Mathematics

Distance correlation is a recent extension of Pearson's correlation, that characterises general statistical independence between Euclidean-space-valued random variables, not only linear relations. This review delves into how and when distance correlation can be extended to metric spaces, combining the information that is available in the literature with some original remarks and proofs, in a way that is comprehensible for any mathematical statistician.

We consider a nonparametric version of the integer-valued GARCH(1,1) model for time series of counts. The link function in the recursion for the variances is not specified by finite-dimensional parameters, but we impose nonparametric smoothness conditions. We propose a least squares estimator for this function and show that it is consistent with a rate that we conjecture to be nearly optimal.

In this short note we study how well a Gaussian distribution can be approximated by distributions supported on [−a,a] . Perhaps, the natural conjecture is that for large a the almost optimal choice is given by truncating the Gaussian to [−a,a] . Indeed, such approximation achieves the optimal rate of e −Θ( a 2 ) in terms of the L ∞ -distance between characteristic functions. However, if we consider the L ∞ -distance between Laplace transforms on a complex disk, the optimal rate is e −Θ( a 2 loga) , while truncation still only attains e −Θ( a 2 ) . The optimal rate can be attained by the Gauss-Hermite quadrature. As corollary, we also construct a ``super-flat'' Gaussian mixture of Θ( a 2 ) components with means in [−a,a] and whose density has all derivatives bounded by e −Ω( a 2 log(a)) in the O(1) -neighborhood of the origin.

Partial orderings and measures of information for continuous univariate random variables with special roles of Gaussian and uniform distributions are discussed. The information measures and measures of non-Gaussianity including third and fourth cumulants are generally used as projection indices in the projection pursuit approach for the independent component analysis. The connections between information, non-Gaussianity and statistical independence in the context of independent component analysis is discussed in detail.

There is a well-known equivalence between avoiding accuracy dominance and having probabilistically coherent credences (see, e.g., de Finetti 1974, Joyce 2009, Predd et al. 2009, Schervish et al. 2009, Pettigrew 2016). However, this equivalence has been established only when the set of propositions on which credence functions are defined is finite. In this paper, we establish connections between accuracy dominance and coherence when credence functions are defined on an infinite set of propositions. In particular, we establish the necessary results to extend the classic accuracy argument for probabilism originally due to Joyce (1998) to certain classes of infinite sets of propositions including countably infinite partitions.

Optimality results for two outstanding Bayesian estimation problems are given in this paper: the estimation of the sampling distribution for the squared total variation function and the estimation of the density for the L 1 -squared loss function. The posterior predictive distribution provides the solution to these problems. Some examples are presented to illustrate it. The Bayesian estimation problem of a distribution function is also addressed.

In this paper we present new theoretical results for the Dantzig and Lasso estimators of the drift in a high dimensional Ornstein-Uhlenbeck model under sparsity constraints. Our focus is on oracle inequalities for both estimators and error bounds with respect to several norms. In the context of the Lasso estimator our paper is strongly related to [11], who investigated the same problem under row sparsity. We improve their rates and also prove the restricted eigenvalue property solely under ergodicity assumption on the model. Finally, we demonstrate a numerical analysis to uncover the finite sample performance of the Dantzig and Lasso estimators.

The reversed aging intensity function is defined as the ratio of the instantaneous reversed hazard rate to the baseline value of the reversed hazard rate. It analyzes the aging property quantitatively, the higher the reversed aging intensity, the weaker the tendency of aging. In this paper, a family of generalized reversed aging intensity functions is introduced and studied. Those functions depend on a real parameter. If the parameter is positive they characterize uniquely the distribution functions of univariate positive absolutely continuous random variables, in the opposite case they characterize families of distributions. Furthermore, the generalized reversed aging intensity orders are defined and studied. Finally, several numerical examples are given.

The research in this paper gives a systematic investigation on the asymptotic behaviours of four inverse probability weighting (IPW)-based estimators for conditional average treatment effect, with nonparametrically, semiparametrically, parametrically estimated and true propensity score, respectively. To this end, we first pay a particular attention to semiparametric dimension reduction structure such that we can well study the semiparametric-based estimator that can well alleviate the curse of dimensionality and greatly avoid model misspecification. We also derive some further properties of existing estimator with nonparametrically estimated propensity score. According to their asymptotic variance functions, the studies reveal the general ranking of their asymptotic efficiencies; in which scenarios the asymptotic equivalence can hold; the critical roles of the affiliation of the given covariates in the set of arguments of propensity score, the bandwidth and kernel selections. The results show an essential difference from the IPW-based (unconditional) average treatment effect(ATE). The numerical studies indicate that for high-dimensional paradigms, the semiparametric-based estimator performs well in general {whereas nonparametric-based estimator, even sometimes, parametric-based estimator, is more affected by dimensionality. Some numerical studies are carried out to examine their performances. A real data example is analysed for illustration.

We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems of the form min x?�X E[F(x,ξ)] , when the given data is a finite independent sample selected according to ξ . The procedure is based on a median-of-means tournament, and is the first procedure that exhibits the optimal statistical performance in heavy tailed situations: we recover the asymptotic rates dictated by the central limit theorem in a non-asymptotic manner once the sample size exceeds some explicitly computable threshold. Additionally, our results apply in the high-dimensional setup, as the threshold sample size exhibits the optimal dependence on the dimension (up to a logarithmic factor). The general setting allows us to recover recent results on multivariate mean estimation and linear regression in heavy-tailed situations and to prove the first sharp, non-asymptotic results for the portfolio optimization problem.