E L S E V I E R Statistics & Probability Letters 25 (1995) 309-316

STATISTIC8 &

Restricted tests for and against the increasing failure rate ordering on multinomial parameters

Bhaskar Bhattacharya Department of Mathematics, Southern Illinois University, Mailcode 4408. Carbondale, IL 62901-4408. USA

Received April 1994; revised October 1994

Abstract

We consider the likelihood ratio tests for (i) testing a constant failure rate (truncated geometric) against the alternative of increasing (nondecreasing) failure rate ordering of a collection of multinomial parameters, and for (ii) testing the null hypothesis that this parameter vector satisfies increasing failure rate ordering against all alternatives (unrestricted). For both tests the asymptotic distribution of the test statistic under the null hypothesis is shown to be of the chi-bar square type. A numerical example is presented to illustrate the procedure.

Keywords: Isotonic regression; Increasing failure rate order; Likelihood ratio tests; Chi-bar square; Multinomial

1. Introduction

Although the length of life of an object is a continuous random variable, it is often measured as discrete due to limitations of monitoring devices. For example, if a system is monitored periodically, length of life can be defined as the maximum number of time periods successfully completed, or, if a machine operates in cycles, length of life refers to the number of cycles successfully completed prior to failure. Thus, continuous type data are often collected at finite number of specified time points.

Hence we assume that the time to failure, T, is discrete with k possible values labeled t l , t2 . . . . . tk. For 1 ~< i ~< k, let Pl denote the probability that an object chosen at random will fail at time ti (p~ > 0, ~ = 1 P~ = 1) and p = (p l , P2 . . . . . Pk). Let 0~ be the conditional probability that an object will fail at time t~ given that it has not failed before; thus 0i is the discrete failure rate at ti and can be expressed as 0~ =p~(~=~pi) -1, 1 < ~ i < ~ k - l . L e t O = ( O l , 0 2 , . . . , O k _ t ) .

In Section 2, we consider the likelihood ratio test of the null hypothesis that 0~'s are equal against the restriction

0a ~< 02 ~< ... ~< 0k-i (1)

(often called increasing fai lure rate in reliability literature). The asymptotic distribution of the test statistic under the null hypothesis is deduced and is shown to be of the chi-bar-square type, which is a weighted combination of chi-square random variables mixed over their degrees of freedom. Such weighted chi-square

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distributions are common in many order-restricted inference problems (see, e.g., Robertson et al., 1988) and were first found by Bartholomew (1959). We also consider the likelihood ratio test of the restriction (1) as a null hypothesis against the alternative of no restriction. In this case the asymptotic distribution of the test statistic under the null hypothesis is shown to be a convolution of independent chi-bar- square distributions (see Dykstra et al., 1991, for a similar distribution). Hypotheses testing involving the restrictions of decreasing (nonincreasing) failure rates can be handled by some minor changes to the results of Section 2.

An extensive literature exists on inferences for increasing (decreasing) failure rate distributions in the reliability theory (see, e.g., Tenga and Santner, 1984; Lawless, 1980). Inferences under increasing failure rates using isotonic regression techniques are considered by Barlow (1968) and Grenander (1956) for continuous data. In a multinomial setting Barlow et al. (1972) have found the maximum likelihood estimates of the probability vectorp under the restriction (1) which is reproduced in Section 2 for completeness. However, the likelihood ratio tests as considered in this paper are not derived. We refer the reader to Robertson et al. (1988) for other references on this topic.

In Section 3, we consider an example.

2. Main results

Suppose n objects are put on test, and let/~ be the relative frequency of observations that fail at time tg, 1 ~< i ~< k (Y.~I/~i = 1). The likelihood function is proportional to

k

L{p) = I-I p~'. i = 1

Consider a change of variable as 0, = p,(Y.~=,pj) a, 1 ~< i ~< k - 1, or, equivalently, p~ = 0,1-]~7__11 (1 - 0r), k - 1 1 ~< i ~< k - 1, and Pk = Iqj= ~ (1 -- 0r). Then the likelihood function may be rewritten as

k - I

L(O) = 11 0~'( 1 - 0,) "El=':' - "#' (2) i = 1

Maximizing (2) subject to the constraint (1) is precisely the bioassay problem as discussed in Example 1.5.1 of Robertson et al. (1988). The solution /J=(01,02 . . . . . /7k_1 ) is the isotonic regression of the vector ~J = (01,02 . . . . ,0k-1 ) with weight f = (z31, t32 . . . . . t3k_ 1 ), where 0i = /~ , /~- , /~ j , 13, = Y~=i/~J. The "pool adjac- ent violators algorithm" (PAVA) provides an easy method for obtaining 0. Evaluation ofp at 0 = / ig ives the maximum likelihood estimate o f p under the restriction (1).

If we assume that the failure rate is constant over the time points, then the maximum likelihood estimate of 0 is given by 0ol, where 1 is the row vector of l's of length k - 1, and 0o = k- 1

Consider the problem of testing the hypothesis Ho: 01 = 02 . . . . . Ok-1 against the hypothesis of increasing failure rate Hi: 01 ~< 02 <~ ... <<, Ok- 1. By some algebra, it can be shown t h a t p ~ H o if and only if it is a k-cell truncated geometric distribution of the form Pi = p(1 - p)~- 1, 1 ~< i ~< k - 1, and Pk = (1 -- p)k- 1, for 0~<p ~<1. We let Aol denote the likelihood ratio for testing Ho versus H a - Ho, and let To1 = - 2 In Aol. Using the maximum likelihood estimates as constructed above, it is straightforward to show that

To 1 2n /~i In + ~ " \ 1 - 01 "1 = P j ) l n l - - ~ o ~. i = 1 j=i+

B. Bhattacharya / Statistics & Probability Letters 25 (1995) 309-316 311

Expanding I n 0 / a n d In00 about 0i, and In(1 - Oi) and ln(1 - 0o) about (1 - 0/) via Taylor 's Theorem with a second degree remainder term, we obtain

To, = 2n k-l,~=l Pi ( ~ i ~0 ' -00 -~ (00--0i)2262 (0' ----0')2~2fl 2 J + 2n k - l ( E _ ~ ^ x~ f 0 ° - - (~i -4- ( 0 0 - - 0 ' ) 2 (0i--02~ "= / = 1 j = i + l P J ) ~ i 2~ )2 ~i~i 2 3'

where ei is between 1 - 0i and 1 - 0, fli is between 0, and Oi. Vi is between 1 - 0o and 1 - Oi and 6i is between 0o and 0,.

Noting that the first terms within braces cancel, we obtain

~Oo (0, Z_O,)2~ / = I ~ fl' 3 -~- ni~=l tj=~i+l ~j

: E J = ' + 1 P j __ E J = ' + ' Pj k ' 8,+ e,fi(Oo O,)y - E + - - E ~ ( O , - O,)Y. / = I

(3)

/)i

and v, = ~ = , p ~ , 1 ~< i ~ k - 1. Thus, N//n0i, 1 ~< i ~< k - 1 are asymptotical ly independent. F rom (4), it is s traightforward to show that for any 0,'s (0, # 0, 1),

k - Pi ~j=i+,Pj

w, ' = ~ ~ -(f--- -Ory"

0 , ( 1 - - Oi) W, -- (4)

where W is a diagonal matrix with entries as

,,//n(0 - 0) -~ MVN(0, W),

Note that 0 = Pe (0, 3g) is the least square projection of 0 on to ~ with weight ~ where Jg is the cone of nondecreasing vectors of length k - 1. Let U= (U, .. . . . Uk-~) "~ MVN(0, W). Then, under Ho, using continuity of the projection operator , it follows that

and

t , f ~ i 0 , - 0ot . . . . . , / ~ t 0 ~ _ , - 0o)t -~ i t / , - o . . . . . v k _ , - c) ,

where/,7 k- 1 k - , : wiU,/~i=l W, W ~,=, a n d = (w, . . . . . Wk_l). Also ,

k - , k - , w,(U,-/_7) 2 -- Y' w,[U, - Pw(U[ iF), + P. , (U[X) , - 032

i = , /=1

k - 1 k -1 = ~' w , [ U , - Pw(U[ JF),] 2 + ~ w,[P,,,(UI ~ ) , - 0 ] 2,

, = , /= ,

since the cross-product term is zero by Theorems 1.3.2 and 1.3.3 of Rober tson et al. (1988).

~/~O(O- 0) 5% V - Pw(Ul~),

By the law of large numbers, ~i, Vi ~ 1 - 0,, and fli, (~i ~ 0i with probabil i ty 1. By condit ioning on ny~=i~ ~, it can be shown that E(0,) = Oi. Let p = (/~1,/~2 . . . . . /~k). Using the multi-

variate central limit theorem to ,~/n(/~ - p ) and, then using delta method, it follows that

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Let Yi Ui/x/fffli(1 Oi), 1 ~< i ~< k 1, and 17 k- t k- 1 . . . . v~Y~/y~= ~ Using the above facts, it follows from ~,i = 1 Vi"

(3) that, under Ho, Tol converges in distribution to

k 1

~, viEP~(YI o,~. )~ - 17] 2, (5) i = 1

where Y = ( Y ~ . . . . . Yk- 1 ) and v = (v 1 . . . . . Vk- 1 ). The asymptotic distribution of To 1, under Ho, can now be obtained by using the Corollary on p. 70 of Robertson et al. (1988). The probability that P ~ ( Y I ~ ) has exactly l distinct values (level sets) is denoted by P(l, k - 1, v). Also, it follows that, under Ho, vi = (1 - p)~- x for 1 ~< i ~< k - 1. A least favorable distribution can be obtained by using Theorem 3.6.1 of Robertson et al. (1988).

The above developments are summarized in the following theorem.

T h e o r e m 2.1. When H0 is true, then for any real number c,

k - 1

lim P(To~ > c) = ~ P( l , k - 1,v)P(z~_~ > c), n ~ , 1 = 1

and the asymptotic least favorable distribution is 9iven by

sup lim P(Tol > c) = 2-k+Ep(z2-1 > C), p e r t o n~oo I= 1

(6)

where Z 2 is a chi-square random variable with v degrees o f freedom (Z g - 0). The least favorable distribution in (6) is obtained as v ~ (1,0 . . . . . 0) which occurs when p ~ 1.

The asymptotic distribution depends upon the true unknown p only through the level probabilities P(l, k - l, v). For equal weights, the level probabilities can be calculated recursively (Corollary A on p. 81 of Robertson et al., 1988). However, when the weights are not equal (as in this case), the level probabilities are much more difficult to compute. Robertson et al. (1988, p. 77-79) (see references therein) also provides the calculations of P(l, k, w) for k ~< 5 and arbitrary w. An important contribution is made by Bohrer and Chow (1978) by using weights based on/~.

Critical values for the test based upon the least favorable distribution given in (6) can be found in Table 5.3.1 of Robertson et al. (1988), for k -- 4 . . . . . 16 and a = 0.10, 0.05 and 0.01. Usually, a test based upon the least favorable distribution is quite conservative. A more appropriate way is to estimatep from the data, and then use this value to approximate the asymptotic p-value (see Example, procedure (b)).

Turning to the problem of testing H~ as a null hypothesis and H E : n o restriction as the alternative the test statistic is T 1 2 = - - 2 In A 1 2 where A12 is the corresponding likelihood ratio. As before, expanding In 0~ about 0i, and ln(1 - 01) about (1 - 0i), we obtain

T 1 2 = n -t EJ=i+lP! (Oi--Oi) 2, i = 1 72

where 61 is between 01 and 01 and yi is between 1 - 0i and 1 - 0~. Supposep e H 1 so that the restriction (1) holds among 0~'s. In particular, let us suppose the form of 0 is such

that there are t blocks (1 ~< t ~< [(k - 1)/2]) of constant values:

Oq+j . . . . . Oil+k, < "'" <0i,+1 . . . . . Oi,+k,,

B. Bhattachar ya / Statistics & Probabili ty Let ters 25 (1995) 309 - 316 313

where 0 ~< ix < ... < i, ~< k - 2, ki >1 2, gi. Also, let

o~'rO = {Xcz ~ k - I: Xi, + l ~ " " ~ Xil + k "~ . . . "~X.i, + l ~ "" ~ Xi, + k,}.

Since t~ -+ 0 almost surely,

0 = P~tOl * ) - P~(Ol ~0),

where - represents equality for sufficiently large n with probability 1. Then TI 2 - ~ = 1 Tj 2, where

• Yj=i+ 1/~j T~2 = n ~ -+ (0i - - 0 i ) 2.

i = i j + 1 ~2

The asymptotic distribution of T12 is the same as the distribution of

ij -I- kj ~, v i[r i -- Pv,(YJl ~f~j)i]2,

i = i j + l

where Ya = ( Yij + 1 . . . . . Yij + k), VJ = (Vi~ + l . . . . . Vi, + k) and OF; is the cone of nondecreasing vectors of length kj. By using the Corollary on p. 70 of Robertson et al. (1988), the asymptotic distribution of Ti2 is given by the following:

kj lim P(T{2 > c) ~ e(l, kj, v j) 2 = P(Zk;- ! > C).

Since x/~/~'s are asymptotically independent, it follows that the asymptotic distribution of T~2 is a con- volution of independent chi-bar-square random variables. The expression for this asymptotic distribution is given in Theorem 2.2 below.

To show that Ho is least favorable in H~, note that

T12 ~ m i n II Y J - zll~ = m i n II g - zlb. (7) j = 1 Z e , .~ j Z ~ ,~/'O

The last quantity in (7) is largest when ~r 0 is smallest (and thus equal to J(), which occurs when 0eHo. These facts are summarized in the folowing theorem.

Theorem 2.2. Suppose p e H1 such that the corresponding 0 has t blocks with at least two constant values within each block. Then for any real number c,

b - t ~ I lim P(T12 > c ) = ~ ~ P(lj, kj, vJ)P()~_l_t >c),

n ~ o o l = t I~ + . . . + l , = l j = l

where b = ~ = 1 kj. The asymptotic least favorable distribution is given by

sup lim P(T12 > c) = sup lim P(T12 > c). (8) peH~ n ~ p~Ho n ~

The author strongly conjectures that the least favorable distribution in (8) is obtained as v ~ (1, 1 . . . . . l) which occurs when p ~ 0, and, thus the level probabilities in the least favorable case can be obtained using the recursive relation of the equal weight situation, but is unable to prove this result for a general k. Lee et al. (1993) derived the bounds on the chi-bar-squared distributions for arbitrary weights which are monotone in

314 B. Bhattacharya / Statistics & Probability Letters 25 (1995) 309 316

Table 1 Frequency distribution of failure and maximum likelihood estimates (unrestricted = ()i, and restricted = 01) of the failure rates

Using (a) Using (b)

hour ni O, Oi 6, Oi

0- 41 0.1262 0.1262 0.1111 0.1111 50- 44 0.1549 0.1549 0.1341 0.1341

100- 50 0.2083 0.2083 0.1761 0.1645 150- 48 0.2526 0.2289 0.2051 0.1645 20(0 28 0.1972 0.2289 0.1505 0.1645 250- 29 0.2544 0.2362 0.1835 0.1645 300- 18 0.2118 0.2362 0.1395 0.1645 350 16 0.2388 0.2388 0.1441 0.1645 400- 15 0.2941 0.2941 0.1579 0.1645 450- 11 0.3056 0.2951 0.1375 0.1645 500 7 0.2800 0.2951 0.1014 0.1645 550- 11 0.6111 0.6111 0.1774 0.1645 600-629 7 - - 0.1373 0.1645 Not ~iled at 630 hr 44

nature. In our case, although the weights are nonincreasing for 0 < p < 1, they are also functions of one variable p, which makes the result of Lee et al. (1993) applicable, but conservative. Nonetheless, it is shown in the literature (e.g., Robertson and Wright, 1982, 1983) that the level probabilities are fairly robust in nature and the equal weight case provides adequate approximation in most cases. So the equal weight approxima- tion can still be used in this situation.

3. Example

The failure times of 369 radio transmission receivers are reported in Table 1 (see Cox, 1958; Mendenhall and Hader, 1958). Forty four of these 369 receivers did not fail during the test period (630 h). We analyze these data in two ways as follows. (a) Assuming that they provide no information regarding the failure rate of radio transmitters during the test period, we discard these 44 items from our analysis. (b) We consider a time point t14 large enough such that n~4 = 44. The unrestricted and restricted maximum likelihood estimates of the failure rates are listed in Table I.

Using the procedure (a), under Ho, the estimated constant failure rate is 0.2016. To test Ho: 01 = 02 . . . . . 012 against all alternatives (unrestricted) the value of the likelihood ratio test statistic is 42.2413 and the corresponding p-value is P(Z2~ > 42.2413)= 0.0006. This analysis seems to reject the composite null hypothesis that the failure rate is constant (or equivalently, the data are from a 13-cell truncated geometric distribution).

From Table 1, it seems plausible that the failure rate of transmission receivers be nondecreasing as the number of hours in operation increases. So we consider the problem of testing Ho: constant failure rate against Hi: failure rate is nondecreasing over hours. For the restricted test of Ho versus H1 the value of the likelihood ratio test statistic is 40.2707. The p-value computed under the least favorable distribution given in

B. Bhattacharya / Statistics & Probability Letters 25 (1995) 309-316 315

Theorem 2.1 is

~ ( 11 ) 2 _ t i p ( z 2 t > 40.2707)= 0.0000. 1=1\1- 1

Thus, we would reject the null hypothesis of constant failure rate in favor of the failure rates being nondecreasing (H t ).

To determine if the data are compatible with the hypothesis 0t ~< 02 ~< .-- ~< 0t2, we consider testing H1 against H2 (no restriction). The computed value of T12 is 1.9705. The approximate asymptotic p-value based on the equal weight case is

12

P(l, 12)P(z~2_ , > 1.9705) = 0.9856, I = l

where the equal weight level probabilities P(l, 12) are given in Table A.10 of Robertson et al. (1988). Thus, we fail to reject the hypothesis of nondecreasing failure rate.

Using procedure (b), under Ho, the estimated constant failure rate of 0.1507. To test Ho: 01 = 02 . . . . . 0t 3 against all alternatives (unrestricted) the value of the likelihood ratio test statistic is 15.3802 and the corresponding p-value is P(Z22 > 15.3802) = 0.2213. Thus, we would fail to reject the null hypothesis that the failure rate is constant.

When testing Ho versus Ht, the calculated value of the test statistic is 7.7390. The p-value using the least favorable distribution is 0.2741. However, the true asymptotic null distribution of Tot depends upon the unknown weights vi, 1 ~< i ~< 13. A p-value computed from this distribution is

13

P(l, 13, v)P(z~_ 1 > 7.7390). / = 1

We estimate this p-value by replacing v by its estimate ~. We conducted a simulation study by first estimating p by/~ = nt/n, and then generating the weights using the relation fi = (1 -/~)~- t, for 1 ~< i ~< 13. We obtained 20000 isotonic regressions for these ~ and recorded the number of level sets in each case. The estimated values of P(l, 13, ~) are found to be: 0.0726, 0.2378, 0.3074, 0.2292, 0.1071, 0.0372, 0.0074, 0.0011, 0.0002, 0.0000, 0.0000, 0.0000, 0.0000. The p-value associated with the simulated P(l, 13, f) is 0.0392 and this would reject the null hypothesis of constant failure rate in favor of the failure rates being nondecreasing (Ht).

When testing Ht versus H2, the calculated value of the test statistic is 7.6412. As before, the p-value computed using the equal weights approximation is 0.6396. So we would fail to reject the hypothesis of nondecreasing failure rate.

Acknowledgements

The author would like to thank Professor Richard Dykstra for very helpful discussions concerning Theorem 2.2. The comments of the referee are also appreciated which led to an improved manuscript.

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