﻿ Selecting a Component with Longer Mean Life Time in Bivariate Pareto Models

Open Journal of Statistics
Vol.05 No.05(2015), Article ID:58220,4 pages
10.4236/ojs.2015.55037

Selecting a Component with Longer Mean Life Time in Bivariate Pareto Models

Parameshwar V. Pandit, Shubhashree Joshi

Department of Statistics, Bangalore University, Bangalore, India

Received 10 June 2015; accepted 20 July 2015; published 23 July 2015

ABSTRACT

In any parallel system, selecting a component with longer mean lifetime is of interest to the researchers. Hanagal (1997) [1] discussed selection procedures for a two-component system with bivariate exponential (BVE) models. In this paper, the problem of selecting a better component with reference to its mean life time under bivariate Pareto (BVP) models is considered. Three selection procedures based on sample proportions, sample means and maximum likelihood estimators (MLE) are proposed. The probability of correct selection for the proposed procedures is evaluated through Monte Carlo simulation using normal approximation. The asymptotic relative efficiency (ARE) of the proposed procedures is presented to facilitate the evaluation of the performance of procedures.

Keywords:

Asymptotic Relative Efficiency (ARE), Better Component, Bivariate Pareto, Probability of Correct Selection

1. Introduction

The problem of determining the component with longer life time in a two-component parallel system when the two components are dependent is of interest in the present context. The component which has longer mean life time is considered to be a better component. Hanagal [1] considered selecting the better component in a parallel system with two dependent components when the joint distribution life time of the components is bivariate exponential (BVE) distribution. Hanagal [1] considered BVE distribution proposed by Freund [2] , Marshall-Olkin [3] and Block-Basu [4] . Hyakuntake [5] considered the above problem when (X1, X2) follows BVE distribution of Marshall-Olkin. However selection of the better component when (X1, X2) follows other than BVE has not been considered in the literature.

The main aim of this paper is to select a best component with reference its life length in a two component parallel system developing a proper statistical tool. Here, the components of the system are assumed to be dependent and their lifetimes follow bivariate Pareto distribution.

The problem of selecting the component in a two dependent component parallel system when life times (X1,X2) of two components follow bivariate Pareto (BVP) distribution is considered in this paper. Three selection procedures are proposed and their probabilities of correct selection are evaluated.

2. Selection Procedures

Veenus and Nair [6] proposed BVP model with survival function

(1)

where In this paper, assume β = 1, we get the survival function of (X1, X2) given by

(2)

The pdf of (X1, X2) is given by

where

The above BVP model is not absolutely continuous with respect to Lebesgue measure on and has a posi-

tive probability on the diagonal i.e.,. The random variables X1 and X2 are independent iff

and X1 and X2 have identical marginal’s iff.

Let be a random sample of size n from BVP and let n1(n2) be the number of obser-

vations with X1 < X2 (X1 > X2) in the sample of size n. The distribution of (n1, n2) is trinomial

where.

We propose three selection procedures:

The first selection procedure R1 is based on counts

R1: Select C1 as better component if n2 > n1 and select C2 when n2 < n1.

The second selection procedure is based on the sample means of two lifetimes of the components

R2: Select C1 as better component if and select C2 as better component when

where and are the sample means of the lifetimes of the two components C1 and C2 respectively.

The third selection procedure R3 is based on MLE’s

R3: Select C1 as better component if and select C2 when, where and are the MLE’s of θ1 and θ2 respectively. There are no closed form expressions for MLE’s and so Hanagal [7] obtained MLE’s by either Newton-Raphson procedure or Fisher’s method of scoring. The derivation of MLEs for the parameters is given in Appendix.

By the assumption θ1 < θ2 (selecting the component C1) the probability of correct selection based on three procedures are

The exact distribution of, and are difficult to obtain but their asymptotic normal distributions can be obtained. By central limit theorem, and have asymptotic standard normal with distribution

where, ,

where, ,

and

are (i,j)th elements of the inverse of Fisher information matrix

Hence

, i = 1, 2, 3 where is the cumulative distribution func-

tion of standard normal distribution,. If ci > cj, then the selection procedure Ri is better than Rj,

.

3. Determination of Minimum Sample Size and Asymptotic Relative Efficiency (ARE)

The probability requirement based on the selection procedure Ri, i = 1, 2, 3 is where

is fixed constant.

That is, or or; where Zp is the solution of.

The minimum sample size required for the selection procedure Ri is.

The ARE of the selection procedure Ri with respect to the selection procedure Rj is given by

.

The AREs are presented in Table 1 for some combinations of (θ1, θ2, θ3)

Table 1 gives the efficiency of three procedures R1, R2 and R3. The efficiency comparison would be useful in choosing an appropriate procedure.

4. Some Remarks and Conclusions

1) It is observed from the table that the selection procedure R2 based on sample means performs better than the other two selection procedures R1 and R3.

Table 1. The AREs for some combinations of (θ1, θ2, θ3).

2) The selection procedures R1 and R3 are equally efficient.

3) The probability of correct selection under selection procedures is computed when the sample size is large and the result is similar to that obtained through AREs.

4) The problem of selecting the best component in multi components parallel system is under progress for multivariate exponential (MVE) and multivariate Pareto (MVP) distributions.

Cite this paper

Parameshwar V.Pandit,ShubhashreeJoshi, (2015) Selecting a Component with Longer Mean Life Time in Bivariate Pareto Models. Open Journal of Statistics,05,355-359. doi: 10.4236/ojs.2015.55037

References

1. 1. Hanagal, D.D. (1997) Selection of a Better Component in Bivariate Exponential Models. Journal of the Italian Statistical Association, 5, 453-460.

2. 2. Freund, J.E. (1961) A Bivariate Extension of the Exponential Distribution. Journal of the American Statistical Association, 56, 971-977.
http://dx.doi.org/10.1080/01621459.1961.10482138

3. 3. Marshall, A.W. and Olkin, I. (1967) A Multivariate Exponential Distribution. Journal of the American Statistical Association, 62, 30-44.
http://dx.doi.org/10.1080/01621459.1967.10482885

4. 4. Block, H.W. and Basu, A.P. (1974) A Continuous Bivariate Exponential Extension. Journal of the American Statistical Association, 69, 1031-1037.

5. 5. Hyakuntake, H. (1992) Selecting the Better Component of a Bivariate Exponential Distribution. Statistics and Decisions, 10, 153-162.

6. 6. Veenus, P and Nair, K.R.M. (1994) Characterization of a Bivariate Pareto Distribution. Journal of the Indian Statistical Association, 32, 15-20.

7. 7. Hanagal, D.D. (1997) Note on Estimation of Reliability under Bivariate Pareto Stress-Strength Model. Statistical Papers, 38, 453-459.
http://dx.doi.org/10.1007/BF02926000

Appendix

Maximum Likelihood Estimators of the parameters (θ1, θ2, θ3) of BVP distribution

The likelihood of the sample of size n is

where n1 be the number of observations with X1i < X2i in the sample of size n and.

The log likelihood of (X1i, X2i) is

The likelihood equations are

Maximum Likelihood Estimators are obtained solving above likelihood equations simultaneously. One can generate some consistent estimators say for the parameters and use

as initial solution in Newton-Raphson procedure or Fisher’s method of scoring to obtain MLE’s

.

So we choose some consistent estimators as follows

where.

Hence it is easy to check that The elements of Fisher information matrix are given by

Thus has an asymptotic multivariate normal with mean vector zero and variance-covariance ma-

trix where and.