Certain Computational Aspects of Power Efficiency and of State Space Models

Certain Computational Aspects of Power Efficiency and of State Space Models

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##### Date

2005-04-05

##### Authors

Gagnon, Richard Edward

##### Advisor

Kedem, Benjamin

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##### 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.