Estimation of missing values in the general gauss-maekoff model

Abstract

The problem of estimating parameters in the case of missing values up today was connected with the standard linear models with covariance matrix ?²I (cf. DODGE, 1984) where ?² is an unknown scalar and I-the unit matrix. In this paper the results of the standard model are generalized for the linear GAUSS-MARKOFF (GM) model and ZYSKIND-MABTIN model (particular case of GM) with missing values and with covariance matrix ?²Vt where V≧0.

R. A. FISHEB (I960) and many authors used improper standard models because the covariance matrix introduced by tiiero. corresponded to the covariance matrix witli lull observations but not, as it should be, to the complete model in which the estimator of missing values is substituted in the place of missing observations, In this paper the model with the oroper covariance matrix is considered in the case of the GM model and also in the case of ZYSKIND-MARTIN model (cf. def. 4.1). The invariance of some matrix expressions is proved (cf. lemma 2.4).

In the general GATJSS-MABKOFF model the estimator y₁ of the vector of missing values y₁ is obtained (cf. 3.3) in th. 3.1), This is the generalization of OKTABA and JAGIELLO result (1973, 1982). By using the symmetric c-inverse matrix T* the analogue estimator y₁ is given (th. 3.2, formula (3.28)). Two models are considered: 1) with actual observations and 2) with substitution vector y₁ in missing vector y₁. If 1) is ZYSKIND-MABTIN (1969) model then 2) is the same one (th. 4.1). The quadratic forms (4.13) and (4.14) in models 1) and 2) are identical (generalization of R. A. FISHEB'S (1960) rule) and the solutions ? of the normal equations in 1) and 2) ZYSKIND-MABTIN models are identical. This paper is a preliminary report

Keywords:

• General GAUSS-MABKOFF model
• estimator of vector of missing values
• c-inverse matrix
• non-negative definite matrix
• unified least square theory