Estimation of P[Y<X] for generalized Pareto distribution

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Abstract

This paper deals with the estimation of P[Y<X] when X and Y are two independent generalized Pareto distributions with different parameters. The maximum likelihood estimator and its asymptotic distribution are obtained. An asymptotic confidence interval of P[Y<X] is constructed using the asymptotic distribution. Assuming that the common scale parameter is known, MLE, UMVUE, Bayes estimation of R and confidence interval are obtained. The ML estimator of R, asymptotic distribution and Bayes estimation of R in general case is also studied. Monte Carlo simulations are performed to compare the different proposed methods.

Introduction

In reliability contexts, inferences about R=P[Y<X], where X and Y independent distributions, are a subject of interest. For example, in mechanical reliability of a system, if X is the strength of a component which is subject to stress Y, then R is a measure of system performance. The system fails, if at any time the applied stress is greater than its strength.

The problem of estimating of R have been widely used in the statistical literature. The maximum likelihood estimator (MLE) of P[Y<X], when X and Y are normally distributed, has been considered by Downtown (1973), Govidarajulu (1967), Woodward and Kelley (1977) and Owen et al. (1977). Tong, 1974, Tong, 1977 considered the estimation of P[Y<X] when X and Y are independent exponential random variables. Awad et al. (1981) considered the MLE of R, when X and Y have bivariate exponential distribution. Ahmad et al. (1997) and Surles and Padgett, 1998, Surles and Padgett, 2001 considered the estimation of P[Y<X], where X and Y are Burr type X random variable. The gamma case is studied in Constantine and Karson (1986). The theoretical and practical results on the theory and applications of the stress–strength relationships in industrial and economic systems during the last decades are collected and digested in Kotz et al. (2003). The class of life-time distributions (in particular, exponential and gamma) is considered by Nadarajah (2003). Estimation of P[Y<X] from logistic (Nadarajah, 2004a), Laplace (Nadarajah, 2004b), exponential case with common location parameter (Baklizi and El-Masri, 2004), Burr type III (Mokhlis, 2005), beta (Nadarajah, 2005a), gamma (Nadarajah, 2005b), bivariate exponential (Nadarajah and Kotz, 2006) and Weibull (Kundu and Gupta, 2006) distributions are also studied. Kundu and Gupta (2005) considered the estimation of P[Y<X], when X and Y are independent generalized exponential distribution. Recently, inferences on reliability in two-parameter exponential stress–strength model (Krishnamoorthy et al., 2007) and ML estimation of systemreliability for Gompertz distribution (Saraçoglu and Kaya, 2007) are considered. Kakade et al. (2008) studied exponentiated Gumbel case and Abd Elfattah and Marwa (to appear) studied exponential case based on censored samples.

In this paper, we focus on estimation of R=P[Y<X], where X and Y follow the generalized Pareto (GP) distribution with different parameters. We obtain the MLE of R and its asymptotic distribution. The asymptotic distribution is used to construct an asymptotic confidence interval. Two bootstrap confidence intervals of R are also proposed. Assuming that the common scale parameter is known, the maximum likelihood estimator of P[Y<X], confidence intervals, UMVUE and Bayes estimation of R are obtained.

This paper is organized as follows: In the next Section, the GP distribution is introduced. Estimation of R with common scale parameter is given in Section 3. In this section, the ML estimator of R, asymptotic distribution and bootstrap confidence intervals are presented. Estimation of R if the common scale parameters are known is discussed in Section 4. In this section MLE, UMVUE and Bayes estimation of R are discussed. In Section 5, the estimation of R in general case is studied. The ML estimator of R, asymptotic distribution and Bayes estimation of R are presented in Section 5. The different proposed methods have been compared using Monte Carlo simulations and their results have been reported in Section 6. In Section 7, a numerical example is illustrated and the results of different methods are compared.

Section snippets

Generalized Pareto distribution

A random variable X is said to have generalized Pareto distribution, if its probability density function (pdf) is given by f(ξ,μ,σ)(x)=1σ1+ξx-μσ-(1/ξ+1),where μ,ξR and σ(0,+). For convenience, we reparametrized this distribution by defining ξ/σλ, 1/ξα and μ0. Therefore, f(x)=αλ(1+λx)-(α+1);x>0.The cumulative distribution function is defined by F(x)=1-(1+λx)-αfor λ>0 and α>0. Here α and λ are the shape and scale parameters, respectively. It is also well known that this distribution has

Estimation of R with common scale parameter

In this section, we investigate the properties of R, when the common scale parameter λ, is the same, and the general case is studied in Section 5.

Estimation of R if λ is known

In this section, we consider the estimation of R when λ is known. Without loss of generality, we can assume that λ=1.

Estimation of R in general case

Computing the R when the scale parameter is different is considered, in this section. Surles and Padgett, 1998, Surles and Padgett, 2001 considered this case, also. In Surles and Padgett (2001) there is no exact expression for R, but they presented a bound for it.

Simulation results

In this section, we present some results based on Monte Carlo simulations to compare the performance of the different methods mainly for small sample sizes.

We consider tree cases separately to draw inference on R, namely when (i) the common scale parameter λ is unknown, (ii) the common scale parameter λ is known and (iii) the scale parameter λ1 and λ2 are unknown. In the first two cases, we consider the following small sample sizes: m,n=15,25 and 50. We take different values for α,β and λ, also.

Numerical example

Since the confidence intervals based on the asymptotic results for small sample sizes do not perform very well results, so we present an analysis based on two bootstrap methods. To do this, we simulate 20 numbers from GE(1,1) and 20 numbers from GE(1.4,1), reported in Table 4. A complete analysis of these data are presented in this section. In the first row of Table 5, it is assumed that λ is unknown. The MLE of (α,β,λ) is obtained using (6), (7), (10). The iterative procedure is stops whenever

Conclusion

In this paper, we have addressed the problem of estimating P[Y<X] for the generalized Pareto distributions. We consider the cases when the common scale parameter is known or unknown and when the scale parameters are different. When the common scale parameter is unknown, it is observed that the maximum likelihood estimator works quite well. We can use the asymptotic distribution of the maximum likelihood estimator to construct confidence intervals which work well even for small sample sizes.

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