Using Bayesian analysis and maximum entropy to develop non-parametric probability distributions for the mean and variance

Estimation of the population mean, and variance is generally carried out using 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. In addition, the data generated from many real populations may be naturally, bounded, i.e. weights, heights, etc. Thus, the unbounded normal probability model may not be appropriate. Utilizing Bayesian analysis and maximum entropy, procedures are developed which produce nonparametric distributions for both the mean and the mean/standard deviation combination. 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.