Felipe N. Miranda L. Regular Article First Online: 04 July This is a preview of subscription content, log in to check access. Acknowledgements We are very grateful to the associate editor as well as the anonymous referees for fruitful comments and remarks that have improved the final version of the paper. Abar B, Loken E Self-regulated learning and self-directed study in a pre-college sample. Agresti A An introduction to categorical data analysis. Berkson J Minimum chi-square, not maximum likelihood! Biemer P Latent class analysis and survey error. Birch M A new proof of the Pearson-Fisher theorem.

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Bryant F, Satorra A Principles and practice of scaled difference chi-square testing. Clogg C Latent class models for measuring. In: Latent trait and class models. Collins L, Lanza S Latent class and latent transition analysis for the social, behavioral, and health sciences. Wiley, New York Google Scholar.

Cressie N, Pardo L Minimum phi-divergence estimator and hierarchical testing in loglinear models. Following Basu et al. We can treat the two observations at 10 and 12 as mild to moderate outliers.

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We have evaluated the minimum LSD estimator in this case for the full data as well as the outlier deleted data. The estimates are presented shown in Tables 1 and 2 respectively. For each of several male flies one samples about daugh- ter flies, and then determines the frequency of the number of daughter flies having a recessive lethal mutation in its X-chromosome. The data represent a frequency of frequencies; refer to Woodruff et al. Suppose we are given a random sample X1 ,. Theorem 6.

Theorem 7. Hence the desired approximation to the power function follows from the above asymptotic distribution. Title Suppressed Due to Excessive Length 15 6. Maji, Ghosh and Basu have also reported a similar observation for the robustness of the corresponding test statistics. However, the numerical illustrations reported in their paper and in the next section of present paper, this independence is not true for samples with moderate size. Therefore, as in the case of estimation, the robustness of the LSD based test of simple null hypothesis can not e indicated in terms of the influence function analysis even if we even go up to second order.

However, Maji, Ghosh and Basu showed that the robustness of the minimum LSD estimators can be measured quite accurately in terms of the secord order influence function of the estimator. Extending the same idea in case of testing, it is a routine exercise to see that the third order influence function of the test statistics at the null, being a function of the second order influence function of the corresponding estimator, can serve a better measure of robustness in this case.

In this article we have restricted ourselves to the simple null case. However the results may be extended to the case involving nuisance parameters following the same general approach.

## Statistical Inference Of Exponential Record Data Under Kullback-Leibler Divergence Measure

This experiment is available in Woodruff et al. This is a sex- linked recessive lethal experiment in drosophila fruit flies to test chemical mutagenicity.

The responses are the numbers of recessive lethal mutations observed among daughters of these flies. The data are given in Table 5. The two large counts for the treated group appears to be possible outliers. Though the tests are different but the non-robust nature of the likelihood test can be seen under this set-up also. The results are shown in Tables 6 and 7. The outlier deleted p-values and full data p-values are far from close in these cases. On the whole it appears that the two large counts in the treated group indicate a false significance for the likelihood ratio test and some other members of our class, but the more robust members clearly recognize the significance to be false.

Title Suppressed Due to Excessive Length 19 Table 5 Frequencies of the number of recessive lethal daughters for drosophila data x 0 1 2 3 4 5 6 7 Observed Control 15 3 0 0 0 0 0 Observed Treated 11 5 0 0 0 1 1 Table 6 Estimated Poisson parameters for the two-sample drosophila example; the numbers within the parentheses show the corresponding estimates for the treated case after deleting the two outliers.

The theoretical properties of this new family of divergences have been established for discrete models and similar results under continuous set-up can be done in subsequent works. References 1.

Ali, S. Silvey A general class of coefficients of divergence of one distribution from another. Journal of the Royal Statistical Society B, 28, — Basu, A. Harris, N. Hjort, and M.

- The Essential Eugene Field Collection (12 books and collections).
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Jones Robust and efficient esti- mation by minimising a density power divergence. Biometrika, 85, — Shioya, and C. Park Beran, R. Minimum Hellinger distance estimates for parametric models. An- nals of Statistics, 5, — Bregman, L. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming.

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Original article is in Zh. Chung, K. A Course in Probability Theory. Academic Press, New York. Cressie, N. Read Multinomial goodness-of-fit tests.

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Journal of the Royal Statistical Society B, 46, — Information-type measures of difference of probability distributions and indirect observations. Studia Scientiarum Mathematicarum Hungarica, 2, — On topological properties of f-divergences. Studia Sci. Hungarica, 2: Deheuvels, P. Uniform limit laws for kernel density estimators on possibly unbounded intervals. In Recent advances in reliability theory Bordeaux, , Stat. General asymptotic confidence bands based on kernel-type function estimators.

Inference Stoch. Deroye, L. Nonparametric density estimation. The L1 view. Devroye, L.