terms are zero because

k k k

E qi E = i E i3 = I i=~1 i=1 i=l

Now for sufficiently large ns with probability one, ?5

satisfies the hypothesis imposed on U in Lemma A.2 so that pi = E (p C) E (p I D) where represents equality for sufficiently large n with probability one. Thus using Lemma A. 1

k

To 1 p i{aji [VE(\/n(i -qi)] i=l

- E[E(Vn(p - q) I D) i V/n(Ai - qi)]2}

The random vector Vn (A - q) converges in law to a singular normal distribution with zero means and variance-covariance matrix, given by 1M [Emij] where mii = qi(6i- qj) and &, 1 if i - j and 6ij = 0 if i 5- j. Now let Z = (Z1, Z2, . . ., Zk), where Z1, Z, .. . Zk are independent random variables such that Zi is normal with mean zero and variance qi- and let Z = Yk=, qiZi. The random vector (qi(Z1 - Z), q2(Z2 -Z) . * qk(Zk- Z)) has this singular normal distribution. The projection operator E(. ID) is continuous and To, is a continuous function of its arguments so that by Theorem 4.4 and Corollary I of Theorem 5.1 of Billingsley (1968), To1 converges in law to k

E -*-l Eqi(Z* _ ) ]2 i=l

-[E(q(Z -) |D)i - qi(Zi - Z)]2

Using Lemma A.1, the fact that q is constant on the sets Sl, S2, ..., Sh and some algebra, this is equal to

k

qiI(Z* -Z)2 - [E(ZID)i- Z-]2} *=1

However, k k

E, qi(Zi - )2 = E qiZi - E(ZID)J]2 i=1 i~=l

k

+ 2 F? qEZi - E(Z ID) i[E(Z ID)* - Z]

+ qiEE(ZID)i -Z]2

and k

E qiZi - E(Z I D) i]E(Z I D) i i=l1

k

= L qiZi- E(ZjD)i]z = 0 i=1

using Corollary A on page 343 of Barlow et al. (1972) since q is constant on the sets S1, S2, ..., Sh. The distribution of Ek= 1 qEE (Z ID) Z]2 iS given by Theorem 3.1 of Barlow et al. (1972) so that Theorem 1 follows since the probability that E (Z D) assumes exactly t values is the same as the probability that E(X D) assumes t values. (Recall that E(. ID) can be computed independently on the sets CY, 82, ..Y. SI.)

Proof of Theorem 2: The statistic T12 can be written 2 Ek_i npi[ln pi.- ln pi~]. Expanding ln pi~ about pi and




proceeding as in the proof of Theorem 1 we obtain k

T12> qi-'[E(q(Z -) ID)? - qi(Zi - 2)]2 i=l

k

= E [E(v\qZ D)i - VqiZij2 i=l

The conclusion of the theorem follows from Theorem 2.5 of Robertson and Wegman (1978) and the final conclusion follows from the fact that '1 [E(X I XD)i 2 is the square of the distance from X to D and C C D. The assertion that "lim,x Pq[T12 > t] is a nondecreasing function of the number of distinct values of q" follows similarly.

[Received February 1976. Revised August 1977.]

REFERENCES Barlow, R.E., Bartholomew, D.J., Bremner, J.M., and Brunk, H.D.

(1972), Statistical Inference Under Order Restrictions, New York: John Wiley & Sons.

Bartholomew, D.J. (1959a), "A Test for Homogeneity for Ordered Alternatives," Biometrika, 46, 36-48.

(1959b), "A Test for Homogeneity for Ordered Alternatives II," Biometrika, 46, 328-335.

(1961), "A Test for Homogeneity of Means Under Restricted Alternatives (with discussion)," Journal of the Royal Statistical Society, Ser. B, 23, 239-281.

Billingsley, Patrick (1968), Convergence of Probability Measures, New York: John Wiley & Sons.

Bliss, C.I., and Fisher, R.A. (1935), "Fitting the Negative Binomial Distribution to Biological Data and a Note on the Efficient Fitting of the Negative Binomial," Biometrics, 9, 176-200.

Chacko, V.J. (1959), "Testing Homogeneity Against Ordered Alter- natives," Ph.D. dissertation, Department of Statistics, University of California, Berkeley.

(1963), "Testing Homogeneity Against Ordered Alterna- tives," Annals of Mathematical Statistics, 34, 945-956.

(1966), "Modified Chi-Square Test for Ordered Alternatives," Sankhya, Ser. B, 28, Parts 3 and 4, 185-190.

Childs, Donald R. (1967), "Reduction of a Multivariate Normal Integral to Characteristic Form," Biometrika, 54, 1 and 2, 293-299.

Derman, Cyrus, Gleser, Leon J., and Olkin, Ingram (1973), A Guide to Probability Theory and Application, New York: Holt, Rinehart and Winston.

Eeden, C. van (1958), Testing and Estimating Ordered Parameters of Probability Distributions, Ph.D. dissertation, University of Amsterdam, Studentendrukkeri, Poortpens, Amsterdam.

Feller, William (1968), An Introduction to Probability Theory and Its Applications, Vol. I, 3rd Edition, New York: John Wiley & Sons.

Robertson, Tim, and Wegman, Edward J. (1978), "A Likelihood Ratio Test for Testing Order Restrictions for Normal Means," to appear, Annals of Statistics, 6.



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