OF ASYMPTOTIC RELATIVE EFFICIENCY FOR THE MULTIPARAMETER TESTING PROBLEM - wITH APPLICATION TO THE GROWTH CURVE PROBLEM

نویسنده

  • P. K. Sen
چکیده

WOOLSON, ROBERT FRANCIS. Some Alternative Measures of Asymptotic Relative Efficiency for the Multiparameter Testing Problem with Application to the Growth Curve Problem. (Under the direction of PRANAB KUMAR SEN.) Criteria for evaluating the loss in underfitting"or overfitting a growth curve model are proposed. This problem has been formulated as one of comparing two test statistic sequences which have limiting chi-square distributions. As the degrees of freedom of the two competitive chi-square distributions may be different, alternative measures to the standard Pitman asymptotic relative efficiency (ARE) are considered. Two such criteria are the trace asymptotic relative efficiency (TARE) and the curvature asymptotic relative efficiency (CARE). Each of these quantities is sho\VD to be the product of two factors: the first factor reflecting the degrees of freedom of the two tests and the common significance level, while the second factor is a function of the two noncentrality parameters. This second factor for the TARE is the ratio of the traces of the matrices in the nonth centrality parameters while the second factor for the CARE is the q root of the ratio of the determinants of the matrices in the noncentrality parameters, where q is the number of parameters in the common hypothesis of interest. The CARE selects the test whose power function has the greater generalized Gaussian curvature at the null point, and the TARE selects the test with greater average local power, e • where the average is taken as the average over the family of spheres. The CARE and TARE are applied to the one-sample and multi-sample growth curve problems. Bounds for the TARE similar to the bounds which exist for the CARE are derived. It is shown that the c-sample efficiencies are multiples of the one-sample results. Numerical illustrations of the TARE and the CARE are presented for specific covariance matrices.

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ALTERNIrIVE MEASURES OF ASYMPTOTIC RELATIVE EFFiCIENCY FOR THE MULTIPARAMETER . TESTING PROBLEM

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تاریخ انتشار 1972