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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.

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},_{ }X_{2}) follows BVE distribution of Marshall-Olkin. However selection of the better component when (X_{1},_{ }X_{2}) 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 (X_{1},X_{2}) 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 [

where _{1},_{ }X_{2}) given by

The pdf of (X_{1},_{ }X_{2}) is given by

where _{ }

The above BVP model is not absolutely continuous with respect to Lebesgue measure on

tive probability on the diagonal i.e.,_{1} and X_{2} are independent iff

_{1} and X_{2} have identical marginal’s iff

Let _{1}(n_{2}) be the number of obser-

vations with X_{1} < X_{2} (X_{1} > X_{2}) in the sample of size n. The distribution of (n_{1}, n_{2}) is trinomial

where

We propose three selection procedures:

The first selection procedure R_{1} is based on counts

R_{1}: Select C_{1} as better component if n_{2} > n_{1 }and select C_{2} when n_{2} < n_{1}.

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

R_{2}: Select C_{1} as better component if _{2} as better component when

where _{ }C_{1 }and C_{2} respectively.

The third selection procedure R_{3} is based on MLE’s

R_{3}: Select C_{1} as better component if _{2} when_{1} and θ_{2} respectively. There are no closed form expressions for MLE’s and so Hanagal [

By the assumption θ_{1} < θ_{2} (selecting the component C_{1}) the probability of correct selection based on three procedures are

The exact distribution of

where

where

^{th} elements of the inverse of Fisher information matrix

Hence

tion of standard normal distribution,_{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

That is, _{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 _{1}, θ_{2}, θ_{3})

_{1}, R_{2} and R_{3}. The efficiency comparison would be useful in choosing an appropriate procedure.

1) It is observed from the table that the selection procedure R_{2} based on sample means performs better than the other two selection procedures R_{1} and R_{3}.

Parameters | ARE (R_{3}, R_{1}) | ARE (R_{2}, R_{3}) | ARE (R_{2}, R_{1}) |
---|---|---|---|

θ_{3} = 1.01 | |||

θ_{1} = 1.02, θ_{2} = 1.05 θ_{1} = 1.03, θ_{2} = 1.02 θ_{1} = 1.04, θ_{2} = 1.00 | 1.0950 1.0952 1.0954 | 4.6641 5.0025 7.2516 | 5.1072 5.4794 7.9455 |

θ_{3} = 1.02 | |||

θ_{1} = 1.02, θ_{2} = 1.05 θ_{1} = 1.03, θ_{2} = 1.02 θ_{1} = 1.04, θ_{2} = 1.00 | 1.0953 1.0954 1.0956 | 4.1597 4.4267 5.4259 | 4.5561 4.8471 5.9464 |

θ_{3} = 1.03 | |||

θ_{1} = 1.02, θ_{2} = 1.05 θ_{1} = 1.03, θ_{2} = 1.02 θ_{1} = 1.04, θ_{2} = 1.00 | 1.0954 1.0955 1.0957 | 3.8080 4.0225 4.6125 | 4.1711 4.4064 5.0521 |

2) The selection procedures R_{1} and R_{3} 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 (θ_{1}, θ_{2}, θ_{3}) of BVP distribution

The likelihood of the sample of size n is

where n_{1} 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})

The likelihood equations are

Maximum Likelihood Estimators are obtained solving above likelihood equations simultaneously. One can generate some consistent estimators say

So we choose some consistent estimators as follows

where

Hence it is easy to check that

Thus

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