Abstract: I would like to congratulate Sanat Sarkar for an excellent and thorough appraisal of the state of art with the controlling of false discovery rate (FDR), mostly arising in the context of multiple hypothesis testing (MHT) problems. He has indeed focused on all the major statistical aspects of the FDR/MHT complex, elucidating the theoretical and methodological perspectives. There are some genuine statistical challenges embedded in this largely interdisciplinary field of research, and in this discussion I will mostly highlight some of these issues with due emphasis on applications in high-dimensional low sample size (HDLSS) data models, and appraise them in the light of the developments in this seminal work.

Abstract: We consider the question in how fax long memory in volatility affects the asymptotic distribution of location estimators. Specifically, we consider Mestimation for models with finite moments. Under symmetry assumptions, the asymptotic distribution turns out to be the same as under independence, even if a nonlinear estimator is used. However, for nonlinear estimators, deviations from these assumptions can imply a slower rate of convergence and hence asymptotic efficiency zero compared to the sample mean. This means that for long-memory volatility models, estimators that are robust with respect to bias, turn out to be extremely sensitive with respect to the variance. Simulations and data examples illustrate the results by comparing the asymptotic behaviour of the sample mean, the median and a Huber estimator with an intermediate tuning parameter. AMS (2000) subject classification. Primary 62P05.

Abstract: Inference procedures based on the Hellinger distance provide attractive alternatives to likelihood based methods for the statistician. The minimum Hellinger distance estimator has full asymptotic efficiency under the model together with strong robustness properties under model misspecification. However, the Hellinger distance puts too large a weight on the inliers which appears to be the main reason for the poor efficiency of the method in small samples. Here some modifications to the inlier part of the Hellinger distance are provided which lead to substantial improvements in the small sample properties of the estimators. The modified divergences are members of the general class of disparities and satisfy the necessary regularity conditions so that the asymptotic properties of the resulting estimators follow from standard theory. In limited simulations the proposed estimators exhibit better small sample performance at the model and competitive robustness properties in relation to the ordinary minimum Hellinger distance estimator. As the asymptotic efficiencies of the modified estimators are the same as that of the ordinary estimator, the new procedures are expected to be useful tools for applied statisticians and data analysts. AMS (2000) subject classification. Primary 62F10, 62F35; Secondary: 62F12.

Abstract: Exchangeability of observations corresponds to a condition shared by the vast majority of applications of the Bayesian paradigm. By de Finetti’s representation theorem, if exchangeable observations form an infinite sequence of random variables, then they are conditionally independent and identically distributed given some random parameter, which is the main object of statistical inference. Such parameter is a limiting mathematical entity and therefore hypotheses related to it might be not verifiable. For this reason, statistical analysis should be directed toward the prevision of the empirical distribution of N observations. In view of these considerations, specific forms of (finitary) exchangeable laws based on sequences of nested partitions have been introduced and studied in Bassetti and Bissiri (2007). In this paper, we intend to carry on this line of research studying another class of exchangeable laws, which rests on the concept of exchangeable random partition. These distributions are related to species sampling sequences, but allow negative correlation between observations. Marginal and predictive distributions are calculated together with the posterior distribution of the empirical process, and finally, it is shown how the predictive mean can be approximated by importance sampling. AMS (2000) subject classification. Primary 62C10,62F15,60G09.