1. Introduction

The Power function distribution is often used to study the electrical component reliability [3]. A continuous random variable X is said to have Power function distribution if its probability density function is given by [2]

(1)

Where θ is the shape parameter, μ is the location parameter and σ is the scale parameter. We are interested to find Bayes estimator of shape parameter θ under different loss functions. Let μ = 0 and σ = 1 then the form of the density function becomes

(2)

1.1. Prior and Posterior Density Function of

For Bayesian estimation we need to specify a prior distribution for the parameter. Consider a Gamma prior for θ having pdf[6]

(3)

Now the Posterior density function of θ for the given random sample X is given by[4]

(4)

which implies that, since prior and posterior distribution belongs to the same family hence the prior is conjugate prior.

2. Different Estimators of Parameter θ

In this section Bayes estimators of parameter θ for different loss functions along with maximum likelihood estimator has been determined.

2.1. MLE of θ

Let X = be a random sample of size n drawn from the Power function distribution defined in (2). Then the likelihood function of θ for the given random sample X is given by[4]

(5)

Taking log on both sides of (5) we get

The MLE of θ will be the solution of the equation[4]

2.2. Bayes Estimator of θ for Squared Error Loss Function

Here we have determined Bayes estimator of θ for squared error loss function defined as[6]

(6)

For squared error loss function Bayes estimator is the mean of posterior density function. From (4) posterior density function is a Gamma distribution with parameter

. Hence the mean of posterior density function is.Therefore, is the Bayes estimator of θ under SE loss function.

2.3. Bayes Estimator of θ for Quadratic Loss Function

Now suppose the loss function is quadratic, which is defined as[5]

(7)

Under quadratic loss function Bayes estimator of θ is obtained by solving the following equation

Therefore , is the Bayes estimator of θ under quadratic loss function

2.4. Bayes Estimator of θ for MLINEX Loss Function

Now let us consider the MLINEX loss function defined as [5]

(8)

For MLINEX loss function Bayes estimator of θ is obtained from [5]

(9)

Hence from (9) we get

, is the Bayes estimator of θ under MLINEX loss function.

2.5. Bayes Estimator of θ for NLINEX Loss Function

Let us consider the following NLINEX loss function of the form [1]

(10)

where D represents the estimation error .i.e. ; For NLINEX loss function Bayes estimator of θ is [1]

(11)

where stands for posterior expectation. Now,

Therefore

(12)

Again

Using (12) and (13) in (11) we get

is the Bayes estimator of θ under NLINEX loss function.

3. Empirical Study

To compare the estimators and we have considered MSE of the estimators. The MSE of an estimator 𝜃 is defined as

To obtain the variance of θ, we have used the true value of the parameter 𝜃 under consideration. We have obtained the estimated value, MSE and Bias of the estimator by using the Monte Carlo simulation method from the Power function distribution. Five thousand samples have taken for each case. The results and their graphs are presented bellow.

From table 1 we observed that MSE of is very high for small sample but declining sharply and becoming closer to other estimators with increasing sample size. Among Bayes estimators gives smaller value of MSE when sample size is small but for large sample they are almost identical (figure1).

Table 1. Estimated value and MSE of different estimators of the parameter 𝜃 of Power function distribution when α =0.5, β=1, θ= 1 and c=1

Figure 1. Graph of MSEs of different estimates of parameter θ of Power function distribution when α=0.5, β=1, θ =1 and c=1

Table 2. Estimated value and MSE of different estimators of the parameter 𝜃 of Power function distribution when α =1, β=2, θ= 1 and c=1

Table 2 represents largest value of MSE for in all cases. It is also clear from table 2 that MSE of is smallest than others estimators for different sample size (figure 2).

Figure 2. Graph of MSEs of different estimates of parameter θ Power function distribution when α=1, β=2, θ =1 and c=1

Table 3. Estimated value and MSE of different estimators of the parameter 𝜃 of Power function distribution when α =2, β =2, θ =1.5 and c = 2

Table.3. shows the variation in the performance of the estimator for different sample size. More or less similar pattern are observed here as previous tables that is MSE of is higher than all other estimators. MSE of is least in the class of Bayes estimators. Also MSE of are very close to each other for large sample (figure 3)

Figure 3. Graph of MSEs of different estimates of parameter θ of Power function distribution when α=2, β=2, θ =1.5 and c=2

Table 4. Estimated value and MSE of different estimators of the parameter 𝜃 of Power function distribution when α =1.5, β=2, θ= 1.5 and c=2

For different sample size table 4 also shows minimum values of MSE for and some cases it is very near to that of. On the other hand keep its tradition as previous cases.

Figure 4. Graph of MSEs of different estimates of parameter θ of Power function distribution when α=1.5, β=2, θ =1.5 and c=2

Table 5. Estimated value and MSE of different estimators of the parameter 𝜃 of Power function distribution when α =2, β=3, θ= 2 and c=3

Table 5 gives smaller values of MSE for than all other estimators in the study. From the above table according to MSE the relation among the estimators is

Figure 5. Graph of MSEs of different estimates of parameter θ of Power function distribution when α=2, β=3, θ =2 and c=3

The performance of the estimators for different values of parameter θ are also shown in the table 6 along with their graphical presentation.

Table 6. Estimated value and MSE of different estimators of the parameter 𝜃 of Power function distribution when n = 20 , α =2 , β = 3 and c = 2

Figure 6. Graph of MSEs of different estimates of parameter θ of Power function distribution when n=20, α=2, β=3 and c=2

4. Conclusions

From the above analysis and graphical study we can conclude that except for few cases Bayes estimator under NLINEX loss function and Squared Error loss function are better than other estimators in the study.