arXiv subjects

Michael D. Perlman

Publications and source records attributed to Michael D. Perlman.

11 recordsLinked to original sources

arXiv2024

Pure Significance Tests for Multinomial and Binomial Distributions: the Uniform Alternative

A {\it pure significance test} (PST) tests a simple null hypothesis $H_f:Y\sim f$ {\it without specifying an alternative hypothesis} by rejecting $H_f$ for {\it small} values of $f(Y)$. When the sample space supports a proper uniform pmf $f_\mathrm{unif}$, the PST can be viewed as a classical likelihood ratio test for testing $H_f$ against this uniform alternative. Under this interpretation, standard test features such as power, Kullback-Leibler divergence, and expected $p$-value can be considered. This report focuses on PSTs for multinomial and binomial distributions, and for the related goodness-of-fit testing problems with the uniform alternative. The case of repeated observations cannot be reduced to the single observation case via sufficiency. The {\it ordered binomial distribution}, apparently new, arises in the course of this study.

math.ST↗

arXiv2022

Estimating the Ratio of Means in a Zero-inflated Poisson Mixture Model

The problem of estimating the ratio of the means of a two-component Poisson mixture model is considered, when each component is subject to zero-inflation, i.e., excess zero counts. The. resulting {\it zero-inflated Poisson mixture (ZIPM) model} can be treated as a three-component Poisson mixture model with one degenerate component. The EM algorithm is applied to obtain frequentist estimators and their standard errors, the latter determined via an explicit expression for the observed information matrix. Bayes and empirical Bayes estimators also are obtained by means of conjugate priors and their data-based variants. Lastly, the ZIPM distribution and the ZTP (zero-truncated Poisson) distribution are compared.

math.ST↗

arXiv2019

On the structure of essential graphs for AMP chain graph models

Chain graphs (CG) use undirected and directed edges to represent both structural and associative dependences. Like acyclic directed graphs (ADGs), the CG associated with a statistical Markov model may not be unique, so CGs fall into Markov equivalence classes, which may be superexponentially large, leading to unidentifiability and computational inefficiency in model search and selection.Under the Andersson--Madigan--Perlman (AMP) interpretation of a CG, each Markov-equivalence class can be uniquely represented by a single CG, the AMP essential graph, that can be used for efficient inference and model search and selection. Here the graphical structure of an AMP essential graph is studied in detail. This structural information will be useful for algorithmic construction of the essential graph.

math.ST↗

arXiv2019

On the feasibility of parsimonious variable selection for Hotelling's $T^2$-test

Hotelling's $T^2$-test for the mean of a multivariate normal distribution is one of the triumphs of classical multivariate analysis. It is uniformly most powerful among invariant tests, and admissible, proper Bayes, and locally and asymptotically minimax among all tests. Nonetheless, investigators often prefer non-invariant tests, especially those obtained by selecting only a small subset of variables from which the $T^2$-statistic is to be calculated, because such reduced statistics are more easily interpretable for their specific application. Thus it is relevant to ask the extent to which power is lost when variable selection is limited to very small subsets of variables, e.g. of size one (yielding univariate Student-$t^2$ tests) or size two (yielding bivariate $T^2$-tests). This study presents some evidence, admittedly fragmentary and incomplete, suggesting that in some cases no power may be lost over a wide range of alternatives.

math.ST↗

arXiv2019

How Uniform is the Uniform Distribution on Permutations?

For large $q$, does the (discrete) uniform distribution on the set of $q!$ permutations of the vector $(1,2,\dots,q)$ closely approximate the (continuous) uniform distribution on the $(q-2)$-sphere that contains them? These permutations comprise the vertices of the regular permutohedron, a $(q-1)$-dimensional convex polyhedron. Surprisingly to me, the answer is emphatically no: these permutations are confined to a negligible portion of the sphere, and the regular permutohedron occupies a negligible portion of the ball. However, $(1,2,\dots,q)$ is not the most favorable configuration for approximate spherical uniformity of permutations. Unlike the permutations of $(1,2,\dots,q)$, the normalized surface area of the largest empty spherical cap among the permutations of the most favorable configuration approaches 0 as $q\to\infty$. Several open questions are posed.

math.PR↗

arXiv2013

An Alternative Markov Property for Chain Graphs

Graphical Markov models use graphs, either undirected, directed, or mixed, to represent possible dependences among statistical variables. Applications of undirected graphs (UDGs) include models for spatial dependence and image analysis, while acyclic directed graphs (ADGs), which are especially convenient for statistical analysis, arise in such fields as genetics and psychometrics and as models for expert systems and Bayesian belief networks. Lauritzen, Wermuth and Frydenberg (LWF) introduced a Markov property for chain graphs, which are mixed graphs that can be used to represent simultaneously both causal and associative dependencies and which include both UDGs and ADGs as special cases. In this paper an alternative Markov property (AMP) for chain graphs is introduced, which in some ways is a more direct extension of the ADG Markov property than is the LWF property for chain graph.

cs.AI↗

arXiv2013

Enumerating Markov Equivalence Classes of Acyclic Digraph Models

Graphical Markov models determined by acyclic digraphs (ADGs), also called directed acyclic graphs (DAGs), are widely studied in statistics, computer science (as Bayesian networks), operations research (as influence diagrams), and many related fields. Because different ADGs may determine the same Markov equivalence class, it long has been of interest to determine the efficiency gained in model specification and search by working directly with Markov equivalence classes of ADGs rather than with ADGs themselves. A computer program was written to enumerate the equivalence classes of ADG models as specified by Pearl & Verma's equivalence criterion. The program counted equivalence classes for models up to and including 10 vertices. The ratio of number of classes to ADGs appears to approach an asymptote of about 0.267. Classes were analyzed according to number of edges and class size. By edges, the distribution of number of classes approaches a Gaussian shape. By class size, classes of size 1 are most common, with the proportions for larger sizes initially decreasing but then following a more irregular pattern. The maximum number of classes generated by any undirected graph was found to increase approximately factorially. The program also includes a new variation of orderly algorithm for generating undirected graphs.

cs.AI↗

arXiv2012

Reverse Exchangeability and Extreme Order Statistics

For a bivariate random vector (X,Y), symmetry conditions are presented that yield stochastic orderings among |X|, |Y|, |max(X,Y)|, and | min(X, Y)|. Partial extensions of these results for multivariate random vectors (X1,...,Xn) are also given.

math.ST↗

arXiv2012

Reversing the Stein Effect

The Reverse Stein Effect is identified and illustrated: A statistician who shrinks his/her data toward a point chosen without reliable knowledge about the underlying value of the parameter to be estimated but based instead upon the observed data will not be protected by the minimax property of shrinkage estimators such as that of James and Stein, but instead will likely incur a greater error than if shrinkage were not used.

stat.ME↗

arXiv2010

Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas

Do there exist circular and spherical copulas in $R^d$? That is, do there exist circularly symmetric distributions on the unit disk in $R^2$ and spherically symmetric distributions on the unit ball in $R^d$, $d\ge3$, whose one-dimensional marginal distributions are uniform? The answer is yes for $d=2$ and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for $d\ge4$. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in $R^2$ by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in $R^d$ are also described, and determined explicitly for $d=2$.

stat.OT↗

arXiv2008

Multiple Testing and Error Control in Gaussian Graphical Model Selection

Graphical models provide a framework for exploration of multivariate dependence patterns. The connection between graph and statistical model is made by identifying the vertices of the graph with the observed variables and translating the pattern of edges in the graph into a pattern of conditional independences that is imposed on the variables' joint distribution. Focusing on Gaussian models, we review classical graphical models. For these models the defining conditional independences are equivalent to vanishing of certain (partial) correlation coefficients associated with individual edges that are absent from the graph. Hence, Gaussian graphical model selection can be performed by multiple testing of hypotheses about vanishing (partial) correlation coefficients. We show and exemplify how this approach allows one to perform model selection while controlling error rates for incorrect edge inclusion.

math.ST↗