Modified Moment , Maximum Likelihood and Percentile Estimators for the Parameters of the Power Function Distribution

This paper is concerned with the modifications of maximum likelihood, moments and percentile estimators of the two parameter Power function distribution. Sampling behavior of the estimators is indicated by Monte Carlo simulation. For some combinations of parameter values, some of the modified estimators appear better than the traditional maximum likelihood, moments and percentile estimators with respect to bias, mean square error and total deviation.


Introduction
The Power function distribution is a flexible life time distribution model that may offer a good fit to some sets of failure data.Theoretically Power function distribution is the inverse of Pareto distribution.An excellent account of this distribution and its properties is given in Kleiber and Kotz (1970).Meniconi and Barry (1995) discussed the application of Power function distribution.They proved that the Power function distribution is the best distribution to check the reliability of any electrical component.They used Exponential distribution, Lognornal distribution and Weibull distribution and showed from reliability and hazard function that Power function distribution is the best distribution.
The probability distribution of Power function distribution is ( ) With shape parameter and scale parameter , the interval (0, ) Rider (1964) derived Distributions of the product and quotients of the Order Statistics from a Power function distribution.Moments of Order Statistics for a Power function Distribution were calculated by Malik (1967).Lwin (1972) discussed Bayesian estimation for the scale parameter of the Pareto Distribution using a Power function prior.Ahsanullah and Kabir (1975) discussed the Estimation of the location and scale parameters of a Power function distribution.
Cohen and Whitten (1982) used the moment and Modified Moment Estimators for the Weibull Distribution.Samia and Mohammad (1993) used five modifications of moments to estimate the parameters of the Pareto Distribution.Lalitha and Anand (1996) used Modified Maximum Likelihood to estimate the scale parameter of the Rayleigh Distribution.Rafiq (1996) discussed the parameters of the Gamma Distribution.Rafiq (1999) discussed the method of Fractional Moments to estimate the parameters of Weibull Distribution.Kang and Young (1997)  In this paper, we use the percentiles method, maximum likelihood and moment method to estimate the two parameters of the Power function distribution.The present paper introduces the modified estimators for parameters of the Power function distribution.The resulting estimators are easier to calculate than the maximum likelihood and, for certain combinations of parameter values; they improve the estimated values and mean square error.We examine these methods using two parameters Power function distribution to find the most accurate method (the method which has least M.S.E).

Percentile Estimator (P.E)
Let be a random sample of size n drawn from probability density function of Power function distribution.The cumulative distribution function for a Power function distribution with shape and scale parameters , respectively By solving we get Where = ( ) Let P 75 and P 25 are used. ( Dividing ( 4) and ( 5) we get Where H+L = 1.

First Modified Percentile Estimator (M.P.E.1)
In this modification of the percentile estimators, equation ( 5) is replaced by the coefficient of variation of Power function distribution.

̅ √ ( )
Taking square in both sides we get Simplification we get From ( 4)

Second Modified Percentile Estimator (M.P.E.2)
In this modification of the percentile estimators, equation ( 5) is replaced by the median of Power function distribution.

Maximum Likelihood Method (M.L.E)
Let be a random sample of size n drawn from probability density function of Power function distribution.The likelihood function of this random sample is the joint density of the n random variables and is a function of the unknown parameters.Thus ( ) Likelihood function is The maximum likelihood estimator (MLE) of the parameter is the value of the parameter that maximizes L and MLE for 2 parameter of Power function distribution can be obtained by solving the equations resulting from setting the two partial derivatives of L( , ) to zero; (17) Where t n is the largest value in the sample data.By neglecting (17) and replacing

Third Modified Maximum Likelihood Estimator (M.M.L.E.3)
Using c.v of Power function distribution and neglecting √ ( ) Where is standard deviation and ̅ is mean.

Moment Estimators (M.E)
The method of moments is another technique commonly used in the field of estimation of parameters.If the numbers represent a set of data, then an unbiased estimator for the k th origin moment is ∑ Where stands for thr k th sample moment.
The first moment of Power function distribution is Therefore by equating sample and population moments we get And ́= ( ) Put in (31), we get Where and

First Modified Moment Estimator (M.M.E.1)
In this modification of the moment estimators, the second moment of two parameters Power function distribution is replaced by the coefficient of variation of Power function distribution.

̅ √ ( )
Taking square in both sides we get Simplification we get

Second Modified Moment Estimator (M.M.E.2)
In this modification of the moment estimators, the second moment of two parameters Power function distribution is replaced by the variance of Power function distribution. i.e.

Third Modified Moment Estimator (M.M.E.3)
In this modification of the moment estimators, the first moment of two parameters Power function distribution is replaced by the variance of Power function distribution. i.e. (31)

Fourth Modified Moment Estimator (M.M.E.4)
Using co-efficient of variation of Power function distribution and , by neglecting . (31) From (31) Three approximations for ̂( ) based on its being uniformly distributed on interval [0, 1] show in Table.

Performance Indices (Goodness of Fit Analysis)
Some methods of goodness of fit analysis are employed here.Mean square error MSE and total deviation TD are two measurements that give an indication of the accuracy of parameter estimation.AL-Fawzan (2000) referred to the use of the procedure of MSE and TD.

Total Derivation (TD)
The total derivation TD, calculated for each method is as follows Where and are the known parameters, and ̂ n ̂ are the estimated parameters by any method.These techniques are used to measure the variability of parameter estimates for each simulation.These are used to determine the overall "best" parameter estimation method.

Application
A simulation study is used in order to compare the performance of the proposed estimation methods.We carry out this comparison by taking the samples of sizes as n = 20, 60 and 100 with pairs of ( , ) = {(1, 2), (3, 2), (4, 3)}.We generate random samples of different sizes by observing that if R is uniform (0, 1), then ⁄ is the random number generator of Power function distribution with ( , ) parameters.All results are based on 10,000 replications.Such generated data have been used to obtain estimates of the unknown parameters.The results obtained from parameters estimation of the 2-parameters Power function distribution using different sample sizes and different values of parameters with mean square error MSE and total deviation TD.

F
estimated the parameters of a Pareto Distribution by Jackknife and Bootstrap Methods.Neil (2005) estimated the parameters of Weibull Distribution with the help of percentiles.He called it Common Percentile Method.Zaka and Akhter (2013) derived the different estimation methods for the parameters of Power function distribution.Zaka and Akhter (2013) discussed the different modifications of the parameter estimation methods and proved that the modified estimators appear better than the traditional maximum likelihood, moments and percentile estimators.