Pardo Llorente, Leandro and Martín Apaolaza, Níriam
(2011)
*On the comparison of the pre-test and shrinkage phi-divergence test estimators for the symmetry model of categorical data.*
Journal of Computational and Applied Mathematics, 235
(5).
pp. 1160-1179.
ISSN 0377-0427

PDF
Restricted to Repository staff only until 31 December 2020. 340kB |

Official URL: http://www.sciencedirect.com/science/article/pii/S0377042710004243

## Abstract

The estimation problem of the parameters in a symmetry model for categorical data has been considered for many authors in the statistical literature (for example, Bowker (1948) [1], Ireland et al. (1969) [2], Quade and Salama (1975) [3] Cressie and Read (1988) [4], Menendez et al. (2005) [5]) without using uncertain prior information. It is well known that many new and interesting estimators, using uncertain prior information, have been studied by a host of researchers in different statistical models, and many papers have been published on this topic (see Saleh (2006) [9] and references therein). In this paper, we consider the symmetry model of categorical data and we study, for the first time, some new estimators when non-sample information about the symmetry of the probabilities is considered. The decision to use a "restricted" estimator or an "unrestricted" estimator is based on the outcome of a preliminary test, and then a shrinkage technique is used. It is interesting to note that we present a unified study in the sense that we consider not only the maximum likelihood estimator and likelihood ratio test or chi-square test statistic but we consider minimum phi-divergence estimators and phi-divergence test statistics. Families of minimum phi-divergence estimators and phi-divergence test statistics are wide classes of estimators and test statistics that contain as a particular case the maximum likelihood estimator, likelihood ratio test and chi-square test statistic. In an asymptotic set-up, the biases and the risk under the squared loss function for the proposed estimators are derived and compared. A numerical example clarifies the content of the paper.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Minimum phi-divergence estimator; Phi-divergence statistics; Preliminary test estimator; Symmetry model; Marginal homogeneity; Contingency-tables |

Subjects: | Sciences > Mathematics > Mathematical statistics |

ID Code: | 17332 |

References: | A. Bowker, A test for symmetry in contingency tables, Journal of the American Statistical Association 43 (1948) 572–574. C.T. Ireland, H.H. Ku, S. Kullback, Symmetry and marginal homogeneity of an r×r contingency table, Journal of the American Statistical Association 64 (1969) 1323–1341. D. Quade, I.A. Salama, A note on minimum chi-square statistics in contingency tables, Biometrics 31 (1975) 953–956. T.R.C. Read, N.A.C. Cressie, Goodness of Fit Statistics for Discrete Multivariate Data, Springer-Verlag, New York, 1988. M.L. Menéndez, J.A. Pardo, L. Pardo, K. Zografos, On tests of symmetry, marginal homogeneity and quasi-symmetry in two contingency tables based on minimum φ-divergence estimator with constraints, Journal of Statistical Computation and Simulation 75 (7) (2005) 555–580. T.A. Bancroft, On biases in estimation due to the use of preliminary tests of significance, Annals of Mathematical Statistics 15 (1944) 190–204. C.P. Han, T.A. Bancroft, On pooling means when variance is unknown, Journal of the American Statistical Association 63 (1986) 1333–1342. C. Stein, Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, in: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California Press, Berkeley, CA, 1956, pp. 197–206. A.K.Md.E. Saleh, Theory of Preliminary Test and Stein-Type Estimation with Applications, Wiley, 2006. D. Morales, L. Pardo, I. Vajda, Asymptotic divergences of estimates of discrete distributions, Journal of Statistical Planning and Inference 48 (1995) 347–369. S.M. Ali, S.D. Silvey, A general class of coefficients of divergence of one distribution from another, Journal of the Royal Statistical Society, Series B 26 (1966) 131–142. I. Csiszàr, Eine informationstheorestiche Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten, Publications of the Mathematical Institute of Hungarian Academy of Sciences, Series A 8 (1963) 84–108. L. Pardo, Statistical Inference Based on Divergence Measures, Chapman & Hall, CRC, New York, 2006. M.L. Menéndez, L. Pardo, M.C. Pardo, Preliminary test estimators and phi-divergence measures in generalized linear models with binary data, Journal of Multivariate Analysis 99 (2008) 2265–2284. L. Pardo, M.L. Menéndez, On some pre-test and Stein-rule phi-divergence test estimators in the independence model of categorical data, Journal of Statistical Planning and Inference 138 (2007) 2163–2179. D.V. Glass, Social Mobility in Britain, Free Press, Glencoe, Illinois, 1954. M.L. Menéndez, J.A. Pardo, L. Pardo, Tests based on phi-divergences for bivariate symmetry, Metrika 53 (2001) 15–29. |

Deposited On: | 05 Dec 2012 09:18 |

Last Modified: | 05 Dec 2012 17:47 |

Repository Staff Only: item control page