A class of efficient tests for increasing-failure-rate-average distribution under random censoring

602 IEEE TRANSACTIONS ON RELIABILITY, VOL. 41, NO. 4, 1992 DECEMBER

A Class of Efficient Tests for Increasing-Failure-Rate-Average Distribution Under Random Censoring

Ram C. Tiwari

Jyoti N. Zalkikar University of North Carolina, Charlotte

Florida International University, Miami

Key Words - Efficacy, Proportional hazard censoring model, Appliance data

Reader Aids - Purpose: Present a derivation and an example Special math needed for explanations: IFRA tests, randomly right-

Special math needed to use results: Same Results useful to: Reliability analysts and theoreticians

censored data, efficacy calculation

Summary & Conclusions - A class of tests for the increasing failure rate average (IFRA) alternatives under random censoring is proposed. The tests are based on a function of the Kaplan-Meier estimator. Most of the IFRA tests in the literature depend on the (nuisance) parameter that appears in the definition of IFRA, and the choice of this parameter is crucial in performing the tests. The proposed class of tests does not have this disadvantage. Under some regularity conditions, the asymptotic normality of the tests is established and asymptotically distribution-free tests are obtained by using estimators for the null standard deviations. The efficacies of the tests under the proportional hazard censoring model are studied. The proposed test is most efficient for the Weibull family of IFRA alternatives among the existing tests available for the censored data. The test is applied to published appliance data.

1. INTRODUCTION

A life Cdf F ( x ) is increasing failure rate average (IFRA) if - l n (F ( t ) ) is star shaped for t>O, or equivalently iff [2: p84] :

F(bx) r{F(x)}' for all x 2 0, 0 < b < 1, (1-1)

The equality in (1-1) holds iff F is exponential. The dual class of decreasing failure rate average (DFRA) Cdf' s can be defined analogously by reversing the direction of inequality in (1-1). The IFRA class of life distributions is important in reliability investigations since it can be alternatively defined by a condi- tion intuitively related to wear-out as follows [5]. If F is IFRA, for each x r O and A chosen so that F ( x ) = exp( -Ax),

5 exp( -At) , if t 2 x.

This clearly suggests that, if an IFRA population and a nonwear- ing population have the same chance of lasting for [OJ], then the IFRA device has better chance of surviving any shorter period and worse chance of surviving any longer period. The IFRA class is the smallest class of Cdf s which contains exponential Cdf's and is closed under the formation of s-coherent systems [2]. The IFRA Cdf's describe life lengths experienc- ing damage from random shocks under fairly general assumptions [8]. They are used to model the life Cdf's of items subject to overstress or to accelerated environments such as electronic capacitors subject to a high voltage, ball bearings subject to a high load, or a mechanical assembly subject to a strong vibra- tion [3].

This paper considers the problem of detecting a wear-out through IFRA Cdf's that would lead to appropriate engineering changes in design or operation. The following real situa- tion provides strong motivation.

Example

Consider the data on a small electrical appliance reported by Nelson [16: p 1731. These data are collected from life testing of 407 units at various stages in a development program. For each unit, the number of cycles to failure or to removal from test and its failure code (one of 18 causes) are recorded. Units with code 0 were removed from test before failure (right censored). The objective of the program was to make appropriate engineering changes to improve the early- and long-life por- tions of the life distribution. Often it is costly and time con- suming to make these changes and then to collect and analyze

0

The methods developed in this paper enable us to detect the wear-out behavior of a design and thus assess whether an engineering change in the design leading to elimination of some specific failure modes would be worthwhile.

Statistically we formulate the problem as testing the null hypothesis Ho of no wear vs the alternative hypothesis H I of wear-out. Since we confine our attention to wear-out through IFRA distributions, formally

data to determine the effect of these changes.

Ho: F ( x ) = 1 -exp( -x/p), x r O ( p unspecified),

vs H I : F ( x ) is IFRA, but not exponential.

In the uncensored model, where we get a complete sample, tests for Ho vs H1 have been proposed by Deshpande [7], Kochar [13], Link [15], Bandyopadhyay & Basu [l], and Jam- malamadaka, Tiwari & Zalkikar [ 111. For randomly censored model see Gerlach [9], Tiwari, Jammalamadaka & Zalkikar [ 181, and more recently Wells & Tiwari [ 191. We are interested in testing Ho against H1 on the basis of the randomly censored

0018-9529/92$03.00 01992 IEEE

TIWARIIZALKIKAR: EFFICIENT TESTS FOR INCREASING-FAILURE-RAT-AVERAGE DISTRIBUTION UNDER RANDOM CENSORING 603

data { ( Z ; , Si), i = 1, . . . , n} . The yi are treated as the random times to the right censorship. We introduce a class of test statistics (notation is described at the end of this Introduction):

The Kaplan-Meier [12] estimator of F is:

The test based on J,, (9) rejects Ho in favor of H1 for large values. Several remarks are in order.

1. If = 1, M = 6b the degenerate probability measure at b, and 9 (U) = U, then J, (9) is the test statistic proposed by Deshpande [7].

2. If 6 = 1, 9 ( U ) = U, and M is the uniform distribution on [0, 11, then J , (9) is the test statistic proposed by Link [15] and Jammalamadaka, et a1 [ l l ] .

3. If M ' 6 b and *(U) = U, .I,(*) is the test statistic proposed by Gerlach [9] and Tiwari, et a1 [18].

4. If M = Ab, then J, (9) is the test statistic proposed by Wells & Tiwari [ 191:

One disadvantage of the test (1-4) is that it depends on b, and thus, the choice of b is important. However, the proposed test statistic J,( 9) does not suffer this disadvantage.

Section 2 states the asymptotic s-normality of a suitably normalized version of J, (9) under some regularity conditions on the Cdf's F & H, and on 9.

Section 3 computes the efficacy of the test, and discusses the optimal tests for the Weibull alternatives in the proportional hazard censoring model. For any fixed amount of censoring the efficacy of the test is higher than other existing tests for the censored data. Section 4 is an application of the proposed test to the appliance data.

continuous Cdf of uncensored data XI, . . . ,X,, , unknown continuous Cdf of right censored data Yl,. . . , Y,, unknown minimum

ordered Zi S[Xi% Y,]: the value is 1 if the argument is true, else it is 0. 6; corresponding to the Z ( i ) Cdf of Z1, ..., Z, F - H survival functions of X1, Y1 ,Z1, respectively

Xi A Y,

F,, IFRA increasing failure rate average b, M Xi U, 9 suitable weight function, known

Kaplan-Meier estimator of F [12]

a parameter, and its Cdf, both are known i.i.d. r.v.'s having Cdf F ( x ) , i = l , ..., n i.i.d. r.v.'s having Cdf H ( x ) , i = l , ..., n

denotes the complement of a Cdf -

Other, standard notation is given in "Information for Readers & Authors" at the rear of each issue.

2. ASYMPTOTIC NORMALITY AND CONSISTENCY

Let {+ ( t ) ; 0 st< m} be a Gaussian process with mean 0 and covariance kernel

E{4(s)@(t)} = F ( s ) F ( t ) {K(u)F(u)} - ' d F ( u ) ,

for O s s < t < m . (2-1)

Assumptions

A . l Both F 8~ H have a common support [0, CO),

A.2 ~ ~ { ~ ( u ) } - ' d F ( u ) < m . 0

Ref [lo: theorem 2.11 of Gill implies that the weak convergence of $ , , ( X I = ~ { F , , ( X A Z ( , , ) ) -F (xAZ( , ) ) } to + ( x ) on [0, 00) holds as n-00. The asymptotic s-normality of J, , ( \k ) is given by theorem 1.

Theorem 1. Let the function 9 be continuous, nondecreasing, and piecewise differentiable with bounded derivative on [O, 11.

Then, under A. 1 & A.2, the f i { J , , (q) - A ( F ) } converges in distribution to a normal r.v. with mean 0 and finite variance:

Corollary 1. Let *(U) = ua, CY 2 1.


dM( b) . order to perform the test based on J,, , we estimate 02. We can write:

The proofs of theorem 1 and corollary 1 are along the lines of Kumazawa [14] and of [19], and hence are omitted.

We state the following result related to the Kaplan-Meier process {+,(x), O l x < m } which extends the result of Jam- malamadaka, et a1 [ l l ] . Note that -

H * ( x ) = Pr{Z1cx, h l = l } ,

H * ( x ) = A 9[Z,IX, 6j=1]. i = l

We estimate U: by:

. dF(x)dF(y), 0 I bl, b2 I 1 , . {K, (Zj - ) } - 2 dM(b l )dM(b , ) .

Notation

g ( x - ) the left-hand limit of g ( x ) at x ; g is any function

Under H,, and A . l & A.2, the consistency of a:,, can be proved along the lines of [14: lemma 2.41.

Another s-cogsistency estimator of .', can be obtained by replacing p and K in (2 -2 ) by,

+ is defined in (2-1).

The proof of this theorem is technical and is deferred to

Corollary 1 shows that the null asymptotic variance 0: depends on he unknown mean p, and the censoring Cdf H. In

the appendix.

TIWARUZALKIKAR: EFFICIENT TESTS FOR INCREASING-FAILURE-RAT-AVERAGE DISTRIBUTION UNDER RANDOM CENSORING 605

f i n = ( ai)-' Zi and E,, respectively.

This can be proved using the results in [6, 181. From the discussion in this section, and from corollary 1,

it follows that the tests rejecting H, in favor of H , for large values of &[.I,,, - A ( a)]/u,,, are s-consistent against all continuous IFRA alternatives.

i = l i = 1

3. EFFICIENCY LOSS DUE TO CENSORING

We consider the alternatives (FA,} with X, = X, + - y / f i , y>O and F b ( x ) = 1 -exp( - x ) . The alternatives are in the Weibull family of Cdf's:

The calculations yield:

abln(b) d M ( b ) , a 2 1. (3-1)

0 ( l + a b ) 2 P" = -

For the proportional hazard censoring model,

H ( x ) = F q x ) , 0 5 p c 1,

blb2

a2[ (b~+l)(bl+62+2b,b2a)(2b,Q+bl/b2-b,(l+p) +1)

1. - bl b2

( b p + 1) (b2+a) (b,b2+2ab1 -b1( 1 + f i ) + 1)

b:b2 (b , +a) (b,+b2+2a) (b , (b , +b,+2a) -b, (1 + P ) )

+

(3-2)

The Pitman efficacy of J,,, is:

TABLE 1 Optimal Values of a and the Corresponding Efficacy of Jn,.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.03 1.07 1.11 1.16 1.20 1.24 1.28 1.33 1.37 1.41

1.3897 1.3395 1.2922 1.2474 1.2051 1.1650 1.1271 1.0911 1.0570 1.0247

1.07 1.10 1.15 1.19 1.22 1.26 1.30 1.35 1.39 1.43

1.3091 1.2814 1.2548 1.2292 1.2046 1.1809 1.1580 1.1360 1.1148 1.0944

When censoring is light (0 is small), the efficacy of J,,, test is higher for the uniform Cdf compared to Be ( 1 ,5 ) . For heavy censoring, the skewed-to-the-right beta distribution, Be( 1,5), improves the efficacy of Jn,ol test over the uniform distribution. It was observed but not reported here that, for other choices of beta Cdf s, the efficacy of J,,, tests did not improve over the uniform Cdf and Be ( 1,5). It is clear from the table that the reproduction in efficacy values is controlled by a.

Let M = ab in (3-1) and (3-2), then (3-3) gives the efficacy of the Wells & Tiwari [ 191 test. For fixed values of P = 0 (0.1 )O. 9, table 2 gives the optimal values of b & a, viz, bop & ayopt, that max- imize this efficacy. The table also provides the maximum efficacy, e (b,, aopJ. Tables 1 & 2 show that J,,, test has higher efficacy among all the existing tests for the censored data with the choice of M as the uniform for P < O S and as Be( 1, 5) for P 2 0.5.

TABLE 2 Optimal Values of b and a for Weibull Altematives

with Fixed Values of 6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.38 0.31 0.27 0.24 0.22 0.21 0.19 0.18 0.17 0.16

1.05 1.10 1.15 1.20 1.24 1.29 1.33 1.37 1.42 1.46

1.3645 1.3146 1.2727 1.2362 1.2038 1.1748 1.1484 1.1242 1,1020 1.0815

p ' (&) & u;(p) are given by (3-1) & (3-2), respectively. The choice of M is a beta Cdf, Be(a, , a 2 ) , with parameters

a1 and a2 The values of (a,, a2) are (1, l), and (1, 5). [Be( 1, 1 ) is the uniform Cdf on [0, 11 .] The results are presented in table 1 which gives:

the optimal value of a that maximizes e (a , 0) for fixed values

the corresponding maximum efficacy. of P=O(O.l)0.9

4. APPLIANCE DATA

We now apply the J,,,test to (a part 00 the appliance data 1161 given in table 3 and already described in section 1. All units in group 5 were cycled by automatic test equipment and had roughly the


TABLE 3 Number of Cycles to Failure or Removal for 64 Units in Group

5 (Before Modification)

Cycles failure Cycles failure Zi code I zi code

45 8 4395 0

88 1 5 3412 5 1117 5 1789 5 4137 0 4129 8 1585 6 4368 0 3901 0 2432 5 4006 0 4816 0 2070 0 1588 5 546 5

2157 5 3008 2 3 164 5 3688 0 1312 2

687 17 1312 2 1103 0 535 5

1342 5 496 5

2533 0 3319 17 4319 0 2203 17 2899 5

713 5 1321 5 816 5

2762 5 1489 5 4413 0

649 5 1238 5 378 0

1738 0 148 17

Cycles failure Zi code

703 5 487 5 437 17

3892 0 3937 0 2113 5 3552 0 3484 0 3855 0 2757 8 5448 0 4693 0 412 5

4004 5 3830 5 3380 5 470 0

4017 5 3985 0 4056 0

same design. Since units with failure code 0 were removed from the study, their cycles are treated as censored observations. The cycles of all units with code other than 0 are treated as failures. Z , , ) is treated as failure so that 6,,, = 1. We consider the proportional hazard censoring model where = Fp, for some p 2 0. For this model Pr { X < Y } = 1 / (0 + 1 ) , and A.2 implies that P < 1. The appliance data consist of 37.5% of censored observations and the corresponding is 0.6. For this P , the best choice of M from table 2 is Be( 1,5) with cy = 1.30. Then,

J64,1,3 = 0.8456857,

u:,3 (0.6) = 0.0356852,

A(1.3) = 0.8385164.

[Under proportional hazard censoring model, U: ( p ) given by (3-2) can be computed numerically and hence we do not have to use its estimate .',,,I. The 1-sidedp-value of 0.33 indicates no statistical evidence of a wear-out behavior, calling for no design changes to correct wear-out for units in group 5. However, during the development program, this group was modified to increase the number of cycles to failure for code 5 . Using hazard plots, Nelson [ 161 estimated the failure-time distributions before and after the modifications. Both hazard plots show exponential fit, confirming our conclusion.

J,,, was calculated using the representation:

IMSL Subroutines were used for computing F,,, and for the in- tegration w.r.t. M .

ACKNOWLEDGMENT

We are pleased to thank the Associate Editor and referees for their valuable comments and suggestions which improved the presentation of the paper.

APPENDIX

Let D[O, m] be the class of real valued, bounded, and right continuous functions defined on (0, 03) with finite left limits at each x E (0 , m), and finite limits at x = 0, W. D[O, 031 is a metric space with Skorohod metric. Let C[O, m] be the class of continuous functions on (0, 03) with the uniform metric. Similarly, define D[O,l] and C[O, 11. See [4: chapters 2 & 31.

Proof of Theorem 2

Since the first term in (2-3) goes to 0 in probability as n- 00, it suffices to prove that the process E,, defined by:

converges weakly to c. Let E > O be arbitrary small. Since F & G are continuous, there exists 0 < K- ' ( 1) such that F(t9) < E . Again, since F & G are continuous, and $,, E D[O, 031, + E C[O, 031, it is observed that F,, E D[O, 11 and c E

Since +,, converges weakly to 4, by the Skorohod, Dudley, Wichura theorem [17: theorem 4, p 471 there exist pro- cesses +: and +* such that +: = +,,, ,*= + and +: converges weakly to +* and 114; - +*[It - 0 a.s. as n -m.

C[O, 11.

Notation

= equality holds in distribution 11.11; sup-norm over [o, 01.

Define,

S-*(b) = lm [ , * ( b x ) - +* (31 d F ( x ) , O l b l l 0

Then, [; = E n , and = and using the Lebesgue Dominated Convergence theorem, we have

TIWARI/ZALKIKAR: EFFICIENT TESTS FOR INCREASING-FAILURE-RAT-AVERAGE DISTRIBUTION UNDER RANDOM CENSORING 607

I e m I

C, and C2 are constants (independent of n).

Since 114: - $* 11: - 0 a s . as n- 00 and E is arbitrary small, it follows that 11; - 0 a.s. as n - W . Hence, from [17: corollary 1, p 481 3;, converges weakly to r. The covariance kernel of { can be obtained by letting *(U) E U

0

-

in $(\k) in theorem 1.

REFERENCES

[l] D. Bandyopadhyay, A. P. Basu, “A note on tests for exponentiality by Deshpande”, Biometriku, vol 76, 1989, pp 403-405.

[2] R. E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, 1975; Holt Rinehart and Winston.

[3] R. E. Barlow, E. M. Scheuer, “Estimation from accelerated life tests”, Technometrics, vol 13, 1971, pp 145-159.

[4] P. Billingsky, Convergence of Probability Measures, 1968; John Wiley & Sons.

[5] Z. W. Bimbaum, J. D. Esary, A. W. Marshall, “Stochastic characteriza- tion of wear-out for components and systems”, Ann. Math. Statistics, vol

[6] Y. Y . Chen, M. Hollander, N. A. Langberg, “Testing whether new is better than used with randomly censored data”, Ann. Statistics, vol 11,

[7] J. V. Deshpande, “A class of tests of exponentiality against increasing failure rate average alternatives”, Biometriku, vol 70, 1983, pp 514-518.

[8] J. D. Esary, A. W. Marshall, F. Proschan, “Shock models and wear pro- cesses”, Ann. Probability, vol 1, 1973, pp 627-649.

[9] B. Gerlach, “Tests for increasing failure rate average with randomly right censored data”, Statistics, vol 20, 1989, pp 287-295.

[IO] R. Gill, “Large sample behavior of the product-limit estimator on the whole line”, Ann. Statistics, vol 11, 1983, pp 49-58.

1111 S. R. J a m l a m a d a k a , R. C. Tiwari, J . N. Zalkikar, “Testing for exponentiality against IFRA alternatives using a U-statistic process”, Revista Brasileira de Probabilidade e Estatistica, vol 4, 1990, pp 147-159.

[12] E. L. Kaplan, P. Meier, “Nonparametric estimation from incomplete observations”, J. Amer. Statistical Assoc., vol 53, 1958, pp 457-481.

[13] S. C. Kochar, “Testing exponentiality against monotone failure rate average”, Commun. Statistics - Theor. Meth., vol 14, 1985, pp 381-392.

37, 1966, p~ 816-825.

1983, pp 267-274.

(141 Y. Kumazawa, “On testing whether new is better than used using randomly censored data”, Ann. Statistics, vol 15, 1987, pp 420-426.

[15] W. A. Link, “Testing for exponentiality against monotone failure rate average alternatives”, Commun. Statistics - Theor. Meth., vol 18, 1989, pp 3009-3017.

[16] W. Nelson, Applied Life Dura Analysis. 1982; John Wiley & Sons. [17] R. Shorack, 1. A. Wellner, Empirical Process with Applications to

Statistics, 1986; John Wiley & Sons. [18] R. C. Tiwari, S. R. Jammalamadaka, J. N. Zalkikar, “Testing an increas-

ing failure rate average distribution with censored data”, Statistics, vol

[I91 M. T. Wells, R. C. Tiwari, “A class of tests for an increasing failure- rate-average distribution with randomly right-censored data”, ZEEE Trans. Reliability, vol 40, 1991 Jun. pp 152-156.

20, 1989, pp 279-286.

AUTHORS

Dr. Ram C. Tiwari; Department of Mathematics; University of North Carolina, Charlotte, North Carolina 28223 USA.

Ram C. Tiwari was born in Uttar Pradesh, India on 1950 May 18. He received his BSc and MSc in Statistics from the University of Allahabad, Uttar Pradesh and PhD in Statistics from the Florida State University, Tallahassee in 1977 & 1981. From 1972 to 1982 he was a lecturer at the University of Allahahad, India. From 1981 to 1985 he was an Assistant Professor at the In- dian Institute of Technology, Bombay. From 1981 to 1982 and 1985 to 1986 he was a Visiting Assistant Professor at the University of California, Santa Bar- bara. During 1986.1989 he was an Assistant Professor at the University of North Carolina at Charlotte. Since 1989 he has been an Associate Professor at the University of North Carolina at Charlotte. His interests include Bayes nonparametric inference, gaodness-of-fit tests, and reliability theory. He is a member of the American Statistical Association and the Institute of Mathematical Statistics.

Dr. Jyoti N. Zalkikar; Department of Statistics; Florida Intemational Univer- sity; University Park; Miami, Florida 33199 USA.

Jyoti N. Zalkikar was born in Bombay, India on 1962 July 30. She received her BSc and MSc in Statistics from the Bombay University in 1982 & 1984, and the MS in Statistics and PhD in Mathematics (Statistics Track) from the University of California at Santa Barbara in 1987 & 1988. From 1984 July to 1985 December she worked as a research assistant at the Indian In- stitute of Technology, Bombay. Since 1988 she has worked as an Assistant Pro- fessor at the Florida International University. Her areas of interest are Bayes nonparametric inference, reliability, and life testing. She is a member of the American Statistical Association and the Institute of Mathematical Statistics.

Manuscript TR91-106 received 1991 June 26; revised 1992 February 14.

IEEE Log Number 00713 4 T R F

Modification of: Error Log Analysis: Statistical Modeling and Heuristic Trend Analysis

(Continued f rom page 601)

D ~ , ~ i ~ ~ . T i ~ ~ y. Lin; Dept. of Electrical & Computer ~ ~ ~ i ~ ~ ~ ~ i ~ ~ ; univer. sity of California, San Diego; La Jolla, California 92093-0407 USA.

Dr. Daniel P. Siewiorek; Dept. of Electrical & Computer Engineering; Carnegie Mellon University; Pittsburgh, Pennsylvania 15213 USA.

Daniel Pa Siewiorek (S’67,M’72,SM’79,F’81): For biography, see IEEE Trans. Reliability, vol 39, 1990 Oct, p 408.

Manuscript TR91-022 received 1991 February 21; revised 1992 March 26.

IEEE Log Number 01035 4 T R b Ting-Tins Y. Lin (S’84,MW): For biography, see ZEEE Tram. Reliubili-

ty, vol 39, 1990 Oct, p 432.

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A class of efficient tests for increasing-failure-rate-average distribution under random censoring