Abstract: This study deals with the estimation of P = (Y<X) when X and Y are two independent weighted exponential distributions with different parameters. The MLE of the R based on one simple iteration procedure is obtained. Furthermore, a simulation study was conducted to compute Bayesian estimates of the model. Finally, we carried out its Bayesian estimations of parameters with a real data set.
INTRODUCTION
In reliability contexts, inferences about R = (Y<X), where X and Y independent distribution, are a subject of interest. Note that the estimation of R is very common in the statistical literature. 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 Maximum Likelihood Estimator (MLE) of R when X and Y have bivariate exponential distribution has been considered by Awad et al. (1981). Church and Harris (1970), Downtown (1973), Woodward and Kelley (1977) and Owen et al. (1964) considered the estimation of R when X and Y are normally distributed. Similar problem for multivariate normal distribution has been considered by Gupta and Gupta (1990). Kelley et al. (1976) and Sathe and Shah (1981) considered the problem of estimating R when X and Y are independent exponential random variable. Constantine et al. (1986) considered the estimation of R when X and Y are independent gamma random variable. Surles and Padgett (1998, 2001) considered the estimation of R when X and Y are Burr type X random variables. Kundu and Gupta (2005) considered this problem when X and Y are generalized exponential distribution. So Rezaei et al. (2010) obtained estimation of P[Y<X] when X and Y are two independent generalized Pareto distributions with different parameters. Gupta and Kundu (2009) introduced a new class of weighted exponential distribution by using of the idea of Azzalini (1985). In this study, we focus on estimation of R = P[Y<X], where X and Y follow the Weighted Exponential (WE) distribution with different parameters.
WEIGHTED EXPONENTIAL DISTRIBUTION
A random variable X is said to have weighted exponential distribution, if its Probability Density Function (PDF) is given by:
| (1) |
And it’s 0 otherwise. We will denote it as WE(α,λ). Note that in the model (1) the location parameter can be easily incorporated. The cumulative distribution function is defined by:
| (2) |
For λ>0 and α>0. Here, α and λ are the shape and scale parameters, respectively (Gupta and Kundu, 2009).
ESTIMATION OF MAXIMUM LIKELIHOOD OF R WITH COMMON SCALE PARAMETER
Here, we investigate the properties of R, when the common scale parameter λ, is the same. To investigate the properties of R, denote by WE(α,λ) the distribution of reparametrized WE. Let X~WE(α,λ) and Y~WE(β,λ), where X and Y are independent random variables. Therefore:
| (3) |
The result follows after simplification.
To compute the MLE of R, suppose X1,X2,…XN is a random sample from WE(α,λ) and WE(β,λ) is a random sample from WE(β,λ). Therefore, the log-likelihood function of the observed sample is:
| (4) |
The MLEs of α, β and λ say
| (5) |
| (6) |
| (7) |
There is no closed-form solution to this system of equations, so we will solve
for
where, g is the vector of normal equations for which we want:
with:
and G is the matrix of second derivatives:
Where:
The Newton-Raphson algorithm converges, as our estimates of α,β and
λ change by less than a tolerated amount with each successive iteration,
to
| (8) |
BAYESIAN ESTIMATIONS
Here, we obtain the estimation of parameters weighted exponential distribution with using of posterior mode method.
Bayes notation and prior elicitation: Let
| (9) |
Now it is assumed that x1,…,xn is a random sample from f(.|λ,α) as given in Eq. 1. We assume that λ has a prior π1(.) with Gamma (a,b) distribution. Also the prior on α is π2(.) with Gamma (c,d) density and it is independent of π1(.):
| (10) |
| (11) |
The likelihood function of the observed data is:
| (12) |
Therefore, the joint density function of the observed data, α and λ is:
| (13) |
The posterior density function of {α,λ}, given the data is:
| (14) |
The following are different algorithms to generate WE(α,λ).
| • | Interpretation 1: | |
| • | Step 1: |
|
| • | Step 2: |
|
| • | Interpretation 2: | |
| • | Step 1: U~exp(λ) and V~exp(λ(1+α)) (U and V are independent) | |
| • | Step 2: X = U+V~WE(α,λ) | |
Throughout this article, we assume that all parameters follow independent gamma prior distribution.
BAYESIAN ESTIMATION OF PARAMETERS BY POSTERIOR MODE METHOD
Let Xj (i = 1,2,..., n) be a random sample from WE distribution. Using the corresponding pdf (1) and expression (9), the log posterior distribution for a sample of size n with priors Gamma (a,b) and Gamma(c,d) is given by:
| (15) |
where, K does not depend on parameters α and λ the hyper parameters a, b, c and d are fixed. Thus to obtain Bayesian estimation of parameters via posterior mode method, we require to maximize (13). First, we obtain the likelihood equation:
| (16) |
| (17) |
For implementation of the work, we need to use either the scoring algorithm
or the Newton-Raphson algorithm to solve the two non linear Eq.
14 and 15 simultaneously we prefer to use Newton-Raphson
procedure to obtain the approximate roots of the equations. So we will solve
for
In our case we will estimate β = (α,λ) iteratively:
where, g is the vector of normal equations for which we want:
With:
Therefore, the second derivatives are indispensable. Following, are the elements of symmetric observed information matrix or symmetric Hessian matrix:
and G is the matrix of second derivatives:
The Newton-Raphson algorithm converges as our estimates of α and λ
change by less than a tolerated amount with each successive iteration, to
Estimation of hyper parameters of prior distribution via moment of we model: Let x1, x2, ...., xn is a random sample from f(.|λ,α) and denote the prior distribution as Eq. 10 and 11. We obtain the estimate of a,b,c and d from the past data. To estimate of the hyper parameters, first we obtain the moment estimators of parameters of WE distribution, then their corresponding mean and variance of each estimator are obtained by using simulation consequently the estimated value of a,b,c and d are obtained by solving two simple equation:
We can estimate the hyper parameters of prior distribution by moment estimators which is well-known in the literature as empirical Bayes Procedure. For implementation, in a simulation study we can use one set of data to estimate hyper parameters of prior distribution and use another set of data to obtain posterior mode. Empirical Bayes approach is used when we do not have prior knowledge about hyper parameters of prior distribution. In later section we carry out the procedure of computing the moment for WE distribution.
Algorithm of computing for hyper parameters by moment method: If X follows, WE(α,λ) then the MGF of X for -1<t<1 can be obtained as:
Therefore, differentiating MX(t)and having t=0, we obtain:
and:
So:
For λ = 1 it is given by Gupta and Kundu (2009).
Now, we can to obtain the values of α and λ for replication samples and so estimate the hyper parameters of the prior distribution by Mont Carlo simulation by following equations:
Also to computation of estimation of the hyper parameters of a and b we apply this method. In this paper we prefer to consider the hyper parameters as fix but instead we estimate Bayesian parameters of the model in the cases Non Informative priors, informative priors and most informative priors by method namely Posterior mode.
NUMERICAL SIMULATION STUDY FOR THE POSTERIOR MODE METHOD
For WE model, 10000 samples, each of size 10, 30, 50 were generated as past data. In this simulation that has carried out with a SAS code, we follow up priors distribution as non informative , i.e., "a = b = c = d = 0", less informative prior that in the case we take a = 8, b = 16, c = 2, d = 1; for the informative prior we take a = 80, b = 160, c = 20, d = 10 and finally for the most informative prior we get a = 400, b = 800, c = 100, d = 50. The results derived due to the simulation have sited in Table 1 and 2.
| Table 1: | Estimations for α = 0.5 and λ = 2 |
| Table 2: | Estimations for α = 0.5 and λ = 1 |
| Table 3: | Estimates for α and λ |
| Table 4: | Survival time of Guinea pigs |
CASE STUDY
The data set consists of survival times of guinea pigs injected with different amount of tubercle bacilli and was studied by Bjerkedal (1960). Guinea pigs are known to have high susceptibility of human tuberculosis which is one reason for choosing this species. We consider only the study in which animals in a single cage are under the same regimen. Table 4 represents the survival times of Guinea pigs. The data are given below:
In this case n = 71, the sample mean
The maximum likelihood estimate of the parameters are
CONCLUSION
In this article, we have addressed the problem of estimating P(Y<X) for the weighted exponential distribution. Also a Bayesian approach on estimation of the parameters of this model was performed via a simulation study. All results are sited in Table 1 and 2 and they show in case of "Most Informative priors", the estimations are better than the rest.