Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series A, Pt. 3, 431--443

ON THE EXISTENCE OF THE MAXIMUM LIKELIHOOD ESTIMATE IN VARIANCE COMPONENTS MODELS

By

EUGENE DEMIDENKO, * Dartmouth Medical School, New Hampshire*

and

HÈLÉNE MASSAM,
* York University, Toronto and University of Virginia, Charlottesville*

SUMMARY. We consider the variance components model} $ {\bf \by = \bX\bbeta} +\sum_{i=1}^{r} \bZ_{i}\bu_{i}+ \eepsilon $ {\footnotesize where the random effects $\bu_{i}$ are normally distributed as ${\cal N}(0, \sigma _{i}^{2} \bI_{k_{i}}),i=1,...r$ and the common random error} $\eepsilon$ {\footnotesize is normally distributed as ${\cal N}({\bf 0,}\sigma _{0}^{2}{\bI}_{n}).$ The parameters $\{$}$\bbeta,${\footnotesize $\sigma _{0}^{2},...,\sigma _{r}^{2}\}$ are constrained to satisfy the conditions $\sigma _{0}^{2}>0,$ $\sigma_{i}^{2}\geq 0,i=1,...,r$. The main results in this paper are Theorem 3.1 and Theorem 3.4 giving, respectively, necessary and sufficient conditions for the existence of the maximum likelihood estimate and the restricted maximum likelihood estimate of the parameters.

*AMS (1991) subject classification.* 62H12.

*Key words and phrases. *Variance components, existence, maximum likelihood.