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