On detection of outliers in symmetric normal models

Extensions of recent results for detection of mean slippage type outliers from i.i.d. multivariate normal and elliptically symmetric distributions are made to symmetric case, that is, when the observations are equicorrelated. The main tool used is Wijsman's (1967) representation theorem. The results obtained can be viewed as a robustness property of the use of Mardia's multivariate kurtosis as a locally optimal test statistic to detect outliers against equicorrelated distributions.     

On the distributional transform, Sklar's theorem, and the empirical copula process

We review the distributional transform of a random variable, some of its applications, and some related multivariate distributional transformations. The distributional transform is a useful tool, which allows in many respects to deal with general distributions in the same way as with continuous distributions. In particular it allows to give a simple proof of Sklar's theorem in the general case. It has been used in the literature for stochastic ordering results. It is also useful for an adequate definition of the conditional value at risk measure and for many further purposes. We also discuss the multivariate quantile transform as well as the multivariate extension of the distributional transform and some of their applications. In the final section we consider an application to an extension of a limit theorem for the empirical copula process, also called empirical dependence function, to general not necessarily continuous distributions. This is useful for constructing and analyzing tests of dependence properties for general distributions.     

Some critical aspects of the use of exchangeability in statistics

Summary This expository paper provides a framework for analysing de Finetti's representation theorem for exchangeable finitely additive probabilities. Such an analysis is justified by reasoning of statistical nature, since it is shown that the abandonment of the axiom of σ-additivity has some noteworthy consequences on the common interpretation of the Bayesian paradigm. The usual (strong) fromulation of de Finetti's theorem is deduced from the finitely additive (weak) formulation, and it is used to solve the problem of stating the existence of a stochastic process, with given finite-dimensional probability distributions, whose sample paths are probability distributions. It is of importance, in particular, to specify prior distributions for nonparametric inferential problems in a Bayesian setting. Research partially supported by MPI (40% 1990, Gruppo Nazionale ?Modelli Probabilistici e Statistica Matematica?).     

An Elementary Introduction to Maximum Likelihood Estimation for Multinomial Models: Birch's Theorem and the Delta Method

A fairly complete introduction to the large sample theory of parametric multinomial models, suitable for a second-year graduate course in categorical data analysis, can be based on Birch's theorem (1964) and the delta method (Bishop, Fienberg, and Holland 1975). I present an elementary derivation of a version of Birch's theorem using the implicit function theorem from advanced calculus, which allows the presentation to be relatively self-contained. The use of the delta method in deriving asymptotic distributions is illustrated by Rao's (1973) result on the distribution of standardized residuals, which complements the presentation in Bishop, Fienberg, and Holland. The asymptotic theory is illustrated by two examples.     

When is n large enough? Looking for the right sample size to estimate proportions

The central limit theorem indicates that when the sample size goes to infinite, the sampling distribution of means tends to follow a normal distribution; it is the basis for the most usual confidence interval and sample size formulas. This study analyzes what sample size is large enough to assume that the distribution of the estimator of a proportion follows a Normal distribution. Also, we propose the use of a correction factor in sample size formulas to ensure a confidence level even when the central limit theorem does not apply for these distributions.     

Resampling m-Dependent Random Variables with Applications to Forecasting

Resampling methods are proposed to estimate the distributions of sums of m -dependent possibly differently distributed real-valued random variables. The random variables are allowed to have varying mean values. A non parametric resampling method based on the moving blocks bootstrap is proposed for the case in which the mean values are smoothly varying or 'asymptotically equal'. The idea is to resample blocks in pairs. It is also confirmed that a 'circular' block resampling scheme can be used in the case where the mean values are 'asymptotically equal'. A central limit resampling theorem for each of the two cases is proved. The resampling methods have a potential application to time series analysis, to distinguish between two different forecasting models. This is illustrated with an example using Swedish export prices of coated paper products.     

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.     

Optimal equivariant estimator with respect to convex loss function

Consider a family of probability distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. We illustrate this theorem under some conditions on convex loss function.     

Bayesian optimal one point designs for one parameter nonlinear models

For nonlinear one parameter models and concave optimality criteria there always exists a locally optimal one point design. This can be proved by an application of Caratheodory's theorem (Läuter, Math. Operationsforsch. Statist. Ser. Statist. 5 (1974a) 625–636). If prior distributions with densities are used, this theorem gives no useful bound on the number of support points of a Bayesian optimal design. Chaloner (J. Statist. Plann. Inference, 37 (1993) 229–236) gave a sufficient condition on the support of the prior distribution for the existence of a Bayesian optimal one point design. In this article, a condition on the shape of the prior density is given, which is also sufficient for the existence of a Bayesian optimal one point design in nonlinear models with one parameter.     

Time series models based on the unrestricted skew-normal process

The standard location and scale unrestricted (or unified) skew-normal (SUN) family studied by Arellano-Valle and Genton [On fundamental skew distributions. J Multivar Anal. 2005;96:93–116] and Arellano-Valle and Azzalini [On the unification of families of skew-normal distributions. Scand J Stat. 2006;33:561–574], allows the modelling of data which is symmetrically or asymmetrically distributed. The family has a number of advantages suitable for the analysis of stochastic processes such as Auto-Regressive Moving-Average (ARMA) models, including being closed under linear combinations, being able to satisfy the consistency condition of Kolmogorov’s theorem and providing the guarantee of the existence of such a SUN stochastic process. The family is able to be represented in a hierarchical form which can be used for the ease of simulation. In addition, it facilitates an EM-type algorithm to estimate the model parameters. The performances and suitability of the proposed model are demonstrated on simulations and using two real data sets in applications.     

Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

In this paper, we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix‐variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large‐dimensional asymptotic regime, where the dimension p and the sample size n approach infinity such that p/n→c ∈ [0, + ∞) when the sample covariance matrix does not need to be invertible and p/n→c ∈ [0,1) otherwise.     

On Bayes' theorem for fuzzy data

There are some ideas concerning a generalization of Bayes' theorem to the situation of fuzzy data. Some of them are given in the references [1], [5], and [7]. But the proposed methods are not generalizations in the sense of the probability content of Bayes' theorem for precise data. In the present paper a generalization of Bayes' theorem to the case of fuzzy data is described which contains Bayes' theorem for precise data as a special case and allows to use the information in fuzzy data in a coherent way. Moreover a generalization of the concept of HPD-regions is explained which makes it possible to model and analyze the situation of fuzzy data. Also a generalization of the concept of predictive distributions is given in order to calculate predictive densities based on fuzzy sample information.     

Minimax rules under zero–one loss for a restricted location parameter

In this paper, we obtain minimax and near-minimax nonrandomized decision rules under zero–one loss for a restricted location parameter of an absolutely continuous distribution. Two types of rules are addressed: monotone and nonmonotone. A complete-class theorem is proved for the monotone case. This theorem extends the previous work of Zeytinoglu and Mintz (1984) to the case of 2e-MLR sampling distributions. A class of continuous monotone nondecreasing rules is defined. This class contains the monotone minimax rules developed in this paper. It is shown that each rule in this class is Bayes with respect to nondenumerably many priors. A procedure for generating these priors is presented. Nonmonotone near-minimax almost-equalizer rules are derived for problems characterized by non-2e-MLR distributions. The derivation is based on the evaluation of a distribution-dependent function Q_c. The methodological importance of this function is that it is used to unify the discrete- and continuous-parameter problems, and to obtain a lower bound on the minimax risk for the non-2e-MLR case.     

Robustness of confidence intervals for scale parameters based on m-estimators

The robustness of confidence intervals for a scale parameter based on M-esimators is studied, especially in small size samples. The coverage probablity is used as measure of robustness. A theorem for a lower bound of the minimum coverage probability of M-estimators is presented and it is applied in order to examine the behavior of the standard deviation and the median absolute deviation, as interval estimators. This bound can confirm the robustness of any other scale M-estimator in interval estimation. The idea of stretching is used to formulate the family of distributions that are considered as underlying. Critical values for the confidence interval are computed where it is needed, that is for the median absolute deviation in the Normal, Uniform and Cauchy distribution and for the standard deviation in the Uniform and Cauchy distribution. Simulation results have been achieved for the estimation of the coverage probabilities and the critical values.     

Simultaneous equivariant estimation of the parameters of linear models

Consider a family of distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. Then we consider a linear model Y=Xβ+ε, in which ε has a multivariate distribution with mean vector zero and has a density belonging to a scale family with scale parameter σ. Also we assume that the underlying family of distributions is invariant with respect to a certain group of transformations. First, we find the class of all equivariant estimators of regression parameters and the powers of σ. By using the characterization theorem we discuss the simultaneous equivariant estimation of the parameters of the linear model.     

Teaching Bayes’ Theorem: Strength of Evidence as Predictive Accuracy

Although teaching Bayes’ theorem is popular, the standard approach—targeting posterior distributions of parameters—may be improved. We advocate teaching Bayes’ theorem in a ratio form where the posterior beliefs relative to the prior beliefs equals the conditional probability of data relative to the marginal probability of data. This form leads to an interpretation that the strength of evidence is relative predictive accuracy. With this approach, students are encouraged to view Bayes’ theorem as an updating mechanism, to obtain a deeper appreciation of the role of the prior and of marginal data, and to view estimation and model comparison from a unified perspective.     

On convergence of the distributions of random sequences with independent random indexes to variance–mean mixtures

We prove a transfer theorem for random sequences with independent random indexes in the double array limit setting under relaxed conditions. We also prove its partial inverse providing the necessary and sufficient conditions for the convergence of randomly indexed random sequences. Special attention is paid to the case where the elements of the basic double array are formed as cumulative sums of independent not necessarily identically distributed random variables. Using simple moment-type conditions we prove the theorem on convergence of the distributions of such sums to normal variance–mean mixtures.     

Size distributions reconsidered

We consider tests of the hypothesis that the tail of size distributions decays faster than any power function. These are based on a single parameter that emerges from the Fisher–Tippett limit theorem, and discriminate between leading laws considered in the literature without requiring fully parametric models/specifications. We study the proposed tests taking into account the higher order regular variation of the size distribution that can lead to catastrophic distortions. The theoretical bias corrections realign successfully nominal and empirical test behavior, and inform a sensitivity analysis for practical work. The methods are used in an examination of the size distribution of cities and firms.     

Dependent models for observations which include angular ones

This paper discusses some stochastic models for dependence of observations which include angular ones. First, we provide a theorem which constructs four-dimensional distributions with specified bivariate marginals on certain manifolds such as two tori, cylinders or discs. Some properties of the submodel of the proposed models are investigated. The theorem is also applicable to the construction of a related Markov process, models for incomplete observations, and distributions with specified marginals on the disc. Second, two maximum entropy distributions on the cylinder are discussed. The circular marginal of each model is distributed as the generalized von Mises distribution which represents a symmetric or asymmetric, unimodal or bimodal shape. The proposed cylindrical model is applied to two data sets.     

Asymptotic normality for limiting discrete distributions

Three basic asymptotic normality conditions for limiting discrete distributions are given in terms of differencing operators. Under these conditions one can establish asymptotic normality by directly dealing with the limiting behaviors of distribution functions themselves without resorting to the central limit theorem.