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Abstract-We demonstrate that it is possible to approximate the mutual information arbitrarily closely in probability by calculating relative frequencies on appropriate partitions and achieving conditional independence on the rectangles of which the partitions are made. Empirical results, including a comparison with maximum-likelihood estimators, are presented.

Dissimilarities, metric divergences, minimum distance estimators,

The aim of this paper is to introduce new statistical criterions for estimation, suitable for inference in models with common continuous support. This proposal is in the direct line of a renewed interest for divergence based inference tools imbedding the most classical ones, such as maximum likelihood, Chi-square or Kullback Leibler. General pseudodistances with decomposable structure are considered, they allowing to define minimum pseudodistance estimators, without using nonparametric density estimators. A special class of pseudodistances indexed by α > 0, leading for α ↓ 0 to the Kulback Leibler divergence, is presented in detail. Corresponding estimation criteria are developed and asymptotic properties are studied. The estimation method is then extended to regression models. Finally, some examples based on Monte Carlo simulations are discussed.

The purpose of this paper is to introduce an approximation of the kernel-based logoptimal investment strategy that guarantees an almost optimal rate of growth of the capital under minimal assumptions on the behavior of the market. The new strategy uses much less knowledge on the distribution of the market process. It is analyzed both theoretically and empirically. The theoretical results show that the asymptotic rate of growth well approximates the optimal one that one could achieve with a full knowledge of the statistical properties of the underlying process generating the market, under the only assumption that the market is stationary and ergodic. The empirical results show that the proposed semi-log-optimal and the log-optimal strategies have essentially the same performance measured on past nyse data.

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The paper introduces scaled Bregman distances of probability distributions which admit non-uniform contributions of observed events. They are introduced in a general form covering not only the distances of discrete and continuous stochastic observations, but also the distances of random processes and signals. It is shown that the scaled Bregman distances extend not only the classical ones studied in the previous literature, but also the information divergence and the related wider class of convex divergences of probability measures. An information processing theorem is established too, but only in the sense of invariance w.r.t. statistically sufficient transformations and not in the sense of universal monotonicity. Pathological situations where coding can increase the classical Bregman distance are illustrated by a concrete example. In addition to the classical areas of application of the Bregman distances and convex divergences such as recognition, classification, learning and evaluation of proximity of various features and signals, the paper mentions a new application in 3D-exploratory data analysis. Explicit expressions for the scaled Bregman distances are obtained in general exponential families, with concrete applications in the binomial, Poisson and Rayleigh families, and in the families of exponential processes such as the Poisson and diffusion processes including the classical examples of the Wiener process and geometric Brownian motion.