Inferences on Gamma Distributions: Uncensored and Censored Cases

Inferential methods for constructing an upper confidence limit for an upper percentile and for finding confidence intervals for a gamma distribution based on samples with multiple detection limits are proposed. The proposed methods are based on the fiducial approach. Computational algorithms are provided and numerical results are given to assess the performance of the proposed methods, and to make comparisons with competing procedures. It is noted that the fiducial approach provides accurate inference for estimating the gamma mean and percentiles. In general, the fiducial approach is very satisfactory and is applicable to small sample sizes. We also derived the likelihood ratio test (LRT) statistics for testing equality of shape parameters, scale parameters, several gamma means, and homogeneity of several independent gamma distributions. Our extensive simulation studies for each of the testing problems indicate that the percentiles of the null distributions of the LRT statistics are affected mainly by the number of distributions to be compared and the sample sizes, but not much on the parameters. These simulation results simply imply that the null distributions depend on the parameters only weakly. The simulation studies also showed that the procedures are very satisfactory in terms of coverage probabilities and powers. Illustrative examples with practical data sets and simulated data sets are given