Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series B, Pt. 1, pp. 43--55

QUADRATIC ESTIMATING EQUATIONS FOR THE ESTIMATION OF REGRESSION AND DISPERSION PARAMETERS IN THE ANALYSIS OF PROPORTIONS

By

SUDHIR R. PAUL
*University of Windsor, Ontario, Canada*

SUMMARY. In the analysis of proportions often interest is in the estimation of the mean or the regression parameters. The dispersion parameter then plays the role of a nuisance parameter. However , in some instances, in Toxicology and other similar fields, the dispersion parameter or the intraclass correlation parameter is of primary interest. For example, in some binary-data situations the intraclass correlation is interpreted as `heritability of a dichotomous trait'. So, efficient and possibly robust estimation of the dispersion parameter or the intraclass correlation is important. Marginal or conditional estimation of the dispersion parameter is difficult. So we consider joint estimation of the mean( regression) parameters and the dispersion parameter. We consider joint estimation using quadratic estimating functions (QEEs) of Crowder (1987). By varying the coefficients of the QEEs we obtain five sets of estimating equations. We compare large sample relative efficiency of the five sets of estimates obtained by the QEE's and the quasi-likelihood estimates with the maximum likelihood estimates. Estimated large sample relative efficiencies of these estimates are also compared for two real life data sets arising from biostatistical practices. These comparisons show that estimates, using optimal quadratic estimating functions of Crowder (1987) are highly efficient and are the best among all estimates investigated.

*AMS (1991) subject classification.* 62A10, 62F10, 62F12.

*Key words and phrases. *Dispersion parameter, efficiency, Gaussian
likelihood, joint estimation, optimal estimating equations.