Refined Inference on the Scale Parameter of the Generalized Logistic Distribution Based on Adjusted Profile Likelihood Functions

Abstract

We consider inference based on the profile likelihood function for the scale parameter of the generalized logistic distribution. This distribution is a generalization of the logistic distribution, a symmetric distribution like the normal distribution, and it has several applications in various fields. The generalization allows for possible left or right skewness, which makes it more flexible for modeling purposes. Inference procedures based on the profile likelihood of the scale parameter do not perform very well when the sample size is small, therefore, we derived adjustments to the profile likelihood for the generalized logistic distribution using results from higher-order likelihood theory. We obtained an adjustment based on the empirical covariances of certain scores of the profile likelihood function. Another adjustment is derived using ancillary statistics. The performance of the adjustments is investigated for point estimation of the scale parameter of the generalized logistic distribution using the bias and mean squared error criteria. Using an extensive simulation study, we found the adjustments are very successful in reducing the bias and the mean squared error of the maximum profile likelihood estimator in most situations. Moreover, we studied the performance of the profile likelihood ratio test and its adjustments using the criterion of the attainment of nominal sizes. We found that, when the sample size is small, the profile likelihood ratio test has empirical sizes that are highly inflated. Therefore, the test will be invalid in such situations. Simulation results show that the adjusted versions of the profile likelihood produce tests that attain the nominal sizes even for very small samples. This also applies to confidence intervals derived from these tests. In conclusion, both adjustments of the profile likelihood have significantly better performance than the unadjusted profile likelihood and are recommended, especially for small samples. In particular, the adjustment based on ancillary statistics appears to have the best overall performance in all situations considered. We applied the methods in this paper to real data on Carbon fibers.