Abstract
When the variance is a known function of the mean, as in quasi-likelihood 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 quasi-likelihood estimators. For a substantial range of models, the extended estimates offer little improvement when the coefficient of variation is small.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amari, S. (1982). Differential geometry of curved exponential families—curvatures and information loss,Ann. Statist.,10, 357–387.
Amari, S. (1985). Differential-geometrical methods in statistics,Lecture Notes in Statist.,28, Springer, New York.
Bhapkar, V. P. (1972). On a measure of efficiency of an estimating equation,Sankyā Ser. A,34, 467–472.
Bhapkar, V. P. (1991). Sufficiency, ancillarity, and information in estimating functions,Estimating Functions, (ed. V. P. Godambe), 239–254, Oxford Science Publications, Clarendon Press, Oxford.
Crowder, M. J. (1987). On linear and quadratic estimating functions,Biometrika,74, 591–597.
Efron, B. (1975). Defining the curvature of a statistical problem (with discussion),Ann. Statist.,3, 1189–1242.
Efron, B. (1978). The geometry of exponential families,Ann. Statist.,6, 362–376.
Efron, B. (1982). Maximum likelihood and decision theory,Ann. Statist.,10, 340–356.
Firth, D. (1987). On the efficiency of quasi-likelihood estimation,Biometrika,74, 233–245.
Fisher, R. A. (1925). Theory of statistical estimation,Proc. Camb. Phil. Soc.,22, 700–725.
Godambe, V. P. and Kale, B. K. (1991). Estimating functions: an overview,Estimating Functions, (ed. V. P. Godambe), 1–20, Oxford Science Publications, Clarendon Press, Oxford.
Godambe, V. P. and Thompson, M. E. (1989). An extension of quasi-likelihood estimation (with discussion),J. Statist. Plann. Inference,22, 137–172.
Hinkley, D. (1980). Theory of statistical estimation: the 1925 paper in R. A. Fisher: An Appreciation,Lecture Notes in Statist.,1, 85–94, Springer, New York.
Kass, R. E. (1989). The geometry of asymptotic inference,Statist. Sci.,4, 188–234.
Lee, Y. and Nelder, J. A. (1992). The robustness of estimators from quasi likelihood and quadratic estimating functions (unpublished manuscript).
McCullagh, P. (1983). Quasi-likelihood functions,Ann. Statist.,11, 59–67.
McCullagh, P. (1984). Generalized linear models,European J. Oper. Res.,16, 285–292.
McCullagh, P. and Nelder, J. A. (1989).Generalized Linear Models, 2nd ed., Chapman and Hall, New York.
Nelder, J. A. (1989). Comments C to Godambe and Thompson's paper,J. Statist. Plann. Inference,22, 155–158.
Spiegel, M. R. (1968).Mathematical Handbook of Formulas and Tables, Schaum's Outline Series, McGraw-Hill, New York.
Vos, P. W. (1992). Minimumf-divergence estimators and quasi-likelihood functions,Ann. Inst. Statist. Math.,44, 261–279.
Wedderburn, R. W. M. (1974). Quasi-likelihood functions, generalized linear models, and the gauss-newton method,Biometrika,61, 439–447.
Author information
Authors and Affiliations
About this article
Cite this article
Vos, P.W. Quasi-likelihood or extended quasi-likelihood? An information-geometric approach. Ann Inst Stat Math 47, 49–64 (1995). https://doi.org/10.1007/BF00773411
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00773411