Asymptotic expansions of the distributions of test statistics for the slope in a simple linear functional relationship

Abstract

A set of multivariate observations is said to satisfy a linear functional relationship if their mean values satisfy a linear relationship. The objective of this research is to investigate the distributional properties of test statistics for testing hypotheses about the slope coefficient in a simple linear functional model. The performance of test statistics with the same level should be compared based on the power of the test. However, the test procedures based on the above test statistics do not have the prescribed level in small samples, because the prescribed level is based on the limiting distribution of the test statistic. In this situation it may not be appropriate to compare directly the power of the test without any adjustment of the nominal significance level. Therefore, it is important to evaluate how close the actual level is to the nominal level, or, more broadly, how close the distribution of the test statistic in the finite sample is to its limiting distribution under the null hypothesis, before comparing the power properties. This study will derive asymptotic expansions of the distribution of the Studentized t-statistic proposed by Fuller (1981), the squared t-statistic proposed by Gleser (1981), and a "quasi" (or modified) LR-statistic, under the null hypothesis for two alternative parameter sequences. In the first sequence, the noncentrality parameter goes to infinity while the sample size stays fixed. In the second sequence, the noncentrality parameter increases at the same rate as the sample size. The asymptotic expansions provide more accurate approximations to the exact distributions of the test statistics than do the limiting distributions. A simulation study numerically evaluates the accuracy of the asymptotic expansions. We propose a simple method of adjusting the nominal significance level of the tests.