A comparison of quasi-likelihood ratios for general estimating functions

General estimating functions are usually used when one desires to conduct inference about a parameter of interest but either robustness with respect to model misspecification is required or the full distribution of the observations is unknown. Unfortunately, the estimating function approach may have limited utility due to, for instance, a Walds test with poor behaviour even for large samples (see, e.g., Jennings, 1986; Heritier and Ronchetti, 1994; Bellio et al., 2008). The use of a quasi-likelihood ratio test derived from estimating functions seems to be appealing since, as the classical likelihood ratio test, it avoids the drawbacks of Wald’s type test. Moreover, a quasi-likelihood function for the parameter of interest is generally derived in order to have a quasi-likelihood ratio test with the usual chi-squared asymptotic distribution (see McCullagh, 1991; Barndorff-Nielsen, 1995; Adimari and Ventura, 2002). The aim of this paper is to compare two different methods of constructing a quasi-likelihood ratio and to show how they are related. The first method is based on the quasi-likelihood approach discussed in Hanfelt and Liang (1995), while the second one is based on a suitable modification of a general estimating function for a parameter of interest in order to achieve the usual first order asymptotic theory (Adimari and Ventura, 2002). The two techniques are illustrated and compared with applications and simulation studies.