On a general class of probability distributions and its applications

Abstract

A general class of probability distributions is proposed and its properties examined. The proposed family contains distributions of a wide variety of shapes, such as U shaped, uniform and long-tailed distributions, as well as distributions with supports that have finite limits at one or both endpoints. Due to its great flexibility, this parametric class (which we refer to as the class of UIC distributions) can be routinely used to fit empirical data collected in different experimental or observational studies without the need of specifying in prior the type and form of distributions to be fitted. It is also simple and inexpensive to simulate from the proposed class of distributions, making it particularly attractive in simulation based optimization applications involving stochastic components with distributions empirically determined from historical data. More importantly, it is shown both theoretically and empirically that under fairly general conditions the sampling distribution of a standardized sample statistic is approximately an UIC distribution, which provides a much closer approximation than the normal approximation in small to medium sample sizes. Applications in the bootstrap, such as estimation of the variance of sample quantiles and quantile estimation by the "smoothed" bootstrap are discussed. The Monte Carlo studies conducted show encouraging results, even in cases where the traditional kernel density approximations do not perform well.