Previous |  Up |  Next


divergence; disparity; minimum disparity estimators; robustness; asymptotic efficiency
Disparities of discrete distributions are introduced as a natural and useful extension of the information-theoretic divergences. The minimum disparity point estimators are studied in regular discrete models with i.i.d. observations and their asymptotic efficiency of the first order, in the sense of Rao, is proved. These estimators are applied to continuous models with i.i.d. observations when the observation space is quantized by fixed points, or at random, by the sample quantiles of fixed orders. It is shown that the random quantization leads to estimators which are robust in the sense of Lindsay [9], and which can achieve the efficiency in the underlying continuous models provided these are regular enough.
[1] A. Basu, S.  Sarkar: Minimum disparity estimation in the errors-in-variables model. Statist. Probab. Lett. 20 (1994), 69–73. DOI 10.1016/0167-7152(94)90236-4 | MR 1294806
[2] A. Basu, S.  Sarkar: The trade-off between robustness and efficiency and the effect of model smoothing. J.  Statist. Comput. Simulation 50 (1994), 173–185. DOI 10.1080/00949659408811609
[3] M. W. Birch: A new proof of the Pearson-Fisher theorem. Ann. Math. Statist. 35 (1964), 817–824. DOI 10.1214/aoms/1177703581 | MR 0169324 | Zbl 0259.62017
[4] E. Bofinger: Goodness-of-fit using sample quantiles. J.  Roy. Statist. Soc. Ser.  B 35 (1973), 277–284. MR 0336896
[5] H. Cramér: Mathematical Methods of Statistics. Princeton University Press, Princeton, 1946. MR 0016588
[6] N. A. C. Cressie, R. C.  Read: Multinomial goodness-of-fit tests. J.  Roy. Statist. Soc. Ser.  B 46 (1984), 440–464. MR 0790631
[7] R. A. Fisher: Statistical Methods for Research Workers (8th edition). London, 1941.
[8] F.  Liese, I.  Vajda: Convex Statistical Distances. Teubner, Leipzig, 1987. MR 0926905
[9] B. G. Lindsay: Efficiency versus robutness: The case for minimum Hellinger distance and other methods. Ann. Statist. 22 (1994), 1081–1114. DOI 10.1214/aos/1176325512 | MR 1292557
[10] M. L.  Menéndez, D.  Morales, L.  Pardo and I. Vajda: Two approaches to grouping of data and related disparity statistics. Comm. Statist. Theory Methods 27 (1998), 609–633. DOI 10.1080/03610929808832117 | MR 1619038
[11] M. L.  Menéndez, D.  Morales, L.  Pardo and I. Vajda: Minimum divergence estimators based on grouped data. Ann. Inst. Statist. Math. 53 (2001), 277–288. DOI 10.1023/A:1012466605316 | MR 1841136
[12] D. Morales, L.  Pardo and I. Vajda: Asymptotic divergence of estimates of discrete distributions. J.  Statist. Plann. Inference 48 (1995), 347–369. DOI 10.1016/0378-3758(95)00013-Y | MR 1368984
[13] J. Neyman: Contribution to the theory of the $\chi ^2$ test. In: Proc. Berkeley Symp. Math. Statist. Probab., Berkeley, CA, Berkeley University Press, Berkeley, 1949, pp. 239–273. MR 0028003
[14] Ch.  Park, A.  Basu and S. Basu: Robust minimum distance inference based on combined distances. Comm. Statist. Simulation Comput. 24 (1995), 653–673. DOI 10.1080/03610919508813265
[15] C. R. Rao: Asymptotic efficiency and limiting information. In: Proc. 4th Berkeley Symp. Math. Stat. Probab., Berkeley, CA, Berkeley University Press, Berkeley, 1961, pp. 531–545. MR 0133192 | Zbl 0156.39802
[16] C. R.  Rao: Linear Statistical Inference and its Applications (2nd edition). Wiley, New York, 1973. MR 0346957
[17] R. C. Read, N. A. C.  Cressie: Goodness-of-fit Statistics for Discrete Multivariate Data. Springer-Verlag, New York, 1988. MR 0955054
[18] C. A. Robertson: On minimum discrepancy estimators. Sankhyä Ser. A 34 (1972), 133–144. MR 0331606 | Zbl 0266.62021
[19] I.  Vajda: $\chi ^2$-divergence and generalized Fisher information. In: Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, Academia, Prague, 1973, pp. 223–234. MR 0356302 | Zbl 0297.62003
[20] I. Vajda: Theory of Statistical Inference and Information. Kluwer Academic Publishers, Boston, 1989. Zbl 0711.62002
[21] B. L. van der Waarden: Mathematische Statistik. Springer-Verlag, Berlin, 1957.
[22] K. Pearson: On the criterion that a given system of deviations from the probable in the case of correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine 50 (1990), 157–172.
[23] I. Csiszár: Eine Informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, Series A 8 (1963), 85–108. MR 0164374
[24] M. S. Ali, S. D. Silvey: A general class of coefficients of divergence of one distribution from another. J.  Roy. Statist. Soc. Ser. B 28 (1966), 131–140. MR 0196777
[25] A. Rényi: On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Probability Theory and Mathematical Statistics, Vol. 1, University of California Press, Berkeley, 1961, pp. 531–546. MR 0132570
[26] A. W. Marshall, I. Olkin: Inequalities: Theory of Majorization and its Applications. Academic Press, New York, 1979. MR 0552278
Partner of
EuDML logo