Certain Computational Aspects of Power Efficiency and of State Space Models

Abstract

A semiparametric approach to the one-way layout is described, and its efficiency in the two-sample case relative to the common t-test is studied. The power efficiency computed for several special cases points to an intriguing behaviour where one test may be more efficient than the other over a certain parameter range and less efficient over another parameter range. Given two random samples from two distributions, the method holds one distribution as the "reference" and treats the other distribution as a "distortion" of the reference. The combined sample is used in the semiparametric estimation of the reference and distortion distributions, and in testing the hypothesis of distribution equality. In order to calculate relative power efficiencies, the asymptotic distributions of the semiparametric and t-test statistics are used in approximating the finite sample distributions of the statistics. Relative power simulations for several special cases show that the theoretical results compare favorably with the finite sample simulation results.

A likelihood approach is employed in deriving a state space smoother, based on a linear state space model between an unobserved "state" time series and an observed time series. A state space smoother provides an algorithm for calculating the conditional mean of any state given the available observations, called smoother estimate, and for calculating the variance of any residual obtained as the difference between a state and its smoother estimate, called smoother precision. Bounds and asymptotic limits are developed for the smoother precisions under the assumption of a univariate state space model. An extension for missing observations handles the special case of prediction. A partial state space smoother is introduced. It provides a smoother like estimate of each state and relies only on a limited number of future observations.