Exact and approximate REML for heteroscedastic regression

Exact REML for heteroscedastic linear models is compared with a number of approximate REML methods which have been proposed in the literature, especially with the methods proposed by Lee and Nelder (LN98) and Smyth and Verbyla (SV99) for simultaneous mean-dispersion modelling in generalized linear models. It is shown that neither of the LN98 or SV99 methods reduces to REML in the normal linear case. Asymptotic variances and efficiencies are obtained for these and other estimators of the same general form. A new algorithm is suggested, similar to one suggested by Huele et al., which returns the correct REML estimators and an improved approximation to the standard errors. It is possible to obtain REML estimators by alternating between two generalized linear models but the final fitted generalized linear model objects will not return the correct standard errors for the variance coefficients. The true REML likelihood calculations therefore fit only partially into the double generalized linear model framework.