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Biometrika 1994 81(3):585-587; doi:10.1093/biomet/81.3.585
© 1994 by Biometrika Trust
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MISCELLANEA

A note on nonregular likelihood functions in heteroscedastic regression models

ADAM CRISP and JIM BURRIDGE

Department of Statistical Science, University College London Gower Street, London WC1E 6BT, U.K.

Received for publication 1 September 1993. Revision received 1 March 1994.
  Abstract

The existence of maximum likelihood estimates for a class of heteroscedastic regression models is considered. For a given dispersion function we show that, under a weak condition, the likelihood is singular at points corresponding to nonreplicated observations, causing unrestricted maximum likelihood estimation to break down, whilst for an alternative class of dispersion functions we obtain a much stronger linear independence condition for the likelihood to be unbounded.

Key Words: Mean-variance relationship • Multimodality • Normal-mixture • Singularity


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