QUASI-LIKELIHOOD OR EXTENDED QUASI-LIKELIHOOD? AN INFORMATION-GEOMETRIC APPROACH

PAUL W. VOS

Department of Biostatistics, East Carolina University, Greenville, North Carolina 27858, U.S.A.

(Received November 30, 1992; revised May 24, 1994)

Abstract. When the variance is a known function of the mean, as in quasilikelihood applications, the sample variance also contains information about the mean and extensions of quasi-likelihood functions have been suggested that incorporate this additional information. In order to be sure these extensions are an improvement, further assumptions are made typically on the higher moments of the data so that there is a trade-off between the greater robustness of the quasi-likelihood estimates and the potentially improved estimates based on the extended quasi-likelihood functions. Improvement is often measured by relative efficiency but more insight can be gained by considering optimality of estimating functions, information loss, and sufficiency. All these measures can be described using the dual geometries of the quasi- and extended quasilikelihood estimators. For a substantial range of models, the extended estimates offer little improvement when the coefficient of variation is small.

Key words and phrases: Information, sufficiency, efficiency, extended quasilikelihood, generalized linear model, dual geometries, angle, curvature.