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Communications in Statistics - Theory and Methods

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Bayes 2-sample prediction for the inverse weibull distribution R. Calabria a; G. Pulcini a a Dept. of Statistics & - Istituto Motori - CNR, Napoli, Italy

To cite this Article Calabria, R. and Pulcini, G.(1994) 'Bayes 2-sample prediction for the inverse weibull distribution',

Communications in Statistics - Theory and Methods, 23: 6, 1811 — 1824 To link to this Article: DOI: 10.1080/03610929408831356 URL: http://dx.doi.org/10.1080/03610929408831356

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COMMUN. STATIST.-THEORY METH., 23(6), 1811-1 824 (1994)

BAYES 2-SAMPLE PREDICTION FOR THE INVERSE WEIBULL DISTRIBUTION R. Calabria, G. Pulcini Dept. of Statistics & Reliability - Istituto Motori - CNR via Marcorii 8 - 80125 Napoli Italy

Key Words and Phrases: inverse Weibull distribution; Bayes estimation; predictive

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distribution; log inverse Gamma distribution.

ABSTRACT This paper deals with the problem of predicting, on the base of censored sampling, the ordered lifetimes in a future sample when samples are assumed t o follow the inverse Weibull distribution. Bayes prediction intervals are derived, both when no prior information is available, and when prior information on the unreliability level a t a fixed time is introduced in the predictive procedure. A Monte Carlo simulation study has shown that the use of the prior information leads t o a more accurate prediction, also when the choice of the informative prior density is quite wrong.

1. INTRODUCTION In many practical problems, such as life testing, reliability and quality control problems, we are often concerned about future observations of an experiment. Then, we use results from an informative past experiment for obtaining a predictive distribution for a future observation. Information contained in the predictive distribution is generally summarized by a point estimate (often the posterior mean) and a prediction interval that encloses that future random value with a given high probability.

Copyright O 1994 by Marcel Dekker, Inc.

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CALABRIA AND PULCINI

For example, we might wish to predict the failure time of an unused component, or the time needed to complete a future test, or the number of failures of a repairable system in a given future time interval. Moreover, statistical inference generally refer to the point & interval estimation of the unknown distribution parameters (and function thereof), which can never be observed. Hence, we can never known how well such quantities have been estimated. Thus, we should address much attention to observables (i.e. quantities which do not involve any parameter and are capable to be observed), and predictive distribution should be used as often as possible. In this paper, Bayes procedures for deriving the predictive distribution of the failure times of a future sample of unused components, given the first m failure times of n items put on test, are proposed, when the underlying lifetime has the inverse Weibull (IW) distribution: f (t) = pa-@t-P-' exp [-(ta)-P]

t

>O

a ,P > O

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In particular, two Bayes procedures are analysed. The first one is based on prior ignorance and provides prediction intervals which are numerically equal to those obtainable by a classical approach. The second procedure uses a prior knowledge on the unreliability level a t a given time, and a noninformative prior on the shape parameter P. Predictive distribution is given when experimental data are collected under both Type I and Type I1 censored sampling. A large Monte Carlo study has 'shown that, under Type I1 censoring, the in-

formative procedure outperforms the noninformative one, not only when the prior information is correct, but also when a wrong choice of prior is made. Notation ordered lifetimes [r.v., observed value] from an IW distribution IW parameters Log-inverse Gamma parameters size of sample put on test number of observations in censored sampling length of time of the experiment: a prefixed quantity in Type I censoring, a r.v. (T E t,) in Type I1 censoring

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1813

INVERSE WEIBULL DISTRIBUTION N

size of future sample

'8,

s-th ordered future lifetime [r.v., observed value] from an IW dis-

Y,

tribution

Fr

unreliability level a t the prior fixed time r

dk

(-l)k(N;"

(k = 0 , . . . ,N - 5)

4

z+jFP (

j = ~. ., . , n - m )

B

B(s, N

2.

-

s + 1) = r ( s ) . r ( N - s + l ) , T ( N

+ l ) , the Beta function.

BACKGROUND

The IW distribution has been recently derived as a suitable model t o describe degradation phenomena of mechanical components (see Keller and Kamath (1982) and Keller et al. (1985)) such as the dynamic components (pistons, crankshaft, main bearing, etc.) of diesel engines. Other physical failure processes leading t o such a model have been given by Erto (1989). Furthermore, Erto (1989) showed that the IW distribution provides a good fit to several data given in literature, such as the times to breakdown of an insulating fluid subject to the action of a constant tension (Nelson (1982)). The IW distribution generally exhibit a long right tail (compared t o that of the commonly used distributions) and its hazard rate has a behaviour similar t o that of the lognormal and inverse Gaussian distributions. As suggested by a referee, updated information on the IW distribution can be found in Johnson e t al. (1994). Inferential procedures based on maximum likelihood principle and on leastsquares method for estimating the distribution parameters and the reliability function are given in literature (Erto (1989), Calabria and Pulcini (1989, 1990)). Also, Bayes procedures for constructing credibility intervals on the parameters and on the reliability function have been proposed in Calabria and Pulcini (1992), when the IW distribution arises from a load-strength context. Since the r.v. W = 1 / T follows a Weibull distribution with parameters a &

P

(see Keller and Kamath (1982)), then, in case of complete sampling, prediction on y,

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CALABRIA AND PULCINI

could be made by using results given for Weibull samples. In particular, prediction bounds for the first failure time in a future Weibull sample, derived classically by Mann (1970) and Lawless (1973), can be used t o predict the last failure time in a future IW sample, as well as Bayes prediction intervals given by Evans and Nigm (1980) and Nigm (1989, 1990) can be used for any ordered failure time in a future IW sample. However, when Bayes prediction is based on an informative prior, then procedures developed for the Weibull distribution can be hardly (from an engineering viewpoint) extended to the IW distribution, because the IW parameters do not retain the physical meaning of the Weibull parameters. Finally, no result, both in a classical and Bayes framework, appears to be available when the prediction is based on data of a censored sample. 3.

PREDICTIVE DISTRIBUTION OF y,

Let t l

5 tz 5 . . . 5 t, denote the observed ordered lifetimes of the first m compo-

nents to fail in a sample of n components whose lifetimes follow the IW distribution Downloaded At: 11:17 19 January 2010

(1). Such data can be collected under Type I and Type I1 censoring. In a Type I1 censoring, the number m of observed lifetimes is fixed and the experiment is stopped a t the (random) time of the m-th failure

(T = t,). T is

contrary, the length of time of the experiment

In a Type I censoring, on the fixed before the data are col-

lected, and m is random (for more details, refer to Nelson (1882) and Balakrishnan and Cohen (1991)). The Bayes approach allows both Type I and Type I1 data t o be analysed in the same manner. Under Type I or Type I1 censoring the likelihood function results in: ~ ( d a t a a , Pcx) ~ ~ a - ~ @exp(~ - @ ad^) - l (1 - exp [ - ( a ~ ) - @ ] } * -(2) ~ Let g ( a , P ) be the joint prior density on the distribution parameters, measuring the uncertainty about the true values. Then, the joint posterior density for a & 4, is:

where

(4) does not depend upon a & P.

INVERSE WEIBULL DISTRIBUTION


O), the joint

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CALABRIA AND PULCINI TABLE I Values of the parameters in the unified form of g(a:, P ) PRIOR DENSITY

6

a

b

Noninformative on P and a: Noninformative on P and LIG on F,

1

0

0

0

>1 >1

Prior densities (11) and (16) can be easily rewritten in a unified form:

where the value of the parameters 6, a and b are given in Table I. Based on (17) the predictive density on y, is

where I =

Sow pm-l-su-@r-a@C:;?

cjA;m-a d p , and the cumulative distribution

function is: rn

c

p m - 1 - 6 ~ - @n-m N-8

H ( y , (data)= --

~ [ ~ ~ + ( ~ + k ) ~ ; ~ ] Y- , "2-0" d ~

j=o k=O

Simple forms of (19) occur for selected m and s values. Suppose that prediction is based on a complete sample ( m

-

-

(19)

n). Then:

Suppose now one wish to predict the time of the last ordered future failure

(s

N ) based on a censored sample ( m < n ) . Then:

-

Finally, if the prediction of the last future lifetime ( s complete sample ( m

n ) , then:

-

N ) is based on a

INVERSE WEIBULL DISTRIBUTION 4.

SIMULATION RESULTS UNDER TYPE I1 CENSORING

The statistical performances of the Bayes procedure based on informative prior have been compared, via Monte Carlo simulation, to the performances of the Bayes procedure based on prior ignorance. Since this last procedure provides prediction intervals which are numerically equal to the classical ones, the performed comparison is equivalent to compare the informative Bayes procedure to the classical method. Thus, equal-tails intervals (which are generally evaluated in classical approaches) are compared from a frequentistic viewpoint in terms of two quantities: a) the 'covering percentage' (CP), defined as the fraction of times an informative 80% Bayes prediction interval covers the predicted observation in repeated sampling, and b ) the 'ratio of interval widths' (RIW), defined as the ratio of the mean width of the noninformative prediction interval to the mean width of the informative one. Of course, in case of very large simulation size, the covering percentage of the noninformative prediction interval tends t o be equal to the probability content 7.

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Two prior densities on F, have been considered, having mode equal to 0.05 & 0.10, respectively, and same standard deviation equal to 0.15. From (13) and (14), when p~ = 0.05, we have a = 4.126 & b = 2.043, and when p~ = 0.10 we have a = 4.540 & b = 2.537. In order t o evaluate the effect of a correct or a wrong choice of the prior densities on the performances of the informative Bayes procedure, IW samples with several true F, values have been generated, namely:

F, = 0.02, 0.05, and 0.10, for the prior density with p~ = 0.05, F, = 0.05,0.10, and 0.20, for the prior density with p~ = 0.10. A prior density is assumed to be correct if its mode is equal to the true F, value, and wrong in the other cases. In Tables I1 and 111, for

fi

= I and for selected n , m, N , and s values, the sta-

tistical properties of the Bayes procedures, based on 1000 Monte Carlo simulations, are given. Small and moderate samples have been considered because, in large samples, data dominate the prior belief and the informative prediction interval actually coincides with the noninformative one. The results are quite favourable to the informative Bayes procedure. In fact, this procedure outperforms the noninformative one not only when the prior density is

CALABRIA AND PULCINI TABLE I1

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Comparison of informative and noninformative Bayes prediction intervals. The informative prior density on F, has mode ,UF = 0.05 and standard deviation u p = 0.15.

n

m

10

10

F, = 0.02 RIW

N

s

CP

F, = 0.05 RIW

CP

F, = 0.10 RIW

CP

1

1

86.4%

1.03

84.1%

1.15

80.8%

1.22

5

1

81.6%

1.24

83.4%

1.19

82.2%

1.14

5

3

88.1%

1.08

85.9%

1.14

83.5%

1.17

5

5

85.5%

1.16

82.6%

1.43

81.1%

1.60

C P > 80% and RIW > 1 are favourable to informative Bayes procedure

correct (as it could be anticipated), but also when the true F, value differs from the prior mode. The C P values are almost always greater than 80% and the RIW values are greater than 1 in all the cases considered. The gain obtained by introducing the prior knowledge is greater when prediction is based on a censored sample of small size, is made on the last ordered lifetime of a large future sample. Results similar t o those given in Tables I1 and I11 have been obtained when

P differs

from 1 and 7 differs from 80%. 5.

NUMERICAL EXAMPLE

Consider the following times to breakdown of an insulating fluid subject t o the action of two electrodes exerciting a constant tension of 36 KV: t l = 0.35, t 2 = 0.59, t3 = 0.96, tr = 0.99, ts = 1.69, ts = 1.97, t , = 2.07, t s = 2.58, t~ = 2.71, tlo = 2.90,

INVERSE WEIBULL DISTRIBUTION

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TABLE I11 Comparison of informative and noninformative Bayes prediction intervals. The informative prior density on F, has mode

p~ = 0.10

and standard deviation o~ = 0.15.

F, = 0.05

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n

m

N

s

CP

RIW

F, = 0.10 CP

RIW

F, = 0.20 CP

RIW

CP > 80% and RIW > 1 are favourable to informative Bayes procedure

t l l = 3.67, t l z = 3.99, t13 = 5.35, t14 = 13.77, and t 1 5 = 25.50. These times, in minutes, are given in Nelson (1982). Erto (1989) showed that these times are well fitted by an IW distribution. Suppose that censoring has been made at m = 10, so that only the first 10 lifetimes were available. Prediction on the time t o breakdown in a future experiment (s = N = I ) is required, and both the equal-tails and the HPD 80% intervals on yl must be evaluated. If there is no prior information available, then from (9) the equal-tails 80% prediction interval is (0.560, 22.46), whereas from (10) the HPD 80% interval is (0.05, 8.68). Suppose now that the analyst can express his technical knowledge on the failure mechanism in terms of a mode p~ = 0.05 and a standard deviation o~ = 0.15 of the unreliability level at the time

T

= 0.5. On the basis of this prior information,

the equal-tails 80% prediction interval on yl is (0.530, 18.98), and the HPD 80% interval is (0.07, 7.57).

CALABRIA AND PULCINI

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