Measurements of separation among probability densities or random variables

Distance between two probability densities or two random variables is a well established concept in statistics. The present paper considers generalizations of distances to separation measurements for three or more elements in a function space. Geometric intuition and examples from hypothesis testing suggest lower and upper bounds for such measurements in terms of pairwise distances; but also in L_p spaces some useful non-pairwise separation measurements always lie within these bounds. Examples of such separation measurements are the Bayes probability of correct classification among several arbitrary distributions, and the expected range among several random variables.     

Least-square regularized regression with non-iid sampling

We study the least-square regression learning algorithm generated by regularization schemes in reproducing kernel Hilbert spaces. A non-iid setting is considered: the sequence of probability measures for sampling is not identical and the sampling may be dependent. When the sequence of marginal distributions for sampling converges exponentially fast in the dual of a Hölder space and the sampling process satisfies a polynomial strong mixing condition, we derive learning rates for the learning algorithm.     

Gröbner bases and factorisation in discrete probability and Bayes

Gröbner bases, elimination theory and factorization may be used to perform calculations in elementary discrete probability and more complex areas such as Bayesian networks (influence diagrams). The paper covers the application of computational algebraic geometry to probability theory. The application to the Boolean algebra of events is straightforward (and essentially known). The extension into the probability superstructure is via the polynomial interpolation of densities and log densities and this is used naturally in the Bayesian application.     

Independence in Contingency Tables Using Simplicial Geometry

Frequently, contingency tables are generated in a multinomial sampling. Multinomial probabilities are then organized in a table assigning probabilities to each cell. A probability table can be viewed as an element in the simplex. The Aitchison geometry of the simplex identifies independent probability tables as a linear subspace. An important consequence is that, given a probability table, the nearest independent table is obtained by orthogonal projection onto the independent subspace. The nearest independent table is identified as that obtained by the product of geometric marginals, which do not coincide with the standard marginals, except in the independent case. The original probability table is decomposed into orthogonal tables, the independent and the interaction tables. The underlying model is log-linear, and a procedure to test independence of a contingency table, based on a multinomial simulation, is developed. Its performance is studied on an illustrative example.     

A note on bounding monte carlo variances michael lavine

A density bounded class P of probability distributions on a space χ is the set of all probability distributions corresponding to probability densities bounded below by a given subprob-ability density and bounded above by a given superprobability density. Density bounded classes arise in robust Bayesian analysis (Lavine 1991) and also in Monte Carlo integration (Fishman Granovsky and Rubin 1989). Finding upper and lower bounds on the variance over all p? P allows one to bound the Monte Carlo variance. Fishman Granovsky and Rubin (1989) find bounds on the variance over all p ? P and also find the densities in P achieving those bounds in the case where χ is discrete; that is, where P is actually a set of probability mass functions. This article generalizes their result by showing how to bound the variance and find the densities achieving the bounds when χ is continuous.     

D-optimal design for a quadratic log contrast model for experiments with mixtures

A practical method is suggested for solving complicated D-optimal design problems analytically. Using this method the author has solved the problem for a quadratic log contrast model for experiments with mixtures introduced by J. Aitchison and J. Bacon-Shone. It is found that for a symmetric subspace of the finite dimensional simplex, the vertices and the centroid of this subspace are the only possible support points for a D-optimal design. The weights that must be assigned to these support points contain irrational numbers and are constrained by a system of three simultaneous linear equations, except for the special cases of 1- and 2-dimensional simplexes where the situation is much simpler. Numerical values for the solution are given up to the 19-dimensional simplex     

On principal points for location mixtures of spherically symmetric distributions

In this paper, we investigate some properties of 2-principal points for location mixtures of spherically symmetric distributions with focus on a linear subspace in which a set of 2-principal points must lie. Our results can be viewed as an extension of those of Yamamoto and Shinozaki [2000. Two principal points for multivariate location mixtures of spherically symmetric distributions. J. Japan Statist. Soc. 30, 53–63], where a finite location mixture of spherically symmetric distributions is treated. As an extension of their paper, this paper defines a wider class of distributions, and derives a linear subspace in which a set of 2-principal points must exist. A theorem useful for comparing the mean squared distances is also established.     

D-Optimum Designs for Optimum Mixture in a Quadratic Log Contrast Model

Optimal designs for estimating the optimum mixing proportions in a quadratic mixture model was first investigated by Pal and Mandal (2006). In this article, similar investigation is carried out when mean response in a mixture experiment is described by a quadratic log contrast model. It is found that in a symmetric subspace of the finite dimensional simplex, there exists a D-optimal design that puts weights at the centroid of the sub-space and the vertices of the experimental domain. The optimality is checked by numerical computation using Equivalence Theorem.     

The Skew-Normal Distribution on the Simplex

Density functions on the simplex defined with respect to the Lebesgue measure can change from unimodality to multimodality with perturbation. This phenomenon is induced by the incompatibility of the Aitchison geometry and the Lebesgue measure. A Lebesgue-type measure, compatible with the algebraic geometric structure of the simplex, is used here to define the skew-normal density on the simplex as the Radon-Nykodym derivative with respect to it. Similarities and differences between the densities obtained using the different measures are analyzed.     

An alternative approach for compatibility of two discrete conditional distributions

ABSTRACT

Conditional specification of distributions is a developing area with increasing applications. In the finite discrete case, a variety of compatible conditions can be derived. In this paper, we propose an alternative approach to study the compatibility of two conditional probability distributions under the finite discrete setup. A technique based on rank-based criterion is shown to be particularly convenient for identifying compatible distributions corresponding to complete conditional specification including the case with zeros.The proposed methods are illustrated with several examples.     

Estimation suroptimale de la densité par projection

Superefficiency of a projection density estimator The author constructs a projection density estimator with a data‐driven truncation index. This estimator reaches the superoptimal rates 1/n in mean integrated square error and {In ln(n/n}^1/2 in uniform almost sure convergence over a given subspace which is dense in the class of all possible densities; the rate of the estimator is quasi‐optimal everywhere else. The subspace in question may be chosen a priori by the statistician.     

Bounds for moments through a general orthogonal expansion in a pre-hilbert space-I

Bounds are obtained for the product moments of an arbitrary finite number of ordered random variables. These bounds are obtained with the help of a representation of an arbitrary function in terms of a complete orthonormal system in a pre-Hilbert space of square integrable functions defined in a k-dimensional unit cube.     

PROBLEMS WITH BINOMIAL TWO-SIDED TESTS AND THE ASSOCIATED CONFIDENCE INTERVALS

Confidence intervals for parameters of distributions with discrete sample spaces will be less conservative (i.e. have smaller coverage probabilities that are closer to the nominal level) when defined by inverting a test that does not require equal probability in each tail. However, the P‐value obtained from such tests can exhibit undesirable properties, which in turn result in undesirable properties in the associated confidence intervals. We illustrate these difficulties using P‐values for binomial proportions and the difference between binomial proportions.     

Exhaustive k-nearest-neighbour subspace clustering

Cluster analysis is an important technique of explorative data mining. It refers to a collection of statistical methods for learning the structure of data by solely exploring pairwise distances or similarities. Often meaningful structures are not detectable in these high-dimensional feature spaces. Relevant features can be obfuscated by noise from irrelevant measurements. These observations led to the design of subspace clustering algorithms, which can identify clusters that originate from different subsets of features. Hunting for clusters in arbitrary subspaces is intractable due to the infinite search space spanned by all feature combinations. In this work, we present a subspace clustering algorithm that can be applied for exhaustively screening all feature combinations of small- or medium-sized datasets (approximately 30 features). Based on a robustness analysis via subsampling we are able to identify a set of stable candidate subspace cluster solutions.     

Tail Behavior and Limit Distribution of Maximum of Logarithmic General Error Distribution

Logarithmic general error distribution, an extension of the log-normal distribution, is proposed. Some interesting properties of the log GED are derived. These properties are applied to establish the asymptotic behavior of the ratio of probability densities and the ratio of the tails of the logarithmic general error and log-normal distributions, and to derive the asymptotic distribution of the partial maximum of an independent and identically distributed sequence obeying the log GED.     

SB-robustness of directional mean for circular distributions

In this paper we study the robustness of the directional mean (a.k.a. circular mean) for different families of circular distributions. We show that the directional mean is robust in the sense of finite standardized gross error sensitivity (SB-robust) for the following families: (1) mixture of two circular normal distributions, (2) mixture of wrapped normal and circular normal distributions and (3) mixture of two wrapped normal distributions. We also show that the directional mean is not SB-robust for the family of all circular normal distributions with varying concentration parameter. We define the circular trimmed mean and prove that it is SB-robust for this family. In general the property of SB-robustness of an estimator at a family of probability distributions is dependent on the choice of the dispersion measure. We introduce the concept of equivalent dispersion measures and prove that if an estimator is SB-robust for one dispersion measure then it is SB-robust for all equivalent dispersion measures. Three different dispersion measures for circular distributions are considered and their equivalence studied.     

Improvements in the Poisson approximation of mixed Poisson distributions

We consider the approximation of mixed Poisson distributions by Poisson laws and also by related finite signed measures of higher order. Upper bounds and asymptotic relations are given for several distances. Even in the case of the Poisson approximation with respect to the total variation distance, our bounds have better order than those given in the literature. In particular, our results hold under weaker moment conditions for the mixing random variable. As an example, we consider the approximation of the negative binomial distribution, which enables us to prove the sharpness of a constant in the upper bound of the total variation distance. The main tool is an integral formula for the difference of the counting densities of a Poisson distribution and a related finite signed measure.     

Rubbery Polya Tree

Abstract. Polya trees (PT) are random probability measures which can assign probability 1 to the set of continuous distributions for certain specifications of the hyperparameters. This feature distinguishes the PT from the popular Dirichlet process (DP) model which assigns probability 1 to the set of discrete distributions. However, the PT is not nearly as widely used as the DP prior. Probably the main reason is an awkward dependence of posterior inference on the choice of the partitioning subsets in the definition of the PT. We propose a generalization of the PT prior that mitigates this undesirable dependence on the partition structure, by allowing the branching probabilities to be dependent within the same level. The proposed new process is not a PT anymore. However, it is still a tail‐free process and many of the prior properties remain the same as those for the PT.     

Non‐parametric Estimation of the Number of Zeros in Truncated Count Distributions

We present some lower bounds for the probability of zero for the class of count distributions having a log‐convex probability generating function, which includes compound and mixed‐Poisson distributions. These lower bounds allow the construction of new non‐parametric estimators of the number of unobserved zeros, which are useful for capture‐recapture models, or in areas like epidemiology and literary style analysis. Some of these bounds also lead to the well‐known Chao's and Turing's estimators. Several examples of application are analysed and discussed.