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Estimation Method under Three-Parameter Generalized Exponential Model: Consistency, Uniqueness and its Applications

Kiran Prajapat, Sharmishtha Mitra, Debasis Kundu

Abstract

In numerous instances, the generalized exponential distribution can be used as an alternative to the most widely used non-regular family of distributions: Weibull, gamma, lognormal with three-parameters when analyzing lifetime or any skewed continuous data. A non-regular family is a class of probability distributions that do not satisfy the regularity conditions typically assumed in classical statistical inference. Some key features of such family of distributions are: support of its probability density function depends on one its parameters; its likelihood function may not be bounded for a certain range of parameter space, hence maximum likelihood estimators do not exist; the likelihood function even may not be differentiable or integrable as needed, hence Fisher Information may not exist or be infinite. Moreover, standard results like MLE existence, consistency, asymptotic normality may fail. Therefore, specialized or robust inferential techniques are needed. This article offers a consistent method for estimating the parameters of a three-parameter generalized exponential distribution that sidesteps the issue of an unbounded likelihood function. The method is hinged on a maximum likelihood estimation of shape and scale parameters that uses a location-invariant statistic. Important estimator properties, such as uniqueness and consistency, are demonstrated for the first time under this approach. In addition, quantile estimates for the assumed distribution are provided. We present a Monte Carlo simulation study along with comparisons to a number of well-known estimation techniques in terms of bias and root mean square error. For illustrative purposes, a real dataset from reliability engineering, has been analyzed and the goodness of fit along with the bootstrap confidence intervals are compared with existing traditional methods.

Estimation Method under Three-Parameter Generalized Exponential Model: Consistency, Uniqueness and its Applications

Abstract

In numerous instances, the generalized exponential distribution can be used as an alternative to the most widely used non-regular family of distributions: Weibull, gamma, lognormal with three-parameters when analyzing lifetime or any skewed continuous data. A non-regular family is a class of probability distributions that do not satisfy the regularity conditions typically assumed in classical statistical inference. Some key features of such family of distributions are: support of its probability density function depends on one its parameters; its likelihood function may not be bounded for a certain range of parameter space, hence maximum likelihood estimators do not exist; the likelihood function even may not be differentiable or integrable as needed, hence Fisher Information may not exist or be infinite. Moreover, standard results like MLE existence, consistency, asymptotic normality may fail. Therefore, specialized or robust inferential techniques are needed. This article offers a consistent method for estimating the parameters of a three-parameter generalized exponential distribution that sidesteps the issue of an unbounded likelihood function. The method is hinged on a maximum likelihood estimation of shape and scale parameters that uses a location-invariant statistic. Important estimator properties, such as uniqueness and consistency, are demonstrated for the first time under this approach. In addition, quantile estimates for the assumed distribution are provided. We present a Monte Carlo simulation study along with comparisons to a number of well-known estimation techniques in terms of bias and root mean square error. For illustrative purposes, a real dataset from reliability engineering, has been analyzed and the goodness of fit along with the bootstrap confidence intervals are compared with existing traditional methods.
Paper Structure (14 sections, 4 theorems, 47 equations, 7 figures, 10 tables)

This paper contains 14 sections, 4 theorems, 47 equations, 7 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Proposed Estimators and their Properties
  3. Parameter Estimation
  4. Quantile Estimation
  5. Simulation Study
  6. Evaluation of Parameter Estimation
  7. Evaluation of Quantile Estimation
  8. Illustrative Example
  9. Summary
  10. Proof of Theorem \ref{['thm_2.1']}
  11. Boundedness and Differentiablity of the Likelihood Function $\ell_v(\alpha,\beta)$
  12. Boundedness
  13. Differentiablity
  14. Tables

Key Result

Theorem 2.1

The likelihood function of $\alpha$ and $\beta$, given $v_2, v_3, \dots, v_{n}$, is given by with $0 < v_{2} < \dots < v_{n} < \infty,~ v_{1} = 0$.

Figures (7)

  • Figure 1: Plots for bias and RMSE of the estimator of shape parameter based on various estimation methods varying $\beta$ values.
  • Figure 2: Plots for bias and RMSE of the estimator of scale parameter based on various estimation methods varying $\beta$ values.
  • Figure 3: Plots for bias and RMSE of the estimator of location parameter based on various estimation methods varying $\beta$ values.
  • Figure 4: Plots for bias and RMSE based on various estimation methods when $\beta=0.50, ~ 0.75, ~ 1.00$.
  • Figure 5: Plots for bias and RMSE based on various estimation methods when $\beta=1.5, ~ 2.0, ~ 3.0.$
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.1