Yves-Caoimhin Swan

Local Pinsker Inequalities via Stein's Discrete Density Approach

Pinskers inequality states that the relative entropy between two random variables X and Y dominates the square of the total variation distance between X and Y. In this paper, we introduce generalized Fisher information distances and prove that these also dominate the square of the total variation distance. To this end, we introduce a general discrete Stein operator for which we prove a useful covariance identity. We illustrate our approach with several examples. Whenever competitor inequalities are available in the literature, the constants in ours are at least as good, and, in several cases, better.

Parametric Stein operators and variance bounds

Stein operators are (differential/difference) operators which arise within the so-called Steins method for stochastic approximation. We propose a new mechanism for constructing such operators for arbitrary (continuous or discrete) parametric distributions with continuous dependence on the parameter. We provide explicit general expressions for location, scale and skewness families. We also provide a general expression for discrete distributions. We use properties of our operators to provide upper and lower variance bounds (only lower bounds in the discrete case) on functionals h(X) of random variables X following parametric distributions. These bounds are expressed in terms of the first two moments of the derivatives (or differences) of h. We provide general variance bounds for location, scale and skewness families and apply our bounds to specific examples (namely the Gaussian, exponential, gamma and Poisson distributions). The results obtained via our techniques are systematically competitive with, and sometimes improve on, the best bounds available in the literature.

Maximum likelihood characterization of distributions

A famous characterization theorem due to C. F. Gauss states that the maximum likelihood estimator (MLE) of the parameter in a lo- cation family is the sample mean for all samples of all sample sizes if and only if the family is Gaussian. There exist many extensions of this result in diverse directions, most of them focussing on location and scale families. In this paper we propose a unified treatment of this literature by providing general MLE characterization theorems for one-parameter group families (with particular attention on location and scale parameters). In doing so we provide tools for determining whether or not a given such family is MLE-characterizable, and, in case it is, we define the fundamental concept of minimal necessary sample size at which a given characterization holds. Many of the cornerstone references on this topic are retrieved and discussed in the light of our findings, and several new characterization theorems are provided. Of particular interest is that one part of our work, namely the introduction of so-called equivalence classes for MLE characterizations, is a modernized version of Daniel Bernoulli’s viewpoint on maximum likelihood estimation.

A note on the normal approximation error for randomly weighted self-normalized sums

AbstractLet X = {Xn}n≥1 and Y = {Yn}n≥1 be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums

Approximate Computation of Expectations: the Canonical Stein Operator

\psi _n \left( {X,Y} \right) = \sum\nolimits_{i = 1}^n {{{X_i Y_i } \mathord{\left/ {\vphantom {{X_i Y_i } {V_n , V_n }}} \right. \kern-\nulldelimiterspace} {V_n , V_n }}} = \sqrt {Y_1^2 + \cdots + Y_n^2 } .

A continuous time approach to Robbins Problem of minimizing the expected rank

. These rates are seen to hold for the convergence of a number of important statistics, such as for instance Student’s t-statistic or the empirical correlation coefficient.