The quasi-likelihood function proposed by Wedderburn broadened the
scope of generalized linear models by specifying the variance function in-stead
of the entire distribution. However, complete specification of variance
functions in the quasi-likelihood approach may not be realistic. We define a
nonparametric quasi-likelihood by replacing the specified variance function in
the conventional quasi-likelihood with a nonparametric variance function
estimate. This nonparametric variance function estimate is based on squared
residuals from an initial model fit. The rate of convergence of the
nonparametric variance function estimator is derived. It is shown that the
asymptotic limiting distribution of the vector of regression parameter
estimates is the same as for the quasi-likelihood estimates obtained under
correct specification of the variance function, thus establishing the
asymptotic efficiency of the nonparametric quasi-likelihood estimates. We
propose bandwidth selection strategies based on deviance and Pearson’s
chi-square statistic. It is demonstrated in simulations that for finite samples
the proposed nonparametric quasi-likelihood method can improve upon extended
quasi-likelihood or pseudo-likelihood methods where the variance function is
assumed to fall into a parametric class with unknown parameters. We illustrate
the proposed methods with applications to dental data and cherry tree data.