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Target estimation for bias and mean square error reduction. (English) Zbl 0986.62013

Summary: Given a statistical functional \(T\) and a parametric family of distributions, a bias reduced functional \(\widetilde T\) is defined by setting the expected value of the statistic equal to the observed value. Under certain regularity conditions this new statistic, called the target estimator, will have smaller bias and mean square error than the original estimator. The theoretical aspects are analyzed by using higher-order von Mises expansions. Several examples are given, including \(M\)-estimates of location and scale. The procedure is applied to an autoregressive model, the errors-in-variables model and the logistic regression model. A comparison with the jackknife and the bootstrap estimators is also included.

MSC:

62F10 Point estimation
62G05 Nonparametric estimation
62G99 Nonparametric inference
62E20 Asymptotic distribution theory in statistics

Software:

bootstrap
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References:

[1] CABRERA, J., MEER, P. and LEUNG, H. 1994. Simulation methods for bias correction in ellipse estimation. In Proceedings of the Seattle Symposium on Computer Vision and Performance. Z.
[2] CABRERA, J. and MEER, P. 1996. Unbiased estimation of ellipses by bootstrapping. IEEE Trans. Pattern Analysis and Machine Intelligence 18 752 756. Z.
[3] CABRERA, J. and WATSON, G. S. 1997. Simulation methods for mean and median bias reduction in parametric estimation. J. Statist. Plann. Inference 57 143 152. Z. · Zbl 0880.62021
[4] DEBICHE, M. G. and WATSON, G. S. 1996. Confidence limits and bias correction for estimating angles between directions with applications to palaeomagnetism. J. Geophysical Research 100 24, 405 24, 429. Z.
[5] EFRON, B. 1982. The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia. Z. · Zbl 0496.62036
[6] EFRON, B and TIBSHIRANI, R. J. 1993. An Introduction to the Bootstrap. Chapman and Hall, New York. Z. · Zbl 0835.62038
[7] FANKHAUSER, E. 1996. Etude du developpement de von Mises au deuxieme ordre. Diploma thesis. Ecole Polytechnique Federale, Lausanne, Switzerland. Z.
[8] FERNHOLZ, L. T. 1983. Von Mises Calculus for Statistical Functionals. Lecture Notes in Statist. 19. Springer, New York. Z. · Zbl 0525.62031
[9] FERNHOLZ, L. T. 1996. On higher order von Mises expansions. Technical Report 96-04, Dept. Statistics, Temple Univ.
[10] FILLIPOVA, A. A. 1962. Mises theorem on the asymptotic behavior of functionals of empirical distribution function and its statistical applications. Theory Prob. Appl. 7, pp. 24 57. Z. · Zbl 0118.14501
[11] GATTO, R. and RONCHETTI, E. 1996. General saddlepoint approximations of marginal densities and tail probabilities. J. Amer. Statist. Assoc. 91 666 673. Z. JSTOR: · Zbl 0869.62017
[12] HAMPEL, F. 1974. The influence curve and its role in robust estimation. J. Amer. Statist. Assoc. 69 383 393. Z. JSTOR: · Zbl 0305.62031
[13] HAMPEL, F., RONCHETTI, E., ROUSSEEUW, P. and STAHEL, W. 1986. Robust Statistics. The Approach Based on the Influence Function. Wiley, New York. Z.
[14] HUBER, P. 1981. Robust Statistics. Wiley, New York. Z. · Zbl 0536.62025
[15] REEDS, J. A. 1976. On the definition of von Mises functionals. Ph.D. dissertation, Harvard Univ. Z.
[16] ROUSSEEUW, P. and RONCHETTI, E. 1981. Influence curves of general statistics. J. Comput. Appl. Math. 7 161 166. Z. · Zbl 0472.62046
[17] SCHUCANY, W. R., GRAY, H. R., AND OWEN, D. B. 1971. On Bias Reduction in Estimation. J. Amer. Statist. Assoc., 66, pp. 524 533. Z. · Zbl 0226.62022
[18] VON MISES, R. 1947. On the asymptotic distribution of differentiable statistical functions. Ann. Math. Statist. 18, pp. 309 348. · Zbl 0037.08401
[19] NEW BRUNSWICK, NEW JERSEY 08903 SPEAKMAN HALL E-MAIL: CABRERA@STAT.RUTGERS.EDU PHILADELPHIA, PENNSYLVANIA 19122 E-MAIL: fernholz@sbm.temple.edu
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