LEVERAGE ADJUSTMENTS FOR DISPERSION MODELLING IN GENERALIZED NONLINEAR MODELS |
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| Authors: | Gordon K Smyth Arūnas P Verbyla |
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| Affiliation: | 1. Bioinformatics Division, Walter and Eliza Hall Institute of Medical Research, 1G Royal Parade, Parkville VIC 3050, Australia.;2. School of Agriculture, Food and Wine, The University of Adelaide, Private Mail Bag 1, Glen Osmond SA 5064, Australia.e‐mail: or;3. Mathematical and Information Sciences, CSIRO, Private Mail Bag 2, Glen Osmond SA 5064, Australia. |
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| Abstract: | For normal linear models, it is generally accepted that residual maximum likelihood estimation is appropriate when covariance components require estimation. This paper considers generalized linear models in which both the mean and the dispersion are allowed to depend on unknown parameters and on covariates. For these models there is no closed form equivalent to residual maximum likelihood except in very special cases. Using a modified profile likelihood for the dispersion parameters, an adjusted score vector and adjusted information matrix are found under an asymptotic development that holds as the leverages in the mean model become small. Subsequently, the expectation of the fitted deviances is obtained directly to show that the adjusted score vector is unbiased at least to O(1/n) . Exact results are obtained in the single‐sample case. The results reduce to residual maximum likelihood estimation in the normal linear case. |
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| Keywords: | double generalized linear models gamma distribution inverse‐Gaussian distribution maximum likelihood modified profile likelihood residual maximum likelihood saddlepoint approximation |
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