Modified Normal-based Approximation to the Percentiles of Linear Combination of Independent Random Variables with Applications

Confidence estimation based on data from independent studies

The problem of finding confidence intervals based on data from several independent studies or experiments is considered. A general method of finding confidence intervals by inverting a combined test is proposed. The combined tests considered are the Fisher test, the weighted inverse normal test, the inverse chi-square test and the inverse Cauchy test. The method is illustrated for finding confidence intervals for a common mean of several normal populations, common correlation coefficient of several bivariate normal populations, common coefficient of variation, common mean of several lognormal populations, and for a common mean of several gamma populations. For each case, the confidence intervals based on the combined tests are compared with the other available approximate confidence intervals with respect to coverage probability and precision. R functions to compute all confidence intervals are provided in a supplementary file. The methods are illustrated using several practical examples.

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.