Methods for Estimating the Parameters of the Power Function Distribution

In this paper, we present some methods for estimating the parameters of the two parameter Power function distribution. We use the least squares method (L.S.M), relative least squares method (R.L.S.M) and ridge regression method (R.R.M). Sampling behavior of the estimates is indicated by a monte carlo simulation. We use total deviation (T.D) and mean square error (M.S.E) to identify the best estimator among them. We determine the best method of estimation using different values for the parameters and different sample size.


Introduction
Numerous parametric models are used in the analysis of lifetime data and in problems related to the modeling of failure processes.Among univariate models, a few particular distributions occupy a central role because of their demonstrated usefulness in a wide range of situation.Foremost in this category are the exponential, Weibull, gamma and lognormal distributions.
The Power function distribution is also a flexible life time distribution model that may offer a good fit to some sets of failure data.Theoretically, Power function distribution is a special case of Pareto distribution.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, lognormal 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 to .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), Arnold and Press (1983) 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 Mohamed (1993)  In this paper, we use the least squares method, relative least squares and ridge regression to estimate the two parameters of the power function distribution.The present paper introduces the ridge regression estimators by taking different values of "k".Where "k" is the ridge coefficient.Also, we compare these methods using two parameters power function distribution to find the most accurate method (the method which has least M.S.E).

Least Squares Method (L.S.M)
The least square method (LSM) is extensively used in reliability engineering, mathematics problems and the estimation of probability distribution parameters.
The cumulative distribution function of Power function distribution is given by To get a linear relation between the two parameters taking the logarithm of above equation as follows Where = 1, 2 …. n and n is the sample size.
Let be a random sample of and ( ) is estimated and replaced by the median rank method as follows: Because ( ) of the mean rank method ( ) ( ) , may be a larger value for smaller and a smaller value for larger Therefore we use median rank method.
Thus, equation (2.1.2) is a linear equation and is expressed as To compute a and d by simple linear regression we proceed as follows

Let ( ) ∑ ( -)
We obtain the least square estimates (LSE) of a and d as: where ( )

Relative Least Squares Method (R.L.S.M)
The relative least squares estimators of a and d can be obtained by minimizing the sum of squares of the relative residuals, Pablo and Bruce (1992), w.r.t. a and d as follows

Ridge Regression Method (R.R.M)
The ridge regression estimators are given by Where k ≥ 0 is the ridge coefficient, is the p*p identity matrix and p is the number of parameters. [ After simplification we get Where ( ) and Where k ≥ 0 is the ridge coefficient the readers may see Ronald and Raymond (1978) if k=0, we obtain the least square estimates.

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.

Mean Square Error (MSE)
The MSE can be calculated as below

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 generated random samples of different sizes by observing that if R is uniform (0, 1), then ⁄ is the random number generation of power function distribution with ( ) parameters.All results are based on 10,000 replications.It is observed that MSEs and TDs of all estimators of scale parameter as well as all the estimators of shape parameter are decreasing with the increase of sample size.
Consequently, we recommend using the L.S.M method for the parameters estimation of the power function distribution.After L.S.M, the R.R.M (0.1) and R.L.S.M method are best for estimation of scale and shape parameters of the power function distribution.
( ) is the value of the cumulative distribution function of the 2 parameter power function distribution using the estimated parameters, and ̂( ) ( ̂) ̂ Also Standard bias, Bias = E ( ̂) -and M.S.E ( ̂) = *( ̂ -) + Standard bias, Bias = E ( ̂) -and M.S.E ( ̂) = *( ̂-) + Kang and Young (1997)96)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 et al. (1996)discussed the parameters of the gamma distribution.Rafiq (1999)dicussed the method of fractional moments to estimate the parameters of Weibull distribution.Kang and Young (1997)estimated the parameters of a Pareto distribution by jackknife and bootstrap methods.Marks (2005) estimated the parameters of Weibull distribution with the help of percentiles.He called it Common Percentile Method.

Table 3 : Estimation for the parameters β and γ of Power Function Distribution for sample size 100
(3) the results are listed in table (1), (2) and(3).From these tables, we see that the L.S.M estimates of parameters are too close to the true values, and the values of MSE and TD are very small.The parameter estimates from R.L.S.M, R.R.M methods are close to the true values but not as close as L.S.M estimates, because the values of MSE and TD are greater than the corresponding values from L.S.M.