Multi-Censored Sampling i n the Three Parameter Weibull Distribution

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Multi-Censored Sampling i n the Three

Parameter Weibull Distribution”

A. Clifford Cohen Department of Statistics and Computer Science The University of Georgia

Athens, Georgia 30602

In life and fatique testing, multi-censored samples arise when at various stages of a test, some of the survivors are withdrawn from further observation. Sample specimens which remain after each stage of censoring continue to be observed until failure or until a subsequent stage of censoring. In this paper, maximum likelihood estimat,ors and estimators which utilize the first order statistic are derived for the three parameter Weibull distribution. Estimators are also derived for the special case in which the shape parameter is known, a special case which includes the two parameter exponential distribution. An illustrat,ive example is included.

KEY WORDS Weibull Distribution Multi-Censored Samples Progressively Censored Samples Life Testing

1. INTRoCUCTION

In life and fatigue tests, individual observations are time ordered, and it is common practice to cease testing prior to failure of all sample specimens. In a typical case, the test might be terminated with a single stage of censoring. In many cases, however, censoring occurs in several successive stages. Single stage censoring has received the attention of various writers including Hald (1949), the present writer (1950), Gupta (1952), Epstein and Sobel (1953), and others. Multi-stage or progressive censoring has previously been considered by Herd (1957), Roberts (1962), the writer (1963), (1965), Harter and Moore (1965), Ringer and Sprinkle (1972), Wingo (1973), Lemon (1974), and possibly by others.

In this paper, we consider maximum likelihood estimation in the three parameter Weibull distribution using multi-censored samples. We also consider a modification of the maximum likelihood estimators which utilizes the first order statistic in a manner similar to that suggested by Dubey (1966) for use in single stage censoring. Certain special cases of

* Research supported by the National Science Foundation, Grant GP-34318. Presented to the International Statistical Institute, Vienna, Austria, August 1973. Condensed version without illustrative example included in Session Proceedings. (39th Session I.S.I., XLV, Book 1, pp. 277.)

Received July 1973; revised November 1974.

the Weibull distribution, including the exponential distribution are likewise considered.

2. THE SAMPLE

Let N designate the total sample size, and n the number which fail and therefore result in completely determined life spans. Suppose that censoring occurs in k stages at times Ti > TieI , j = 1, 2, . . . , k, and that rj surviving items are removed (censored) from further observation at the jth stage. Thus

N=n+krj.

Two types of censoring are generally recognized. In Type I censoring, which is of primary interest here, the Tj are fixed, and the number of survivors at these times are random variables. In Type II censoring, the number of survivors are fixed and the Tj are random variables. In both types, the Tj are either fixed or determined independently of the life span X.

The likelihood function L(S), where S signifies a k-stage Type I multi-censored sample as just described, is

L(S)

= c fi I(4 Jj D

_i-1

- F(Tj)]“) (2) in which C is a constant while f(x) and F(x) arc density and distribution functions respectively.

For the three parameter Weibull distribution, we write the density function as

f(x; 8, y, 6) = jj (x - y)a-1e-(Z-y)6’e, y < x < a,

zero elsewhere. (3)

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348 A. CLIFFORD COHEN Accordingly, the distribution function becomes

F(x; 8, y, 6) = 1 - e--(Z-y)6’0. ₍₄₎ An alternate parameterization in which

0 = p’, and thus p = e”“, ₍₅₎

is sometimes to be preferred.

The logarithm of the likelihood function (2) with density (3) follows as

In L = n In 6 - n In 0 + (6 - 1) 2 In (5; - 7) 1

- i c* (xi - 7)’ -I- In C, (6) where c* signifies summation over the entire N observations with each of the ri observations censored at time Ti assigned the value xi = T, , so that

C* (x, - r)” = ~ (xi - r)” + ~ rj(Tj - y)‘. 3. MAXI,MUN LIKELIHOOD ESTIIUATORS

For 6 > 1, maximum likelihood estimating equations follow from (6) on differentiating with respect to 6, 0, and y in turn, and equating to zero. Thereby we obtain d In L - = as f + 2 In (5, - Y) 1 - $ C* (xi - y) * In (zi - 7) = 0,

a

In L -- = de -;+j+ X*(x* -7)” = 0, (7)

a

In L __- = ; c* (xi _ ,$I a7 - (6 - 1) 2 (Xi - r>-’ = 0, 1

where again c* signifies summation over the entire sample of size N with each observation censored at time Tj assigned the value Zj = Tj .

For 6 < 1, the likelihood function becomes infinite as y -+ y, , where y1 is the smallest sample observation (i.e., the first order statistic of the sample). Accordingly, in this case, the applicable estimating equations consist of the first two equations of (7) plus + = y1 - q/2, where 7 is the unit of precision with which observations are made. When it is known that 6 = 1, in which case we are dealing with the exponential distribution, the applicable estimators are

8 = c* (zi - yJ/n, and + = y1 . ₍₈₎ Returning to the case in which 6 > 1, we eliminate 0 between the first two equations of (7) to obtain TECHNOMETRICSO, VOL. 17, NO. 3, AUGUST 1975

[

C* (xi - y)* In (2, - r) 1

c* (2, - 7)” - s

1

- i $ In (xci - y) = 0. (9) When y is known and also in those cases where Yl - q/2 is the applicable estimator for y, we need only solve (9) to obtain 8. This can be accomplished using trial and error techniques described by Cohen (1965). With 8 thus determined, 8 follows from the second equation of (7) as

e = c* (2, - y)i/n. ₍₁₀₎

In the more general case of concern here, we choose a first approximation y1 . Then assuming y = y1 , we proceed as described above and determine first approximations a1 and 0, . With these approximations substituted into the third equation of (7), we calculate the first approximation

(a

In L/ay), . If this value differs from zero, we select a second approximation yZ and repeat. The procedure is con- tinued until we obtain two values yi and yj with a difference that is sufficiently small and such that (a In L/d?); 2 0 2 (a In L/&y) j , whereupon we interpolate for the required estimates.

It is to be noted that y1 is an upper bound on y and is thus available as a first approximation y1 for use in the iteration process described above.

Results in this section overlap to a considerable extent certain work of Lemon (1974) and Wingo (1973), but the three investigations have been carried out with complete independence.

4. MODIFIED MAXIIVIUIVI LIKELIHOOD ESTIMATORS

An alternate set of estimators, otherwise desig- nated as modified maximum likelihood estimators

(MMLE) may be obtained by replacing the third equation of (7) with E(Y,) = y1 where Y, is the first order statistic in a sample of size N. The estimating equations accordingly consist of (9) and

(10) plus (1 l), which follows.

y + (O/N)“61’l = yL , where r, = I’(1 + 6-l). (11) The computational procedure is the same as that

for determining maximum likelihood estimates.

Here, however, we interpolate between two approximations yi and yj such that E(Y,) i 2 y1 ? E(Yr) j to determine the required estimates.

5. SOR~E SPECIAL CASES

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349 the modified equations consist of the second equation

of (7) plus equation (11). These results are subsequently reduced to the following forms.

given below. Corresponding results were also in-

dependently obtained by Wingo (1973), and by

Lemon (1974).

a2

In L 11 -2-

_a6

= - ~-

_6"

- Q C* (2, - r)‘[ln (cc, - r)]“, #lnL n --2

_ae

= --

_{82 - $ c* (2" - Yj6,}

a*

In L -ayz = -(6 - 1) $ (Xi - 7))” MLE With 6 Known, (6 > 1).

n 6 c* (2, - q”-’

c* (2, - 3”

- (6 - 1) 2 (2, - +>-I = 0. 1 (12)

e = c* (2; - q>“/n.

In this case, with 6 known, the first equation of (12) can be solved for T. Subsequently, 8 follows from the second of these equations.

MMLE With 6 Known.

N[(Y,

- WrJ” - c* hi - ~9% = 0, c13j

8 = c* (Xi - q*/n.

In this instance, the first equation of (13) can be solved for 5, and f? follows from the second.

These two equations also yield RIMLE for 6 and 0 when y is known. In this latter case, the first equation of (13) is solved for 8, and 8 follows from the

second equation. Maximum Likelihood estimators

(MLE) for this case are given by Cohen (1965). The Exponential Distribution.

With 6 = 1, the Weibull distribution encompasses the exponential as a further special case. Whereas the applicable MLE result from setting y = y1 , and

d

In L/d0

= 0,

the

MMLE

result from setting E(Y,) = y, and d In L/a0 = 0. The MLE in this case are given in (8). The MMLE follow below. MMLE.

4 =

[ $ (~6 - YI) + $ rj(Tj - YI)

1

/(n - 1)) (14) 9 = y1 - 8/N.

It is to be noted that a complete (uncensored) sample might be regarded as a special case of the progressively censored sample in which ri = 0 for all j, and thus n = N. In this case, the estimators of (14) become

e = N@ - ~1) Ny, - z

N-l ’ T=N-lp (15)

which are recognized as the joint best linear unbiased estimators (BLUE) given by Sarhan (1954), and recently discussed by Cohen & Helm (1973).

6. ESTIMATE VARIANCES AND COVARIANCES

The asymptotic variance-covariance matrix of the MLE (a, 8, q) is obtained by inverting the informa- tion matrix in which elements are negatives of expected values of the second partial derivatives of the logarithm of the likelihood function. For sufficiently large samples, these expected values can be approximated by substituting the estimates 8, 6, and 4 directly into the partial derivatives which are

_ c@+l c* (x, _ y)6-2,

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a2

In L

a2

In L

___

_asae

= -~

_{aea 6}

= 3 c* (Xi - 7)” 111 (2:, - y),

a2

In L

a2 111

L

--

c-c

a 6 a7

a7 a 6

- $ (2; - r>-'

+ ; c* (Xi

- yy

+ j C* (2, - y)*-l In (.r; - y),

a2

In L

a2

In L

_{s c* (x, - #'.}

aeay = &ae

= - 82

Here, as employed elsewhere, c* signifies summation over the entire sample with each observation censored at time Tj assigned the value xi = T, .

The asymptotic variances obtained as described above are dependent on several regularity conditions which as Lemon (lot. tit) points out arc satisfied when 6 > 4. Certain related problems arc also considered by Harter and Moore (1967) in connec- tion with singly and doubly censored samples.

7. AN ILLUSTRATIVE EXAMPLE

An illustrative example has been constructed by simulating a Weibull population with y = 100, 0 = 100, and 6 = 2, with 0 = 10,000. Using formulas given by Cohen (1973) in Equations (4), the population mean, standard deviation, and it’s third standard moment are calculated to be pZ = 188.623, us = 46.325, and LYE:= = 0.6311. The resulting

sample might be conceived of as having been

generated in a life test conducted on 100 randomly selected units of a certain electronic device with p.d.f. given by equation (3). Sixty-eight complete life span observations are included, while thirty-two observations have been censored in three separate stages. Following are the life spans in hours, to two places of decimal, of the 68 items which failed during the test.

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350

TABLIC l--Summary of estimates

A. CLIFFORD COHEN

TYPE PARAMETERS ESTIMATED

ESTIMATORS Y e ‘6 6 u 0

a3

MLE 106.93 1635.05 1.638 91.50 188.80 51.27 .9234

MMLE 102.38 3905.30 1.809 96.66 188.32 49.18 .7713

I I I

V(i) = .0447 V(e) = 2.71 x lo6 V(i) = 10.02 cov(i,e, = 345

Cev(6,y) = -.4364 Cov(e,y) = -3558

y = 100 (Assumed to be known)

MLE (100) 5907.23 1.888 99.32 188.15 48.52 .7098

MMLE (100) 11214.27 2.024 100.11 188.70 45.87 .6150

V(i) = .0331 V(i) = 2.51 x lo7 Cov(i,i) - 902

~5 = 2 (Assumed to be known)

MLE 100.68 9749.38 98.74 188.19 45.74 (.6311)

MMLE 100.34 9821.36 99.10 188.16 45.91 (.6311)

V(i) = 2.42 x lo6 V(j) = 23.26 Co&;) = -4.87 x lo3

Population 100 10,000 2 100 188.62 46.33 0.6311

Values

When the sixth failure occurred at time T, = 124.63, ten units randomly selected from the survivors were removed from the test. Fifteen addi- tional units were removed when the fortieth failure occurred at time T, = 174.22, and the test was terminated with seven survivors at time T, = 249.35. In summarizing these data, we record: N = 100, n = 68, x1” ri = 32, yI = 109.12, TI = 124.63, r, = 10, T, = 174.22, rz = 15, T, = 249.35, rR = 7, x1”” x, = 11,577.47, i$, = 170.257, ?J = .Ol.

1635.05 to 11,214.27, the estimates for 0, P, and CT are quite stable. It is also to be noted that the different estimates for olg are relatively stable, vary- ing from a low of .6150 to a high of .9234.

8. ACKNOWLEDGEMENT

The able assistance of Mr. Nicholas Norgaard, who performed all of the programming and carried out all of the calculations presented here, is grate- fully acknowledged.

Since we know 6 > 1, the MLE were accordingly calculated by simultaneously solving the three equations of (7) as described for the general case in Section 3. The MMLE were calculated by simultaneously solving the first two equations of (7) plus equation (11). The resulting estimates are displayed in the following table along with applicable asymptotic variances and covariances. A second set of estimates are displayed for the case in which 6 = 2 (known). These were calculated using equations (12) for the MLE and equations (13) for the MMLE. A third set of estimates are displayed for the case in which y = 100 (known). These were calculated using the first two equations of (7) for the MLE and using the second equation of (7) plus (11) for the MMLE.

In general, the estimates obtained here compare

quite favorabIy with corresponding population

values. It is to be observed that although the different estimates given above for 8, range from TECHNOMETRICSO, VOL. 17, NO. 3, AUGUST 1975

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