Estimating the Reliability Function for Transmuted Pareto Distribution Using Simulation

In this work, the methods (Moments, Modified Moments, L-Moments, Percentile, Rank Set sampling and Maximum Likelihood) were used to estimate the reliability function and the two parameters ( ) of the Transmuted Pareto (TP) distribution. We use simulation to generate the required data from three cases this indicates sample size , and it replicates for the real value for parameters, for reliability times values we take ( ) ( ). Results were compared by using mean square error (MSE), the result appears as follows : The best methods are Modified Moments, Maximum likelihood and L-Moments in first case, second case and third case respectively.


Introduction
Vilfredo Pareto [1] suggested the Pareto distribution, the (pdf) and (cdf) of the Pareto distribution, respectively. Where , and ( ) . Shaw and Buckley [2] introduced the Quadratic Rank Transmutation Map (QRTM) that the relationship satisfies the cumulative distribution function (cdf) approach.
In this paper, we will estimate the parameters and reliability function of the transmuted Pareto (TP) distribution through six methods which are not previously used in estimating by using simulation.

Transmuted Pareto distribution (TP):
Let X be a random variable of the (TP) distribution, then both of the (pdf) and (cdf) for the (TP) distribution are expressed respectively as [8].
Where: is a value of random variable and . Shape parameter and : Transmutation parameter |λ| . : λ : The minimum possible value of . The reliability function is given by

Estimation Methods:
In this section, we will derive some estimation methods for Transmuted Pareto distribution (TP).

Estimate Initial Value for Estimators:
Makki and Jaafer [9] proposed a method for finding initial values for estimating parameters of any distribution, as follows: The median of (TP) distribution that is given in [8] as follows : Taking the natural logarithm for the equation ( ), getting We can use ̂ and ̂ in equations (7) and (8) are the initial value to another parameter estimator formula can be obtained from the generating sample.

Moments Method (MOM):
The idea of this method is to equate the moments of the distribution with the moments of the sample. ̀ ( ) The first estimate can be get it when ( ) in equation (9), then ̀ Since, The second estimate can be get it when ( ) in equation (9) As for the estimator of the approximate reliability function for this distribution, we obtain it by substituting equations (10) and (11) into equation (3) as follows:
As for the estimator of the approximate reliability function for this distribution, we obtain it by substituting equations (16) and (18) into equation (3) as follows:

L-Moment Method (LM):
Hosking [12] proposed the L-Moments method. The L-Moments methods are anticipations of certain linear combinations of order statistics [13]. The distribution parameters can be estimated by the following (1) and (2) into equation (20), then we get If we substitute in equation (21), then we get While if we substitute in equation (21) Now subtract equation (23) from equation (22), we have As for the estimator of the approximate reliability function for this distribution, we obtain it by substituting equations (24) and (25) into the equation (3) as follows:

Percentile Estimation Method (PER):
This method was originally explored by Kao [14]. This method depends on parameter estimation of any distribution on minimized of inverse distribution which represented from the equation (4) as follows: Taking the natural logarithm for equation (27), we get We square the equation (28), and let it equal to zero, we get the following The partial derivative the equation (29) with respect to is The partial derivative the equation (29) with respect to is ) ) ) We multiply equation (31)  As for the estimator of the approximate reliability function for this distribution, we obtain it by substituting equations (30) and (32) into equation (3) as follows:

Rank Set Sampling Method (RSS):
Let ( ) be a random sample. Assumed that ( ( ) ( ) ( ) ) is the order statistics that obtained by ordering the sample in increasing order [15]. The (pdf) of ( ) is: By substitution equations (1) and (2)

Maximum Likelihood Estimation Method (MLE):
This method of estimation is a common and reliable method, and this method characterized by its accuracy compared to other methods, especially when increasing the sample size. Let be a sample of size for (TP) distribution [8].

Experiments and Results:
In this section, we review the simulation steps in terms of selecting sample sizes, real values of parameters and life time values that were used to estimate reliability: 1. The sample size is n=10, 30, 70, 100 , and the sample which is replicated N=1000. 2. Several values of the shape and the transmuted parameters ( ) are shown in Table ( The next Tables 2,3 and 4 show the empirical value for estimation of parameters and reliability function.

Conclusion:
From the results in Tables 2, 3 and 4, we note the following:  In the second and third cases ( ), we note that the parameters ( , ) and the reliability function cannot be estimated for the two methods, namely MOM, and MM, because there is a condition in the estimate formulas, which is .  In the first case ( ), we note that RSS is the best in estimating ( ) in all cases, while for MOM is the best in estimating the ( ), and PER is the best in estimating ( ) in all cases, while for MLE is the best in estimating the ( ). For the reliability function of the distribution the MM is best when while if , then MOM is the best as well as if , then RSS is the best.  In the second case ( ), in the estimation of ( ), the RSS is best when , and if , then MLE is the best, while if , then the LM is the best as well as when the PER is the best. The RSS is the best in estimating ( ) in all cases, while for the MLE is the best in estimating the ( ). In estimating reliability the MLE is best in all cases, while for the PER is the best.  In the third case ( ), we note that the LM is the best in estimating ( ) when , however in the RSS is the best, and when the PER is the best. The LM is the best in estimating ( ) when , but when the MLE is the best, and when the RSS is the best. In estimating reliability the LM is best when , but when the RSS is the best and when the PER is the best.