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Correcting Four Test Statistics for One-Parameter Distributions Using Mathematica

Journal / Anthology

Communications in Statistics—Simulation and Computation
Year: 2008
Volume: 37
Issue: 8
Page range: 1663-1681

Likelihood ratio, Rao score, Wald, and modified Wald statistics are commonly used to test composite hypotheses on the parameters of distributions. We present source code for a Mathematica program (script) that can be used for generating analytic expressions for Bartlett and Bartlett-type corrections to improve all these four statistics for testing the parameter of an arbitrary regular one-parameter distribution. The script is tested on various discrete and continuous distributions, and the results are compared with those published in earlier works, confirming all of the previously reported expressions. The presented script may in principle be applied to calculate Bartlett and Bartlett-type corrections to improve the accuracy of the asymptotic distributions of these four statistics for any regular one-parameter distribution. In particular, new results are derived for distributions outside the exponential family, which have not been previously published. We apply our results on the t distribution in a simulation study to compare the performance of the four classical tests and their corrected versions.


Bartlett-type correction, Exponential family, Mathematica, Maple, Maximum likelihood estimate, Score statistic, t distribution, Wald statistic