Vol: 34 Page: 289

Authors:
Jacek Wawrzosek

Title:
 Some relations between the linear model, restrictions of parameters and a linear statistical hypothesis

Language: Polish

Keywords:
general linear Gauss-Markov model
restrictions of parameters
testability of a linear hypothesis
consistency of model
hypothesis is consistent

Summary:
Consider the general linear Gauss-Markov onevariate model M = {y, Xβ, σ2V}, restrictions R : l = Fβ in model and linear hypotheses H0 : k = Hβ, H1: k ≠ Hβ. y is a nx1 observable random vector with expectation E(y) = Xβ and dispersion matrix D(y) = σ2V, where an nxp matrix X and an nxn nonnegative definite matrix V are known.
Necessary consistency conditions for triplets (M, R, H0) and (M, R, H1), for models (M, R) and (M, H0) and for the single M, for the single R and for the single H0 are given.
Linear estimability Hβ and testability H0 in the linear model (M, R) are presented. Moreover, we discuss conditions for nontriviality of H0.
Nine conditions for correctly statistical analysis in the triplet (M, R, H0) are collected. This is the generalization of Drwięga (1979), Nordström (1985), Kłaczyński and Pordzik (1990), Rao (1982), Seely (1977) and Wawrzosek (1994) results. This conditions become simpler when M is the Zyskind-Martin model, i.e. when R(X) ⊂ R(V) and in particular cares when the matrix V is non-singular or when F = 0. Theoretical considerations are illustrated with a numerical example.