Publication: Minimum divergence estimators based on grouped data
Loading...
Full text at PDC
Publication Date
2001-06
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
Citations
Exportar
Abstract
The paper considers statistical models with real-valued observations i.i.d. by F(x, theta (0)) from a family of distribution functions (F(x, theta); theta is an element of Theta), Theta subset of R-s, s greater than or equal to 1. For random quantizations defined by sample quantiles (F-n(-1)(lambda (1)),..., F-n(-1)(lambda (m-1))) of arbitrary fixed orders 0 < <lambda>(1) < ... < lambda (m-1) < 1, there are studied estimators <theta>(phi ,n) of theta (0) which minimize phi -divergences of the theoretical and empirical probabilities. Under an appropriate regularity, all these estimators are shown to be as efficient (first order, in the sense of Rao) as the MLE in the model quantified nonrandomly by (F-1(lambda (1), theta (0)),..., F-1(lambda (m-1), theta (0))). Moreover, the Fisher information matrix I-m(theta (0), lambda) of the latter model with the equidistant orders lambda = (lambda (j) = j/m : 1 less than or equal to j less than or equal to m-1) arbitrarily closely approximates the Fisher information F(theta (0)) of the original model when m is appropriately large. Thus the random binning by a large number of quantiles of equidistant orders leads to appropriate estimates of the above considered type.
Description
UCM subjects
Unesco subjects
Keywords
Citation
Birch, M. W. (1964). A new proof of the Pearson-Fisher theorem, Ann. Math. Statist., 35, 817-824.
Bofinger, E. (1973). Goodness-of-fit using sample quantiles, J. Roy. Statist. Soc. Ser. B, 35, 277-284.
Cheng, R. C. H. (1975). A unified approach to choosing optimum quantiles for the ABLE's, J. Amer. Statist. Assoc., 70, 155-159.
Cressie, N. A. C. and Read, R. C. (1984). Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. Ser. B, 46, 440-464.
Devroye, L., Györfi, L. and Lugosi, G. (1996). A Probabilistic Theory of Pattern Recognition, Springer, New York.
Ferentinos, K. and Papaioannou, T. (1979). Loss of information due to groupings, Transactions of the Eighth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, 87-94, Prague Academia.
Liese, F. and Vajda, I. (1987). Convex Statistical Distances, Teubner, Leipzig.
Lindsay, B. G. (1994). Efficiency versus robustness: The case for minimum Hellinger distance and other methods, Ann. Statist., 22, 1081-1114.
Menéndez, M. L., Morales, D. and Pardo, L. (1997). Maximum entropy principle and statistical inference on condensed ordered data, Statist. Probab. Lett., 34, 85-93.
Menéndez, M. L., Morales, D., Pardo, L. and Vajda, I. (1998). Two approaches to grouping of data and related disparity statistics, Comm. Statist. Theory Methods, 27(3), 609-633.
Morales, D., Pardo, L. and Vajda, I. (1995). Asymptotic divergence of estimates of discrete distributions, J. Statist. Plann. Inference, 48, 347-369.
Nagahata, H. (1985). Optimal spacing for grouped observations from the information view-point, Mathematica Japonica, 30, 277-282.
Neyman, J. (1949). Contribution to the theory of the X 2 test, Proceeding of the First Berkeley Symposium on Mathematical Statistics and Probability, 239-273. Berkeley University Press, Berkeley, California.
Pötzelberger, K. and Felsenstein, K. (1993). On the Fisher information of discretized data, J. Statist. Comput. Simulation, 46, 125-144.
Rao, C. R. (1961). Asymptotic efficiency and limiting information, Proc. Fourth Berkeley Symp. on Math. Statist. Prob., Vol. 1, 531-545, Berkeley University Press, Berkeley, California.
Rao, C. R. (1973) Linear Statistical Inference and Its Applications, 2nd ed., Wiley, New York.
Tsairidis, Ch., Zografos, K. and Ferentinos, T. (1998). Fisher's information matrix and divergence for finite optimal partitions of the sample space,Comm. Statist. Theory Methods., 26(9), 2271-2289.
Vajda, I. (1973). X 2-divergence and generalized Fisher information, Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, 223-234, Prague Academia.
Vajda, I. (1989). Theory of Statistical Inference and Information, Kluwer, Boston.