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.

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

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

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 c_i > c_j, then the selection procedure R_i is better than R_j,

.

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

is fixed constant.

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

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

The ARE of the selection procedure R_i with respect to the selection procedure R_j 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.

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.

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.

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

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 X_1i < X_2i in the sample of size n and.

The log likelihood of (X_1i, X_2i) 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.