Overview and evaluation of various frequentist test statistics using constrained statistical inference in the context of linear regression

Abstract

Within the framework of constrained statistical inference, we can test informative hypotheses, in which, for example, regression coefficients are constrained to have a certain direction or be in a specific order. A large amount of frequentist informative test statistics exist that each come with different versions, strengths and weaknesses. This paper gives an overview about these statistics, including the Wald, the LRT, the Score, the $\overline{F}$ - and the D-statistic. Simulation studies are presented that clarify their performance in terms of type I and type II error rates under different conditions. Based on the results, it is recommended to use the Wald and $\overline{F}$ -test rather than the LRT and Score test as the former need less computing time. Furthermore, it is favorable to use the degrees of freedom corrected rather than the naive mean squared error when calculating the test statistics as well as using the $\overline{F}$ - rather than the ${\overline{\chi }}^2$ -distribution when calculating the p-values.

Keywords: F¯-distribution; constrained statistical inference; corrected mean squared error; informative hypothesis testing; informative test statistics; naive mean squared error; type I error rates; χ¯2-distribution.