Estimation and Verification of Hypotheses in Some Zyskind‐Martin Models with Missing Values

Several theorems on estimation and verification of linear hypotheses in some Zyskind‐Martin (ZM) models are given. The assumptions are as follows. Let y = X β + e or ( y, X β, σ 2 V ) be a fixed model where y is a vector of n observations, X is a known matrix nXp with rank r(X) = r ≦ p < n , where p is a number of coordinates of the unknown parameter vector β, e is a random vector of errors with covariance matrix σ 2 V , where σ 2 is unknown scalar parameter, V is a known non‐negative definite matrix such that R(X) ⊂ R(V ). Symbol R(A ) denotes a vector space generated by columns of matrix A. The expected value of y is X β. In this paper four following Zyskind‐Martin (ZM) models are considered: ZMd, ZMa, ZMc and ZMqd (definitions in sec. 1) when vector y y 1 y 2 involves a vector y 1 of m missing values and a vector y 2 with ( n — m ) observed values. A special transformation of ZM model gives again ZM model (cf. theorem 2.1). Ten properties of actual (ZMa) and complete (ZMc) Zyskind‐Martin models with missing values (cf. theorem 2.2) test functions F are given in (2.11)) are presented. The third propriety constitutes a generalization of R. A. Fisher's rule from standard model ( y, X β, σ 2 I ) to ZM model. Estimation of vector y 1 (cf. 3.3) of vector β (cf. th. 3.2) and of scalar σ 2 (cf. th. 3.4) in actual ZMa model and in diagonal quasi‐ZM model (ZMqd) are presented. Relation between y̌ 1 and β is given in theorem 3.1. The results of section 2 are illustrated by numerical example in section 4.