Quantifying uncertainty in statistical distribution of small sample data using Bayesian inference of unbounded Johnson distribution

Probabilistic analysis of physical systems requires information on the distributions of random variables. Distributions are typically obtained from testing or field data. In engineering design where tests are expensive, the sample size of such data is small O(10). Identifying correct distributions with small number of samples is difficult. Furthermore, parameters of assumed distributions obtained from small sample data themselves contain some uncertainty. In this study a Johnson SU family distribution function is used to identify shape, location and scale parameters of distribution that can best fit small sample data. A Bayesian inference procedure is used to determine distributions of the parameters. We show that the procedure correctly bounds the tail regions of the distributions and is less conservative than bounds obtained using bootstrap methods.