Comment on A.F. Desmond's paper: Optimal estimating functions, quasi-likelihood and statistical modelling

Abstract

Professor Desmond should be congratulated for providing such a readable review on the theory of estimating function, which has enjoyed and continues to enjoy increasing popularity in statistics as well as in other disciplines of science. In my comment on this paper, I will discuss an application of this estimating function approach that arises in the following scenario. In working with survey data, researchers are often confronted with two data sets. The primary data set contains rich information for each observation. but is very limited in size. In contrast, the secondary data set has only a subset of information contained in the primary data set, but is large in terms of observation. The question is how to optimally combine the auxiliary information from the secondary data set with the information contained in the primary data set. More specifically, let {zi}n=l be n independent realizations of a random variable Z. This (primary) random sample is drawn from the sampled population. with an unknown probability density function (pdf), &(z). Here Z = (Y, X). where Y is a scalar random variable and X is a p-dimensional vector. Our objective is to estimate the parameter 13’ from the generalized linear model Y = X ‘(1” + U, where T denotes a transposition and the variance function of u need not be constant across observations. Let J,(z) be the pdf of the target population, over which distribution we have auxiliary information given by an r-dimensional vector of unbiased estimating functions [k(Z)], i.e., E,,[k(Z)] =O. where k(Z) = k(Z) k”, and E,,[k(Z)] = l%(z)dF,,(z) = k”. These estimating functions represent moments of joint distributions of some variables common to both data sets, the estimates of which can be obtained from the corresponding estimating equations. For the sake of expositional clarity, we initially assume that (i) the target and sample populations are identical, so that f&(z) =.fi,(z) ==f’(z), and E,,[k(z)] = E,,[k(z)] = E[k(z)] =O; and (ii) k, is known. Then the parameter Ho can be estimated by weighted least squares to yield 6 = [I;= l ~ili.~T] ’ J-r= 1 iixiyi. Following Qin and Lawless (1994), the weighting coefficients iii are obtained by maximizing r]:_=, oi subject to restrictions Ui > 0, CT= 1 Ui = 1 and Cr=, Uik(zi) =O. Denoting I as an r-dimensional vector of Lagrange multipliers for the restriction CT=, ajk(zi) =O. the coefficients Ui satisfy