Abstract
In continuous parametrized models with i.i.d. observations we consider finite quantizations. We study asymptotic properties of the estimators minimizing disparity between the observed and expected frequencies in the quantization cells, and asymptotic properties of the goodness of fit tests rejecting the hypotheses when the disparity is large. The disparity is measured by an appropriately generalized φ-divergence of probability distributions so that, by the choice of function φ, one can control the properties of estimators and tests. For bounded functions φ these procedures are robust. We show that the inefficiency of the estimators and tests can be measured by the decrease of the Fisher information due to the quantization. We investigate theoretically and numerically the convergence of the Fisher informations. The results indicate that, in the common families, the quantizations into 10–20 cells guarantees “practical efficiency” of the quantization-based procedures. These procedures can at the same time be robust and numerically considerably simpler than similar procedures using the unreduced data.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Basu A, Sarkar S (1994a) Minimum disparity estimation in the errors-invariables model. Statistics and Probability Letters 20:69–73
Basu A, Sarkar S (1994b) The trade-off between robustness and efficiency and the effect of model smoothing. Journal of Statistics, Computation and Simulation 50:173–185
Birch MW (1964) A new proof of the Pearson-Fisher theorem. Annals of Mathematical Statistics 35:817–824
Brent RP (1973) Algorithms for minimization without derivatives. Prentice Hall
Brown BW, Lovato J, Russell K (1994) Library of routines for cumulative distribution functions, inverses, and other parameters. http://www.stat.unipg.it/stat/dcdflib/
Cheng SW (1975) A unified approach to choosing optimum quantiles for the ABLEs. Journal of the American Statistical Association 70:155–159
Csiszár I (1963) Eine informationstheoretische ungleichung und ihre anwendung auf den beweis der ergodizität on markoffschen ketten. Publications of the Mathematical Institute of the Hungarian Academy of Sciences 8, Ser. A, 84–108
Csiszár I (1967) Information-type measured of difference of probability distributions and indirect observations. Studia Scientiarium Mathematica Hungarica 2:299–318
Ghurye SG, Johnson BR (1981) Discrete approximations to the information integral. The Canadian Journal of Statistics 9:27–37
Inglot T, Jurlewicz T, Ledwina T (1991) Asymptotics for multinomial goodness of fit tests for simple hypotheses. Theory of Probability and Applications 35:771–777
Liese F, Vajda I (1987) Convex statistical distances. Leipzig, Teubner
Lindsay BG (1994) Efficiency versus robustness: The case of minimum Hellinger distance and other methods. Annals of Statistics 22:1081–1114
Menéndez ML, Morales D, Pardo L, Vajda I (1998) Two approaches to grouping of data and related disparity statistics. Communications in Statistics-Theory and Methods 27:609–633
Menéndez ML, Morales D, Pardo L, Vajda I (2001a) Minimum disparity estimators for discrete and continuous models. Applications of Mathematics (in print)
Menéndez ML, Morales D, Pardo L, Vajda I (2001b) Approximations to powers of φ-disparity goodness of fit tests. Communications in Statistics-Theory and Methods 30:105–134
Millar PW (1984) A general approach to the optimality of minimum distance estimators. Transactions of the American Mathematical Society 286:377–418
Morales D, Pardo L, Vajda I (1995) Asymptotic divergence of estimates of discrete distributions. Journal of Statistical Planning and Inference 48:347–369
Park Ch, Basu A, Basu S (1995) Robust minimum distance inference based on combined distances. Communications in Statistics — Simulation and Computation 24:653–673
Pötzelberger K, Felsenstein K (1993) On the Fisher information of discretized data. Journal of Statistics, Computation and Simulation 46:125–144
Powell MJD (1964) An efficient method for finding the minimum of a function of several variables without calculating derivatives. Computation Journal 7:155–162
Rao CR (1961) Asymptotic efficiency and limiting information. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1:531–545. Berkeley, CA, Berkeley University Press
Read RC, Cressie NAC (1988) Goodness-of-fit statistics for discrete multivariate data. New York, Springer Verlag
Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York
Tsaridis Ch, Zografos K, Ferentinos K (1997) Fisher’s information matrix and divergence for finite and optimal partitions of the sample space. Communications in Statistics-Theory and Methods 26:2271–2289
Vajda I (1973) χ α-divergence and generalized Fisher information. Trans. of the Sixth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes. Prague, Academia, pp. 223–234
Vajda I (1989) Theory of statistical inference and information. Boston, Kluwer Academic Publishers
Vajda I (1995) Conditions equivalent to consistency of approximate MLE’s for stochastic processes. Stochastic Processes and their Applications 56:35–56
Vajda I (2001) On the convergence of information contained in quantized observations. IEEE Transactions on Information Theory (in print)
Vajda I, Janžura M (1997) On asymptotically optimal estimates for general observations. Stochastic Processes and their Applications 72:27–45
Zografos K, Ferentinos K, Papaioannou T (1986) Discrete approximations to the Csiszá r, Rényi, and Fisher measures of information. The Canadian Journal of Statistics 14:355–366
Author information
Authors and Affiliations
Corresponding author
Additional information
upported by the grants A1075101 and GV99-159-1-01.
Rights and permissions
About this article
Cite this article
Mayoral, A.M., Morales, D., Morales, J. et al. On efficiency of estimation and testing with data quantized to fixed number of cells. Metrika 57, 1–27 (2003). https://doi.org/10.1007/s001840100178
Published:
Issue Date:
DOI: https://doi.org/10.1007/s001840100178