Summary
This paper is concerned with estimation for a subfamily of exponential-type, which is a parametric model with sufficient statistics. The family is associated with a surface in the domain of a sufficient statistic. A new estimator, termed a projection estimator, is introduced. The key idea of its derivation is to look for a one-to-one transformation of the sufficient statistic so that the subfamily can be associated with a flat subset in the transformed domain. The estimator is defined by the orthogonal projection of the transformed statistic onto the flat surface. Here the orthogonality is introduced by the inverse of the estimated variance matrix of the statistic on the analogy of Mahalanobis's notion (1936,Proc. Nat. Inst. Sci. Ind.,2, 49–55). Thus the projection estimator has an explicit representation with no iterations. On the other hand, the MLE and classical estimators have to be sought as numerical solutions by some algorithm with a choice of an initial value and a stopping rule. It is shown that the projection estimator is first-order efficient. The second-order property is also discussed. Some examples are presented to show the utility of the estimator.
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References
Amari, S. (1985).Differential geometrical methods in statistics, Lecture Note in Statistics, Springer, New York.
Anderson, T. W. (1973). Asymptotic efficient estimator of covariance matrices with linear structure,Ann. Statist.,1, 135–141.
Burn, R. (1982). Loglinear models with composite link functions in genetics, GLIM 82:Proc. Internat. Conf. General. Linear Models (ed. R. Girchrist), 144–154, Springer, New York.
Cepellini, R., Siniscalo, M. and Smith, C. A. B. (1955). The estimation of gene frequencies in random mating population,Ann. Hum. Genet.,20, 97–115.
Draper, N. R. and Smith, H. (1968).Applied Regression Analysis, Wiley, New York.
Eguchi, S. (1983). Second order efficiency of minimum contrast estimators in a curved exponential family,Ann. Statist.,11, 793–803.
Eguchi, S. (1984). A characterization of second order efficiency in a curved exponential family,Ann. Inst. Statist. Math.,36, 199–206.
Gurrero, V. M. and Johnson, R. A. (1982). Use of the Box-Cox transformation with binary response models,Biometrika,69, 309–314.
Haberman, S. J. (1977). Product models for frequency tables involving indirect observation,Ann. Statist.,5, 1124–1147.
Mahalanobis, P. C. (1936). On the generalized distance in statistics,Proc. Nat. Inst. Sci. Ind.,2, 49–55.
Rao, C. R. (1972).Linear Statistical Inference and its Applications, Wiley, New York.
Takeuchi, K., Yanai, H. and Mukherjee, B. N. (1982).The Foundations of Multivariate Analysis, Wiley, Tokyo.
Yasuda, N. (1968). Estimation of inbreeding coefficient from phenotype frequencies by a method of maximum likelihood scoring,Biometrics,24, 915–934.
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Eguchi, S. A projection method of estimation for a subfamily of exponential families. Ann Inst Stat Math 38, 385–398 (1986). https://doi.org/10.1007/BF02482525
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DOI: https://doi.org/10.1007/BF02482525