Non-parametric Distributions for Quantile, Average, and Combined Average-standard Deviation, Using Bayesian Analysis And Maximum Entropy

Estimation of the quantile, mean, and variability of populations is generally carried out by means of sample estimates. Given normality of the parent population, the distribution of sample mean and sample variance is straightforward. However, when normality cannot be assumed, inference is usually based on approximations through the use of the Central Limit theorem. Furthermore, the data generated from many real populations may be naturally bounded; i.e., weights, heights, etc. Thus, a normal population, with its infinite bounds, may not be appropriate, and the distribution of any specified quantile, such as the median, is not obvious. Using Bayesian analysis and maximum entropy, procedures are developed which produce distributions for any specified quantile, the mean, and combined mean and standard deviation. These methods require no assumptions on the form of the parent distribution or the size of the sample and inherently make use of existing bounds.