Symbolic computation and the comparison of traditional and robust test statistics
James E. Stafford · 1992
This thesis discusses two research problems. The first is a statistical problem where inference is to be conducted for a parameter $\theta.$ Three traditional test statistics are compared to test statistics that are robust to model misspecifications in the scalar and multi-parameter settings. The comparisons are made under model assumptions in order to investigate any loss in precision in using the robust statistics when it is not necessary. Comparisons are made through an examination of cumulants and coverage probabilities. Asymptotic expansions for the first four cumulants of each statistic are derived and compared to the cumulants of the standard normal distribution. This is done to assess the normality approximation for each statistic and to see if this approximation is substantially worse for either the robust or traditional statistics. The expansions for the cumulants of each statistic may be inserted into Edgeworth approximations. In this way, coverage probabilities for confidence intervals based on each statistic may be compared. This is done to see if inference procedures are substantially less efficient for either the traditional or robust statistics. The quality of the normal approximation and the efficiency of inference procedures are strongly related. Making comparisons between traditional and robust test statistics involves the derivation and evaluation of a large number of asymptotic expansions. Obtaining these expansions by hand is an exercise that is frustrating, tedious, highly prone to error and at times exceedingly laborious. Symbolic computation provides a practical alternative to doing these expansions by hand. All the clerical detail that is involved in an expansion is left to the computer, allowing the researcher to concentrate on matters of a more theoretical nature. The second research contribution of this thesis involves the development and implementation of complex algorithms for the symbolic computation and evaluation of asymptotic expansions. These procedures are not restricted to the applications found in this thesis, but are general in nature and can be applied to a variety of statistical problems. The procedures provide a quick, reliable method for deriving expansions. They were used to obtain the expansions found in this thesis.