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The use of generalized inverses in restricted maximum likelihood. (English) Zbl 0609.62081
The calculus of generalized inverses and related concepts in matrix algebra is applied to the general restricted maximum likelihood problem. Some new results on g-inverses, Kronecker products, and matrix differentials are presented. For the restricted maximum likelihood problem we obtain generalizations of the well-known results of J. Aitchison and S. D. Silvey [Ann. Math. Stat. 29, 813-828 (1958; Zbl 0092.367)].
We use the approach recently developed by R. D. H. Heijmans and J. R. Magnus [Asymptotic properties of maximum likelihood estimators in the nonlinear regression model when the errors are neither independent nor identically distributed. Rep. AE 20/82, Univ. Amsterdam (1982); On the asymptotic normality of the maximum likelihood estimator with dependent observations. Rep. AE 14/83, Univ. Amsterdam (1983)] to allow for non-i.i.d. observations. A nonlinear seemingly unrelated regressions model with possibly singular covariance matrix and linear restrictions is analyzed, and the linear expenditure system is discussed as a special case.

MSC:
62H12 Estimation in multivariate analysis
62H10 Multivariate distribution of statistics
15A09 Theory of matrix inversion and generalized inverses
15A69 Multilinear algebra, tensor calculus
62P20 Applications of statistics to economics
62E20 Asymptotic distribution theory in statistics
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