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 [
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 [
where
The pdf of (X1, X2) 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.,
Let
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
where
The third selection procedure R3 is based on MLE’s
R3: Select C1 as better component if
By the assumption θ1 < θ2 (selecting the component C1) the probability of correct selection based on three procedures are
The exact distribution of
where
where
Hence
tion of standard normal distribution,
The probability requirement based on the selection procedure Ri, i = 1, 2, 3 is
That is,
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
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.
Parameters | ARE (R3, R1) | ARE (R2, R3) | ARE (R2, R1) |
---|---|---|---|
θ3 = 1.01 | |||
θ1 = 1.02, θ2 = 1.05 θ1 = 1.03, θ2 = 1.02 θ1 = 1.04, θ2 = 1.00 | 1.0950 1.0952 1.0954 | 4.6641 5.0025 7.2516 | 5.1072 5.4794 7.9455 |
θ3 = 1.02 | |||
θ1 = 1.02, θ2 = 1.05 θ1 = 1.03, θ2 = 1.02 θ1 = 1.04, θ2 = 1.00 | 1.0953 1.0954 1.0956 | 4.1597 4.4267 5.4259 | 4.5561 4.8471 5.9464 |
θ3 = 1.03 | |||
θ1 = 1.02, θ2 = 1.05 θ1 = 1.03, θ2 = 1.02 θ1 = 1.04, θ2 = 1.00 | 1.0954 1.0955 1.0957 | 3.8080 4.0225 4.6125 | 4.1711 4.4064 5.0521 |
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 X1i < X2i in the sample of size n and
The log likelihood of (X1i, X2i)
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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