Tests of Hypotheses for the General Gauss‐Markov Model
Wiktor Oktaba · Biometrical Journal · 1984
Abstract Some theorems connected with testing the linear hypotheses in the case of the fixed general Gauss‐Markov model are given. The assumptions are as follows. Let y = Xβ + e be a fixed model where residuals ea, …, en being coordinates of the random column vector e are normally distributed with means zero and a known covariance matrix α2V non‐singular or singular and an unknown scalar parameter α2. Suppose that the known matrix X is n × p with rank r(X) = r⩽pn, where pis a number of coordinates of the unknown parameter vector β. The expected value of the random vector y with n coordinates constitutes the vector α(y) = Xβ. The formulas of the test functions (cf. theorems 2.1, 2.2, 2.3) contain c‐inverse matrices T− (defined by the condition TT−T = T), and some others when T = V + XBX′ and B = B′ is any symmetric matrix such that (cf. RAO (1971), OKTABA (1982)) r (V:X) = r (V + XBX′); (V:X) denotes matrix with two submatrices V and X. Mutual orthogonal conditions for a set of estimable parametric functions are introduced and used in the decomposition of the sum of squares for the linear hypothesis into orthogonal sums of squares each on one degree of freedom. Each of these hypotheses can be verified by the random variable F as in the analysis of variance. Some applications (cf. sections 4.1 and 4.2) of three theorems are given. Calculations without an electronic computer are rather tedious.