Efficient Estimation of the PDF and the CDF of a Generalized Logistic Distribution

The generalized logistic distribution is a useful extension of the logistic distribution, allowing for increasing and bathtub shaped hazard rates and has been used to model the data with a unimodal density. Here, we consider estimation of the probability density function and the cumulative distribution function of the generalized logistic distribution. The following estimators are considered: maximum likelihood estimator, uniformly minimum variance unbiased estimator (UMVUE), least square estimator, weighted least square estimator, percentile estimator, maximum product spacing estimator, Cramér–von-Mises estimator and Anderson–Darling estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies are also carried out to show that the maximum-likelihood estimator is better than the UMVUE and that the UMVUE is better than others. Finally, a real data set has been analyzed for illustrative purposes.