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Exact and asymptotic distributions of the maximum likelihood estimate in a simple factor analysis model. (English) Zbl 0698.62058

Summary: The exact and asymptotic distributions of the maximum likelihood estimate of a factor loading are studied. We study the distribution in a factor analysis model with two normally distributed observations and one factor. The calculation of the exact density utilizes the density of the Wishart distribution that was studied by E. Sverdrup [Skand. Aktuar. Tidskr. 30, 151-168 (1947)] for the central case.
First, we derive the exact density of the maximum likelihood estimate of the factor loading. Second, we propose to use an arc tan transformation of the factor-loading estimate to improve the approximation to the normal density. Finally, we compare the exact densities with the densities obtained in cases of large sample sizes. Graphical displays of the exact densities and the densities of the asymptotic distribution are presented for a specific example.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
62E15 Exact distribution theory in statistics
62E20 Asymptotic distribution theory in statistics
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