CHENGConghua(程從華)

School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, China

Abstract: The reliability of a system is discussed when the strength of the system and the stress imposed on it are independent and non-identical exponentiated Pareto distributed random variables with progressively censored scheme. Different interval estimations are proposed. The interval estimations obtained are exact, approximate and bootstrap confidence intervals. Different methods and the corresponding confidence intervals are compared using Monte-Carlo simulations. Simulation results show that the confidence intervals (CIs) of exact and approximate methods are really better than those of the bootstrap method.

Key words: stress-strength model; exponentiated Pareto distribution; interval estimation; progressively censored scheme

Introduction

Several researchers have considered different choices for stress and strength distributions. In the early stage, the maximum likelihood estimator (MLE) of a stress-strength parameterR=P(X

This exponentiated Pareto distribution has been extensively used in the analysis of extreme events[10], especially in hydrology[11], as well as in reliability studies when robustness is required against heavier-tailed or lighter-tailed alternatives to an exponential distribution. The exponentiated Pareto distribution can have decreasing and upside-down bathtub shaped failure rates depending on its shape parameters. Modeling survival data by non-monotonic failure rates are desirable, for example, when the course of the disease is such that mortality reaching a peak after some finite periods[12]. Daragahi-Noubary[13]advocated the use of the exponentiated Pareto distribution function for fitting a distribution of annual maximum of the wind speed and that of maximum floods of the Feather river. Some recent applications of exponentiated Pareto distribution function include the estimation of the finite limit of human lifespan. The estimation of parameters has also been attempted by Afify[14]under type-I and type-II censoring schemes. Singhetal.[15]developed estimators for the parameters of the exponentiated Pareto distribution under the progressive type-II censoring with binomial removals.

There are several types of censoring schemes and in particular, the doubly type-II censoring scheme is the most common one. Doubly censoring samples have been considered by a lot of authors[16-18]. The progressively type-II censoring will allow such intermediate removals of units. This type of censoring includes conventional censoring schemes as a special case. Some early works on the progressive censoring can be found in Refs. [19-20]. Since then, numerous articles have been published about the MLE of parameters for a wide range of lifespan models[21-23].

It is clear that there is no exact expression forRwhen the parameters are all different with different censoring schemes. The properties ofRcannot be easily found. In this paper, we consider the problem of estimating the stress-strength parameterR=P(X

The rest of the paper is organized as follows. In section 1, the probability density function (PDF) and the cumulative distribution function (CDF) of the exponentiated Pareto distribution are presented and the explicit expression ofRis derived. In section 2, different interval estimators ofRare presented, including exact, approximate and bootstrap confidence intervals. Some numerical experiments and discussion are presented in section 3. In section 4, an example is provided to illustrate the methods proposed in this paper.

1 Exponentiated Pareto Distribution and Reliability

A random variableXis said to have the exponentiated Pareto distribution, if its PDF is given by

f(x;α,λ)=αλ[1-(1+x)-λ]α+1(1+x)-(λ+1),
x,α,λ>0，

(1)

whereαandλare shape parameters. The CDF is given by

F(x;α,λ)=[1-(1+x)-λ]α.

(2)

An exponentiated Pareto distribution will be denoted byEP(α,λ).

LetYbe the strength of a system andXbe the stress acting on it. Assume thatX～EP(β,λ),Y～EP(α,λ) , andXandYare independent. Therefore, the reliability of the system will be

R=P(X

(3)

2 Interval Estimation of R

2.1 Exact confidence interval based on pivotal quantity

Now, we briefly express the problem that we intend to study. With the pre-determined number of removals(s1,s2, ...,sm1), letX1:m1:n1

Based on results in Refs. [20, 24], it is clear that (U,V) is a complete sufficient statistic for (β,α). It is immediate thatUhas aγdistribution with the shape parameterm1and the scale parameterβ, namely,U～γ(m1,β). By the same way, we haveV～γ(m2,α).

.

The 100(1-γ)% confidence interval with equal-tails ofR, namely, [L1,U1], can be obtained as

whereF(2m1, 2m2)(γ/2)andF(2m1, 2m2)(1-γ/2)are the lower and upperγpercentile points of anFdistribution, respectively. The expected widthEW) of the CI is given by

2.2 Approximate confidence interval

Theorem1 Under extensive regular conditions, asm→∞,

ProofThe proof follows from the asymptotic normality of the MLE. From the well-known asymptotic normality of the MLE, we have

LetI(β) be the observed Fisher information quantity, and then

where the log likelihood function for the observed sample is

lnL(β)=C+m1lnβ+m1lnλ+

(7)

Through some simple computations, the expected Fisher information quantity is

(8)

where,

Ai(β)=

ln2[1-(1+y)-λ]·

{1-[1-(1+y)-λ]-β}n1-i-2(1+y)-(λ+1)dy.

Letω=(1+y)-λ, and then

Applying the same way, the expected Fisher information quantityI(α) ofαis

(9)

where,

Then, we have

Based on these asymptotic results, we can obtain the asymptotic confidence interval ofR, namely [L2,U2]， is

2.3 Bootstrap methods

The bias corrected and accelerated BCA method is described for constructing approximated confidence intervals based on the bootstrap re-sampling method. Further details on bootstrap confidence intervals can be seen in Ref. [29].

(10)

(11)

where [x] denotes the largest integer, which is smaller than or equal tox;

3 Simulations and Discussion

In this section, the Monte-Carlo simulation is conducted to compare the performances of point estimations and interval estimations of different methods under different censoring schemes. We will see the estimation accuracy of the methods in different situations. For different choices of sample sizes and censoring schemes, we generated progressively type-II censoring samples from the exponentiated Pareto distribution using the algorithm presented in Ref. [30]. All the computations are implemented by using MATLAB. Without loss of generality, we setλ=1 and (α,β)=(5, 5), (5, 10), (5, 15). For givennandm, the same three different censoring schemesA,BandCas Basiratetal.[31]are used to generate the progressively censored samples.

Table 1 Censoring schemes

Suppose (n1,m1) and the progressive censoring scheme (S1,S2, ....,Sm1) for the first population, and (n2,m2) and the progressive censoring scheme (S1,S2, ....,Sm2) for the second population. For easy calculation, we suppose thatn1=2m1andn2=2m2. For different choices of sample sizes and censoring schemes, we report the 95% CI and coverage probabilities(C.P.) for all sample sizes and parameter values are also provided in Table 2. For each combination of the sample size and the censoring scheme, we repeat the calculation 1 000 times and 5 000 bootstrap replications.

Table 2 CI and C. P. for different methods

(Table 2 continued)

(Table 2 continued)

By comparing the 95% CI and C.P. in Table 2, we see that the CIs of exact and approximate methods are really better than Bootstrap method. In viewing the table, it is clear that all the estimators work quite well. In all the cases, it is observed that as the effective sample size increases, the performances become better. The lengths of the exact confidence intervals (ECIs) and the corresponding approximate confidence intervals (ACIs) are very close in all cases. The performance of the CIs of BCA bootstrap method is very poor in terms of C.P. in almost cases. As the censoring number increasing, the performance is getting worse. From the computational point of view, the ECIs are the easiest to obtain. Therefore, it is suggested to use the ECIs for all practical purposes.

4 Data Analyses

In this section, we present the analysis of real data, partially considered in Ref. [32] for illustrative purposes. The data represent the waiting times (in minutes) before customer services in two different banks as shown in Tables 4-5.

Al-Mutairietal.[33]analyzed these data sets using the Lindley distribution. Basiratetal.[31]analyzed these data sets using the Weibull distribution. We are interested in estimating the stress-strength parameterR(X

Based on the complete data set, the 95% ACI is (0.640 3, 0.773 0) and the 95% ECI is (0.637 4, 0.769 7).

For illustrative purposes, we apply the three different censoring schemes as Basiratetal.[31]to generate samples from the above data sets withm1=40 andm2=80 as the censoring scheme (S,S′)= (A,A) and (S,S′) = (B,B). We also setm1=30 andm2=50 as the censoring scheme (S,S′)=(C,C).

Based on the censoring scheme (A,A), the 95% ACI and ECI are (0.639 0, 0.773 0) and (0.770 0, 0.889 8), respectively.

Based on the censoring scheme (B,B), the 95% ACI and ECI are (0.670 4, 0.809 9) and (0.668 6, 0.812 4), respectively.

Based on the censoring scheme (C,C), the 95% ACI and ECI are (0.667 5,0.800 1) and (0.648 3, 0.821 1), respectively.

Table 4 Data set Ⅰ-waiting time before customer service in bankⅠ

Table 5 Data set Ⅱ-waiting time before customer service in bank Ⅱ

5 Conclusions

The reliability of a system is discussed with progressively censoring scheme. We apply different interval estimations. Some Monte-Carlo simulations are provided to show the different methods and the corresponding confidence intervals. From the simulation results, it is clear that the CIs of exact and approximate methods are really better than those of bootstrap method.