Karl Mosler

A Cautionary Note on Likelihood Ratio Tests in Mixture Models

We show that iterative methods for maximizing the likelihood in a mixture of exponentials model depend strongly on their particular implementation. Different starting strategies and stopping rules yield completely different estimators of the parameters. This is demonstrated for the likelihood ratio test of homogeneity against two-component exponential mixtures, when the test statistic is calculated by the EM algorithm.

The Lorenz Zonoid of a Multivariate Distribution

Abstract The article extends the usual Lorenz curve and Lorenz order of univariate distributions to the multivariate case. For a given probability distribution in nonnegative d space, d ≥ 1, we define and investigate the Lorenz zonoid and the Lorenz surface, which are sets in (d + 1) space. The surface equals the usual Lorenz curve when d = 1. Included is the definition of the Lorenz surface of a finite number of vectors that may be interpreted as the endowments of economic units in d commodities. As a notion of increasing multivariate disparity, we introduce the set inclusion of Lorenz zonoids and show that it is equivalent to directional majorization.

Lift zonoids, random convex hulls and the variability of random vectors

For a d-variate measure a convex, compact set in Rd+1, its lift zonoid, is constructed. This yields an embedding of the class of d-variate measures having finite absolute first moments into the space of convex, compact sets in [Rd+1. The embedding is continuous, positive homogeneous and additive and has useful applications to the analysis and comparison of random vectors. The left zonoid is related to random convex sets and to the convex hull of a multivariate random sample. For an arbitrary sampling distribution, bounds are derived on the expected volume of the random convex hull. The set inclusion of lift zonoids defines an ordering of random vectors that reflects their variability. The ordering is investigated in detail and, as an application, inequalities for random determinants are given.

Stochastic Orders and Applications: A Classified Bibliography

Main List (by authors).- Field Lists.- GMT: General Mathematical Theory.- MAJ: Majorization.- ECT: Economic Theory.- INM: Inequality Measurement.- FIN: Finance.- INS: Insurance.- CHR: Choice under Risk.- DEM: Decision Methodology.- SEC: Special Applications in Economics.- MPR: Mathematical Programming.- QUE: Queueing.- REL: Reliability.- OOR: Other Applications in Operations Research.- STP: Stochastic Processes.- DEP: Dependence of Random Variables.- PRI: Probability Inequalities.- OPR: Other Applications in Probability.- TES: Statistical Tests.- EST: Statistical Estimation.- SEL: Statistical Selection, Ranking, and Classification.- EXP: Experimental Design.- OST: Other Applications in Statistics.- MPH: Mathematical Physics.- OTH: Other Applications.- COM: Algorithms and Computation.- Appendix: Supplementary List.

Entscheidungsregeln bei Risiko : multivariate stochastische Dominanz

Nachdem in den letzten Jahren das Konzept der stochasti schen Dominanz fur univariate Prospekte breite Anwendung in der Entscheidungsanalyse gefunden hat, scheint es an der Zeit zu sein, dies auch fur multivariate und allge meinere Prospekte zu versuchen. Die vorliegende Monogra phie behandelt die Theorie der stochastischen Dominanz fur univariate und besonders fur multivariate Prospekte sowie fur Prospekte in allgemeineren Raumen. Fur eine groBe Anzahl okonomisch relevanter Nutzenklassen werden die Dominanzrelationen durch Bedingungen an die Wahrschein lichkeitsverteilungen der Prospekte charakterisiert. Der abschlieBende Anwendungsteil enthalt auBer durchgerechne ten Beispielen auch Hinweise auf weitere Anwendungen in der Stochastik und im Operations Research. Die Arbeit ist die leicht erweiterte Fassung eines Manu skripts, das im November 1981 yom Fachbereich Wirtschafts und Organisationswissenschaften der Hocbschule der Bundes wehr Hamburg als Habilitationsschrift angenommen worden ist. Zu danken habe ich vielen und fur vieles: als erster sei Harry Hauptmann genannt, ohne des sen stetige Anregung und fordernde Geduld die Arbeit nicht zustande gekommen ware; Friedrich Schmid verdanke ich haufige und intensive Gesprache; Gunter Bamberg, Martin Beckmann, Harald Scherf und Norbert Schmitz haben wertvolle Diskussionen und Bemer kungen beigesteuert, und Frau Sigrid Jensen-Mundt hat in muhevoller Arbeit das Manuskript geschrieben. Ihnen und allen nicht Genannten und nicht zuletzt dem Verlag gilt mein herzl icher Dank. K. C. M. INHALT EinfUhrung 1.

Zonoid Data Depth: Theory and Computation

A new notion of data depth in d-space is presented, called the zonoid data depth. It is affine equivariant and has useful continuity and monotonicity properties. An efficient algorithm is developed that calculates the depth of a given point with respect to a d-variate empirical distribution.

Computing zonoid trimmed regions of dimension d>2

A probability distribution on Euclidean d-space can be described by its zonoid regions. These regions form a nested family of convex sets around the expectation, each being closed and bounded. The zonoid regions of an empirical distribution introduce an ordering of the data that has many applications in multivariate statistical analysis, e.g. cluster analysis, tests for multivariate location and scale, and risk analysis. An exact algorithm is developed to constructing the zonoid regions of a d-variate empirical distribution by their facets when d>=3. The vertices of the region and their adjacency are characterized, and a procedure is suggested by which all vertices and facets can be determined. The algorithm is available as an R-package.

Majorization in economic disparity measures

Abstract This survey presents an account of univariate and multivariate majorization orderings and their characterization by various classes of economic disparity indices. First, a concise treatment of classical univariate results is given, including majorization with different means and different population sizes, as well as Lorenz orderings of relative and absolute disparity. Second, alternatives to the Pigou-Dalton principle of transfers are discussed which are based on transfers about a given threshold. Third, disparity in several attributes and multivariate majorization are investigated, and a multivariate version of the Lorenz curved is introduced.

Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models

The paper compares several versions of the likelihood ratio test for exponential homogeneity against mixtures of two exponentials. They are based on different implementations of the likelihood maximization algorithm. We show that global maximization of the likelihood is not appropriate to obtain a good power of the LR test. A simple starting strategy for the EM algorithm, which under the null hypothesis often fails to find the global maximum, results in a rather powerful test. On the other hand, a multiple starting strategy that comes close to global maximization under both the null and the alternative hypotheses leads to inferior power.

Orthant orderings of discrete random vectors

Abstract We investigate four orthant stochastic orderings between random vectors X and Y that have finitely discrete probability distributions in R k. They have applications to multiattribute decision under risk, dependency of random vectors, and the statistical comparison of two k-variate samples. For each of the orderings we present conditions that are necessary and sufficient for dominance of Y over X. Given their distributions numerically, these conditions can be checked in an efficient way.