A comparison of the two shape parameters estimation methods for exponentiated inverted Weibull distribution

Introduction

The Exponentiated Inverted Weibull (EIW) distribution represents a lifetime distribution that is extensively used across different sectors of reliability analysis. The Exponentiated Inverted Weibull (EIW) distribution has been developed as an appropriate framework for explaining the war and tear processes of mechanical parts, including the moving parts of diesel engines. The Exponentiated Inverted Weibull (EIW) distribution has garnered significant interest in research works. Exponentiated Inverted Weibull (EIW) distribution is a particular case of Generalized Exponentiated Weibull distribution. If a random variable X has Exponentiated Weibull distribution to 1 𝑋 said to have Generalized Exponentiated Weibull distribution. Flaih et.al. (2012) [7] presented the two shape parameters EIWD of the moments, median, survival function, hazard function, Maximum Likelihood Estimation, Least Square Estimation, Fisher Information matrix and asymptotic confidence interval. Bayes and classical estimators have been obtained for two parameters exponentiated inverted Weibull distribution when sample is available from complete and type II censoring scheme were used methods of the Lindley and Markov chain Monte Carlo method (MCMC) methods for obtaining Bayes estimates under these loss functions by Aljouharah Aljuaid (2013). The estimation of population parameters for the exponentiated inverted Weibull distribution based on grouped data with equi and unequi-spaced grouping worked by Amal S. Hassan et al., (2014) [9] . Pedro Luiz Ramos and Francisco Louzada (2016) wrote a paper Comments on "The Exponentiated Inverted Weibull Distribution" in this relation between the inverse Weibull (IW) distribution and Exponentiated Inverted Weibull (EIW) distribution. Hassan et. al. 2014 [9] new distribution present is Odd Generalization Exponential Inverse Weibull Distribution, three parameter distribution here fitted Maximum Likelihood estimation, Percentile Estimation, Least Square Estimation and fit a real data set. The type-II censored value of Exponentiated Weibull Distribution presented by V Sastry (2023) for different sample size for different parameter values of Scale and Shapes. Dubey (1967) [11] derived the percentile estimators (Percentile) which uses 4 different percentiles to estimate the shape and scale parameters.

Exponentiated Inverted Weibull (EIW) Distribution

Exponentiated Inverted Weibull (EIW) Distribution cumulative distribution function is ()

Probability density function of EIWD is

The distribution function of x takes the presented in equation (1) where and  both are shape parameters and 0 and 0, 0.

The probability density function of EIWD in (2).

1,   this EIWD becomes the standard Inverted Weibull distribution

Characteristics

In this section presented the properties for Exponentiated Inverted Weibull (EIW) Distribution.

The Reliability function

Inverse transformation the method for random variable

Estimation methods

This study concerned with detailed comparison of two parameter estimation methods for Exponentiated Inverted Weibull (EIW) distribution. The Method of Moments (MOM), Maximum Likelihood Estimation (MLE), the least-squares method (LSM) and Percentile Methods are the most well-known parameter estimation methods included in the comparison.

Method of Moments

The Method of Moments (MOM) is a technique for constructing estimators of the parameters that is based on matching the sample moments with the corresponding distribution moments. The rth raw moments for the kth component in the Exponentiated Inverted Weibull (EIW) distribution is

The k th moments of the EIW distribution is given as follows:

Let k=1 in Eq (7), we obtain the Mean as:

Maximum Likelihood Estimation

If 12 , ,..., n X X X is ordered random sample from Exponentiated Inverted Weibull distribution given by pdf   fX , then the Log-Likelihood function (LL) becomes:

There the MLE of  and  which maximize equation ( 9) must satisfy the nonlinear normal equations given by 1 0

Least Squares Estimation

The LSM based on relationship between the empirical cumulative distribution function and the order statistics is frequently used to estimate parameters of distributions. The cumulative function of the kth components in the Exponentiated Inverted Weibull (EIW) distribution is given as follows:

Necessary and sufficient condition for function   , S A B to achieve the maximum is expressed by equation:

from where a system of linear algebraic equations for determination of coefficients A and B is gained:

By determination of determinants of equation systems above equations:

values of required coefficients are gained:

Based on relationship between distribution parameters (α, β) and the coefficients of straight line equation, the expressions are given below:

Percentiles Estimation

In this study Two parameters are Shape parameters. Use the following formula estimating the parameters. Methodology is adopted from Florence (2014) studied the five different methods for joint estimation for scale and shape parameters of two parameter Weibull distribution. The initial Percentile estimators for both shape and scale parameters were derived by Dubey (1967) [11] . The proposed an estimator based on 17 th and 97 th percentiles for shape parameter. The formulae for the shape percentile estimators are presented.

Computational Results

Simulation study A simulation study has been conducted to explore the performances of the different methods discussed in this article. The simulation was done using statistical software R version 4.3.1. The 10000 simulation runs at random samples of size n =5(5)25, 50 (25)100(50)250) are generated from the EIW distribution with different values of shape parameters were performed to compare the performance of the proposed estimation methods.

The main objective of this study is to compare the performance of the methods to estimate the shape parameters of Exponentiated Inverted Weibull (EIW) distribution. The best method parameter estimates are obtained by mentioned methods for each data set.

No restriction was imposed on the maximum number of iterations. The estimates are compared using the values of mean squared error (MSE) only.

Mean Square Error (MSE)

Here, evaluate mentioned methods, through a comprehensive simulation study considering different parameter values and sample sizes. In order to understand the accuracy of the Exponentiated Inverted Weibull distribution, most of the studies address the question as a statistical error and calculate the Mean Square Error (MSE) based on the measured data.

where   ˆi Fx is the measured value and   i Fx is the EIW distribution initial parameter value.

References

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