Confidence interval, prediction interval and tolerance limits for a two-parameter Rayleigh distribution

Confidence Interval, Prediction Interval and Tolerance Interval for the Skew Normal Distribution: A Pivotal Approach

The class of skew normal distributions, introduced by Azzalini (1985), which is an asymmetric distribution and allows the presence of skewness. In this paper, we propose the pivotal quantity approach to construct the confidence interval for the mean, prediction interval for the mean of the future sample, and tolerance interval for the quantile. The fiducial distribution is also studied. Moreover, the performances of all the proposed confidence intervals are investigated through the Monte Carlo simulation. The pivotal quantity is a common method for calculating confidence intervals, which is used to construct confidence intervals in this paper. And the convergence of the obtained confidence interval is illustrated by the figures. Finally, a real data is used to explain proposed intervals in real life.

Estimation of the probability content in a specified interval using fiducial approach

Statistical methods for constructing confidence intervals for the probability content in a specified interval are proposed. Exact and approximate solutions based on the fiducial approach are described when the measurements on the variable of interest can be modelled by a location-scale (or log-location-scale) distribution. Methods are described for the normal, Weibull, two-parameter exponential and two-parameter Rayleigh distributions. For each case, the solutions are evaluated for their merits. Three examples, where it is desired to estimate the percentages of engineering products meet the specification limits, are provided to illustrate the methods.