Inference on Pr X < Y in the Two-Parameter Weibull Model Based on Records

We consider the stress-strength reliability based on record values from the Weibull distribution. The Bayes estimator based on squared error loss and the maximum likelihood estimator are derived and their bias and mean squared error performance are studied. Likelihood-based confidence intervals as well as some bootstrap intervals are developed. We derived also the highest posterior density interval. Simulation studies are conducted to investigate and compare the performance of the estimators and intervals.


Introduction
Chandler 1 introduced and studied some properties of record values.Since then a considerable amount of the literature is devoted to the study of records.Ahsanullah  1.1

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Let X 1 , X 2 , . . .be an infinite sequence of iid random variables.An observation X j is called a record if its value is greater than all previous observations, that is X j > X i for every i < j.We want to estimate the stress-strength reliability Pr X < Y using the data on records.The stress-strength reliability Pr X < Y arises in life-testing experiments when X and Y represent the lifetimes of two devices, and it gives the probability that the device with life time X fails before the other.As an example, Hall 6 studied a situation where the breakdown voltage Y of a capacitor must exceed the voltage output X of a power supply in order for the component to work properly.Weerahandi and Johnson 7 considered another example on rocket motors.This probability has other interpretations in other disciplines, for example, in medical sciences; it is used as a measure of treatment effectiveness in studies involving comparison between control and treatment groups.Various other examples may be found in Kotz et al. 8 .
Estimation of the stress-strength reliability based on record data was considered by Baklizi 9 for the exponential distribution with record values and by Baklizi 10 for the generalized exponential distribution.Kundu and Gupta 11 investigated this problem for simple random samples from the Weibull distribution.In Section 2 we derive the maximum likelihood estimator and the associated large sample intervals.Bayesian procedures based on records are derived in Section 3. Simulations to investigate the performance of the asymptotic inference procedures and to compare them with the bootstrap intervals are described in Section 4. The results and conclusions are given in Section 5.

Likelihood Inference
We shall consider the case when the shape parameters are equal.Let X ∼ W θ 1 , β and Y ∼ W θ 2 , β be independent random variables.Let R Pr X < Y be the stress strength reliability.Let r r 0 , . . ., r n be a set of records from W θ 1 , β , and let s s 0 , . . ., s m be an independent set of records from W θ 2 , β .The likelihood functions are given by 3 as follows: where f and F are the pdf and cdf of X ∼ W θ 1 , β and g and G are the pdf and cdf of Y ∼ W θ 2 , β .The likelihood function of β, θ 1 , θ 2 based on r, s is given by

2.2
Taking the natural logarithm we get the log-likelihood function

2.3
The first partial derivatives are given by

2.5
Equating these partial derivatives to zero and solving simultaneously we obtain

2.6
Hence the MLE of R is given by R θ 2 / θ 1 θ 2 .The study of the exact distribution of R is apparently rather complicated, so we will consider the asymptotic distribution.We need the asymptotic joint distribution of β, θ 1 and θ 2 .The second partial derivatives of the log-likelihood function are given by

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The Fisher information matrix is given by Now we will find the entries of the information matrix.We need to find E r

2.13
A 1 − α % confidence interval for R based on this asymptotic result is given by where η is obtained by substituting m/n for p and the MLEs of θ 1 and θ 2 in the asymptotic standard deviation η.Another interval can be obtained by using the matrix of minus the second partial derivatives of the log-likelihood function.This matrix can be used in place of the Fisher information matrix to obtain η as an estimator of the variance of R. The new interval is given by

2.15
Many authors suggested that parameter transformation may improve the performance of intervals based on the asymptotic normality of the maximum likelihood estimator.For parameters representing probabilities, the logit transformation seems appropriate.Let λ ln R/1 − R , and the maximum likelihood of λ is given by λ ln R/1 − R .The asymptotic variance of √ n λ − λ is given by η2 dλ/dR 2 .This variance can be estimated by substituting the maximum likelihood estimator instead of the parameter.A 1 − α % confidence interval for λ is given by λ L , λ U , where 2.16

Bayesian Inference
As in Kundu and Gupta 11 , we will use the conjugate gamma priors for the scale parameters θ 1 and θ 2 and a squared error loss function.The prior density of θ j therefore is given by where a j > 0, b j > 0 are the parameters of the prior distributions.Assuming that θ 1 and θ 2 are independent, the joint prior distribution of θ 1 and θ 2 is given by π θ 1 , θ 2 Assume that the prior distribution of β, denoted by ζ β , has a support on 0, ∞ .Recall that the likelihood function of β, θ 1 , θ 2 based on r r 0 , . . ., r n and s s 0 , . . ., s m is given by

3.2
The joint posterior density function of β, θ 1 , θ 2 therefore is given by; where the numerator is given by

3.4
This expression for π * β, θ 1 , θ 2 is difficult or even impossible to find in closed form.A simulation technique is needed.It is clear that the conditional posterior distributions of θ 1 and θ 2 given β are given by

3.7
Taking a 1 a 2 0 and b 1 b 2 0 in the prior distributions we obtain

3.10
We can use the following Monte Carlo method to find approximate point estimates.
4 Calculate the approximate Bayes estimator and the approximate posterior variance;

3.11
Approximate highest posterior density HPD intervals for the stress-strength reliability R may be found using the algorithm of Chen and Shao 12 .

A Simulation Study
We conducted simulations to compare the performance of the point and interval estimators developed in this paper.In our simulations we also included some bootstrap intervals 13 , namely, the percentile interval and the bootstrap-t interval based on the logit transformation of R. We used n m 6, 8, 10, 15.We fixed θ 1 1 and used θ 2 1, 3, 5, 7, 9.The confidence level taken is 1−α 0.95.In each simulation run we generated 2000 samples of records from the distributions of X and Y .For each pair of samples we calculated the following intervals: 1 ANE: the interval based on the asymptotic normality of the MLE with variance estimate based on the Fisher information matrix, given by 2.14 , 2 ANO: the interval based on the asymptotic normality of the MLE with variance estimate based on the observed information matrix, given by 2.15 , 3 ANT: the interval based on the asymptotic normality of the MLE of the transformed parameter with variance estimate based on the observed information matrix, given by 2.16 , 4 HPD: the approximate highest posterior density interval of Chen and Shao 12 , 5 Perc: the percentile interval, 13 , and 6 Boot: the bootstrap-t interval based on the logit transformation of the stress-strength reliability parameter, 13 .
The biases and mean squared error of the Bayes and the maximum likelihood estimator are simulated and the results are given in Table 1.The lower L , upper U , and total T error rates and the expected widths W of the intervals are approximated using the results of the 2000 simulation replications.For the percentile and bootstrap-t intervals we used 1000 bootstrap resamples.The results of our simulations are given in Table 2.
2 and Arnold et al. 3 provided a detailed account of theory of records and the inference issues associated with records.In this paper, we will consider confidence interval estimation of the stress-strength reliability based on record data when the underlying distribution is Weibull.The Weibull model is quite popular in analyzing reliability and life-testing data.Its flexibility and capability of modeling various forms of failure mechanisms and hazard function types gave him an important role in reliability literature; see Johnson et al. 4 and Murthy et al. 5 .The probability density function pdf and the cumulative distribution function cdf of the Weibull distribution W θ, β are given by f x θβx β−1 e −θx β , x > 0, θ > 0, β > 0 F x 1 − e −θx β .

Table 1 :
Simulated biases and mean squared errors of the estimators, θ 1 1.

Table 2 :
Simulated error probabilities and expected width of the intervals, θ 1 1.
L: lower error probability, U: upper error probability, T: total error probability, W: expected width of the interval.