We examine robust estimators and tests using the family of generalized negative exponential disparities, which contains the Pearson's chi-square and the ordinary negative exponential disparity as special cases. The influence function and α-influence function of the proposed estimators are discussed and their breakdown points derived. Under the model, the estimators are asymptotically efficient, and are shown to have an asymptotic breakdown point of 50%. The proposed tests are shown to be equivalent to the likelihood ratio test under the null hypothesis, and their breakdown points are obtained. The competitive performance of the proposed estimators and tests relative to those based on the Hellinger distance is illustrated through examples and simulation results. Unlike the Hellinger distance, several members of this family of generalized negative exponential disparities generate estimators which also possess excellent inlier-controlling capability. The corresponding tests of hypothesis are shown to have better power breakdown than the Hellinger deviance test in the cases examined.

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The celebrated U-conjecture states that under the N n (0, I n ) distribution of the random vector X = (X 1 , . . . , X n ) in R n , two polynomials P (X) and Q(X) are unlinkable if they are independent [see Kagan et al., Characterization Problems in Mathematical Statistics, Wiley, New York, 1973]. Some results have been established in this direction, although the original conjecture is yet to be proved in generality. Here, we demonstrate that the conjecture is true in an important special case of the above, where P and Q are convex nonnegative polynomials with P (0) = 0.