Confidence regions and minimax rates in outlier-robust estimation on the probability simplex

We consider the problem of estimating the mean of a distribution supported by the $k$-dimensional probability simplex in the setting where an $\varepsilon$ fraction of observations are subject to adversarial corruption. A simple particular example is the problem of estimating the distribution of a discrete random variable. Assuming that the discrete variable takes $k$ values, the unknown parameter $\boldsymbol \theta$ is a $k$-dimensional vector belonging to the probability simplex. We first describe various settings of contamination and discuss the relation between these settings. We then establish minimax rates when the quality of estimation is measured by the total-variation distance, the Hellinger distance, or the $\mathbb L^2$-distance between two probability measures. We also provide confidence regions for the unknown mean that shrink at the minimax rate. Our analysis reveals that the minimax rates associated to these three distances are all different, but they are all attained by the sample average. Furthermore, we show that the latter is adaptive to the possible sparsity of the unknown vector. Some numerical experiments illustrating our theoretical findings are reported

Outliers in Garch models and the estimation of risk measures

In this paper we focus on the impact of additive level outliers on the calculation of risk measures, such as minimum capital risk requirements, and compare four alternatives of reducing these measures' estimation biases. The first three proposals proceed by detecting and correcting outliers before estimating these risk measures with the GARCH(1,1) model, while the fourth procedure fits a Student’s t-distributed GARCH(1,1) model directly to the data. The former group includes the proposal of Grané and Veiga (2010), a detection procedure based on wavelets with hard- or soft-thresholding filtering, and the well known method of Franses and Ghijsels (1999). The first results, based on Monte Carlo experiments, reveal that the presence of outliers can bias severely the minimum capital risk requirement estimates calculated using the GARCH(1,1) model. The message driven from the second results, both empirical and simulations, is that outlier detection and filtering generate more accurate minimum capital risk requirements than the fourth alternative. Moreover, the detection procedure based on wavelets with hard-thresholding filtering gathers a very good performance in attenuating the effects of outliers and generating accurate minimum capital risk requirements out-of-sample, even in pretty volatile periodsMinimum capital risk requirements, Outliers, Wavelets