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A minimax approach to consistency and efficiency for estimating equations. (English) Zbl 0906.62022
Summary: The consistency of estimating equations has been studied, in the main, along the lines of H. Cramér’s [Mathematical methods of statistics. Princeton N. J.: Princeton University Press (1946; Zbl 0063.01014)] classical argument, which only asserts the existence of consistent solutions. The statement similar to that of J. L. Doob [Trans. Am. Math. Soc. 36, 759–775 (1934; Zbl 0010.17303)] and A. Wald [Ann. Math. Statist. 20, 595–601 (1949; Zbl 0034.22902)], which identifies the consistent solutions, has not yet been established. The obstacle is that the solutions of estimating equations cannot in general be defined as the maximum of likelihood functions.
We demonstrate that the consistent solutions can be identified as the minimax of a function $$R$$, whose properties resemble those of a log likelihood ratio, but which exists in a much wider context. Furthermore, since we do not need $$R$$ to be differentiable, the minimax is consistent even when the estimating equation does not exist. In this respect, the minimax is a new estimator. We first convey the idea by focusing on the quasi-likelihood estimate, and then indicate its full generality by providing a set of sufficient conditions for consistency and studying a number of important cases. Efficiency will also be verified.

##### MSC:
 62F12 Asymptotic properties of parametric estimators 62J12 Generalized linear models (logistic models) 62C20 Minimax procedures in statistical decision theory
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##### References:
  BILLINGSLEY, P. 1968. Convergence of Probability Measures. Wiley, New York. Z. · Zbl 0172.21201  COX, D. R. and HINKLEY, D. V. 1974. Theoretical Statistics. Chapman and Hall, London. · Zbl 0334.62003  CRAMER, H. 1946. Mathematical Methods of Statistics. Princeton Univ. Press. Ź. · Zbl 0060.30513  CROWDER, M. 1986. On consistency and inconsistency of estimating equations. Econometric Theory 3 305 330. Z.  CROWDER, M. 1987. On linear and quadratic estimating functions. Biometrika 74 591 597. Z. JSTOR: · Zbl 0635.62077 · doi:10.1093/biomet/74.3.591 · links.jstor.org  DOOB, J. S. 1934. Probability and statistics. Trans. Amer. Math. Soc. 36 759 775. Z. JSTOR: · JFM 60.0467.02 · doi:10.2307/1989822 · links.jstor.org  FAHRMEIR, L. and KAUFMANN, H. 1985. Consistency and asy mptotic normality of the maximum likelihood estimator in generalized linear models. Ann. Statist. 13 342 368. Z. · Zbl 0594.62058 · doi:10.1214/aos/1176346597  FIRTH, D. and HARRIS, I. R. 1991. Quasi-likelihood for multiplicative random effects. Biometrika 78 545 555. Z. JSTOR: · Zbl 1193.62064 · doi:10.1093/biomet/78.3.545 · links.jstor.org  GODAMBE, V. P. 1960. An optimum property of regular maximum likelihood estimation. Ann. Math. Statist. 31 1208 1211. Z. · Zbl 0118.34301 · doi:10.1214/aoms/1177705693  GODAMBE, V. P. and HEy DE, C. C. 1987. Quasi-likelihood and optimal estimation. Internat. Statist. Rev. 55 231 244. Z. Z JSTOR: · Zbl 0671.62007 · doi:10.2307/1403403 · links.jstor.org  GODAMBE, V. P. and THOMPSON, M. E. 1989. An extension of quasi-likelihood estimation with. discussion. J. Statist. Plann. Inference 22 137 172. Z. · Zbl 0681.62036 · doi:10.1016/0378-3758(89)90106-7  JARRETT, R. G. 1984. Bounds and expansions for Fisher information when moments are known. Biometrika 74 233 245. Z. JSTOR: · Zbl 0549.62024 · doi:10.1093/biomet/71.1.101 · links.jstor.org  LI, B. 1993. A deviance function for the quasi likelihood method. Biometrika 80 741 753. Z. JSTOR: · Zbl 0796.62025 · doi:10.1093/biomet/80.4.741 · links.jstor.org  LI, B. and MCCULLAGH, P. 1994. Potential functions and conservative estimating equations. Ann. Statist. 22 340 356. Z. · Zbl 0805.62003 · doi:10.1214/aos/1176325372  MCCULLAGH, P. 1983. Quasi-likelihood functions. Ann. Statist. 11 59 67. Z. · Zbl 0507.62025 · doi:10.1214/aos/1176346056  MCCULLAGH, P. 1990. Quasi-likelihood and estimating functions. In Statistical Theory and Z. Modelling: In Honour of Sir David Cox D. V. Hinkley, N. Reid and E. J. Snell, eds.. Chapman and Hall, London. Z.  MCCULLAGH, P. and NELDER, J. A. 1989. Generalized Linear Models, 2nd ed. Chapman and Hall, London. Z. · Zbl 0744.62098  MCLEISH, D. L. 1984. Estimation for aggregate models: the aggregate Markov chain. Canad. J. Statist. 12 265 282. Z. JSTOR: · Zbl 0574.62084 · doi:10.2307/3314810 · links.jstor.org  MCLEISH, D. L. and SMALL, C. G. 1988. The Theory and Applications of Statistical Inference Functions. Lecture Notes in Statist. 44. Springer, New York. Z. · Zbl 0654.62001  MCLEISH, D. L. and SMALL, C. G. 1992. A projected likelihood function for semiparametric models. Biometrika 79 93 102. Z. JSTOR: · Zbl 0753.62034 · doi:10.1093/biomet/79.1.93 · links.jstor.org  POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, New York. Z. · Zbl 0544.60045  SERFLING, R. J. 1980. Approximation Theorems of Mathematical Statistics. Wiley, New York. Z. · Zbl 0538.62002  WALD, A. 1949. Note on the consistency of maximum likelihood estimate. Ann. Math. Statist. 20 595 601. Z. · Zbl 0034.22902 · doi:10.1214/aoms/1177729952  WEDDERBURN, R. W. M. 1974. Quasi-likelihood, generalized linear models, and the Gauss Newton method. Biometrika 61 439 447. Z. JSTOR: · Zbl 0292.62050 · links.jstor.org  WOLFOWITZ, J. 1949. On Wald’s proof of the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 601 602. Z. · Zbl 0034.22903 · doi:10.1214/aoms/1177729953  WONG, W. H. 1986. Theory of partial likelihood. Ann. Statist. 14 88 123. Z. · Zbl 0603.62032 · doi:10.1214/aos/1176349844  WOODROOFE, M. 1972. Maximum likelihood estimation of a translation parameter of a truncated distribution. Ann. Math. Statist. 43 113 122. · Zbl 0251.62018 · doi:10.1214/aoms/1177692707  UNIVERSITY PARK, PENNSy LVANIA 16802 E-MAIL: bing@stat.psu.edu
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