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# Abstract: Multidimensional indices and orders of diversity (Karl Mosler)

This paper presents several indices to describe multivariate diversity and evenness. Multivariate generalizations of the Gini-Simpson index and the Rosenbluth index are proposed to measure diversity. A multivariate Gini ratio is also presented to measure evenness. These indices extend the usual univariate measures; they reflect not only the diversity of marginal distributions but also the dependence structure of abundance. The indices fulfill desirable measurement properties and are consistent with certain orderings of multivariate distributions. An order of concentration surfaces and a majorization order are also surveyed shortly.

Keywords: Concentration order, Gini-Simpson index, majorization, multivariate diversity and evenness indices.

# Abstract: Testing for homogeneity in an exponential mixture model (Karl Mosler and Wilfried Seidel)

Diagnostic procedures are studied to test for homogeneity against unobserved heterogeneity in an exponential mixture model. The procedures include a dispersion score test, a likelihood ratio test, a moment likelihood approach and several goodness-of-fit tests. The empirical power of these tests is compared on a broad range of alternatives. We propose a new test, which combines the dispersion score test with a properly chosen goodness-of-fit procedure; its empirical power comes close to the power of the best of the other tests.

Keywords: Mixture diagnosis, frailty model, survival analysis, hazard models, overdispersion, goodness-of-fit, likelihood tests.

# Abstract: A cautionary note on likelihood ratio tests in mixture models (W. Seidel, K. Mosler and M. Alker)

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.

Keywords: EM algorithm, exponential mixture models, initial values, stopping criteria, maximum likelihood estimation, likelihood ratio test.

# Abstract: Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models (Wilfried Seidel, Karl Mosler and Manfred Alker)

The paper compares several versions of the likelihood ratio test for exponential homogeneity against two exponentials. They are based on different implementations of the likelihood maximization algortihm. 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 multuple starting strategy that comes close to global maximization under both the null and the alternative hypotheses leads to inferior power.

Keywords: Maximum likelihood, likelihood ratio tests, mixtures of exponentials, homogeneity tests, EM algorithm.

# Abstract: Depth of hyperplanes and related statistics (Gleb Koshevoy and Karl Mosler)

We introduce a new notion of multivariate depth, the depth of a hyperplane in a sample. For two samples, the mean hyperplane depth (MHD) of one with respect to the other is calculated. We explore the combinatorial and other properties of these notions and, in particular, their relation to majority depth. Affine invariant statistics based on the MHD are introduced to test for homogeneity against dispersion and location alternatives.

# Abstract: Measuring Multidimensional Concentration: A Geometric Approach (Gleb Koshevoy and Karl Mosler )

We present a new approach to measure industrial concentration when more than one characteristic of firm size is relevant. The concentration curve and order, the concentration rates and the Rosenbluth index are extended to a multidimensional setting. An illustrative application to the German mineral oil market is included.

# Abstract: Welfare means and equalizing transfers (Karl Mosler and Pietro Muliere)

Economic disparity and social welfare are measured by real-valued indices and orderings. Often a welfare index can be seen as a proper mean value, based on equivalent equally distributed income, while a welfare ordering is constructed from inequality reducing transfers. This paper surveys welfare and disparity indices from the view of quasilinear means and their generalizations and gives an account of recent notions of equalizing transfers.

# Abstract: Specification analysis in mixed hazard models and a test of crossing survival functions (Karl Mosler and Tomas Philipson)

The paper investigates diagnostic procedures for the specification of common hazard models in duration analysis. It is shown that under mixed hazard specifications the survival functions of different subgroups cannot cross. A nonparametric test for the crossing of two survival functions is provided and its applications in duration analysis are discussed. In particular, the proportional hazard model with unobserved heterogeneity (PHU) is investigated, and procedures are developed to test whether given data are consistent with the PHU model and whether they contain unobserved heterogeneity within the PHU specification. Examples in which crossing survivals are of substantive concern are discussed, including the dynamics of infectious diseases and the demand for vaccination.

Keywords: Survival functions, demand for vaccines, proportional hazard models, unobserved heterogeneity, specification tests.

# Abstract: Lift zonoids, random convex hulls and the variability of random vectors (Gleb Koshevoy and Karl Mosler)

For a d-variate measure a convex, compact set in $\mathbb{R}^{d+1}$, its lift zonoid, is constructed. This yields an embedding of the class of d-variate measures having finite first moments into the space of convex, compact sets in $\mathbb{R}^{d+1}$. The embedding is continuous, positive homogeneous and additive and has useful applications to the analysis and comparison of random vectors. The lift 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.

# Abstract: Orthant orderings of discrete random vectors (Rainer Dyckerhoff and Karl Mosler)

We investigate four orthant stochastic orderings between random vectors X and Y that have finitely discrete probability distributions in $\mathbb{R}^{k}$ . 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.

Keywords: Multivariate stochastic orders, dependence ordering, decision under risk, bivariate risk aversion, comparison of empirical distribution functions.

# Abstract: A bounding procedure for expected multiattribute utility (Hartmut Holz and Karl Mosler)

The paper presents an interactive procedure for expected utility decisions based on a multiattribute utility function. The utility function is a weighted sum of single-attribute utilities. In each stage the decision maker is asked about his or her preference between a given pair of lotteries. The answer is used to derive inequalities either for the attribute weights or the single-attribute utility functions, to calculate bounds for the expected utility of all decision alternatives, and to determine, after this stage, a nondominated set of alternatives. The procedure can be stopped at any instance. It has been efficiently implemented on a PC.

Keywords: Decision under risk, multiple objectives, multiattribute utility technique (MAUT), interactive elicitation of utility functions, additively separable utility function, partial information, efficient sets of alternatives.

# Abstract: Multivariate Gini indices (Gleb Koshevoy and Karl Mosler)

Two extensions of the univariate Gini index are considered: $R_D$ (Arnold, 1987) based on expected distance between two independent vectors from the same distribution with finite mean $\mu \in \mathbb{R}^d$ and $R_V$ (Wilks, 1960) related to the expected volume of the simplex formed from d+1 independent such vectors. A new characterization of $R_D$ as proportional to a univariate Gini index for a particular linear combination of attributes relates it with the Lorenz zonoid. The Lorenz zonoid was suggested by Koshevoy and Mosler (1996) as a multivariate generalization of the Lorenz curve. $R_V$ is, up to scaling, the volume of the Lorenz zonoid plus a unit cube of full dimension. When d=1, both $R_D$ and $R_V$ equal twice the area between the usual Lorenz curve and the line of zero disparity. When d>1, they are different, but inherit properties of the univariate Gini index and are related via the Lorenz zonoid: $R_D$ is proportional to the average of the areas of some two-dimensioned projections of the lift zonoid, while $R_V$ is the average of the volumes of projections of the Lorenz zonoid over all coordinate subspaces.

Keywords and -phrases: dilation, disparity measurement, Gini mean difference, lift zonoid, Lorenz order.

# Abstract: Inequality Indices and the Starshaped Principle of Transfers (Karl Mosler and Pietro Muliere)

The evaluation of income distributions is usually based on the Pigou-Dalton (PD) principle which says that a transfer from any people to people who have less decreases economic inequality, i.e., increases the social evaluation index. We introduce two weaker principles of transfers which refer to a parameter $\theta$. With the new principles, only those PD transfers increase the social evaluation index which take from the class of incomes above $\theta$ and give to the class below $\theta$. The relative positions of individuals remain unchanged, and either no individual may cross the line $\theta$ (principle of transfers about $\theta$ ) or some may do who have been situated next to it (starshaped principle of transfers at $\theta$ ). $\theta$ may be a given constant, a function of mean income, or a quantile of the income distribution. The classes of indices which are consistent with these transfers are completely characterized, and examples are given.

Keywords: Inequality measurement, relative concentration, economic disparity, Pigou-Dalton transfers, starshaped functions.

# Abstract: Zonoid Data Depth: Theory and Computation (Rainer Dyckerhoff, Gleb Koshevoy and Karl Mosler)

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.

# Abstract: The Lorenz Zonoid of a Multivariate Distribution (Gleb Koshevoy and Karl Mosler)

The paper extends the usual Lorenz curve and Lorenz order of univariate distributions to the multivariate case. For a given probability distribution in nonnegative d-space, $d\geq 1$, 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 which 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.

Keywords: Multivariate disparity, Lorenz curve, Lorenz surface, Gini measure, economic inequality measurement.

# Abstract: An algorithm for multivariate weak stochastic dominance (Rainer Dyckerhoff, Hartmut Holz and Karl Mosler)

The paper addresses the computational problem of comparing discrete probability distributions in k-space with respect to two stochastic orderings. Both orderings are extensions of the usual univariate stochastic order and are weaker orderings than Lehmann's multivariate stochastic order. An algorithm is presented which operates efficiently on a semilattice generated by the support of the two distributions. Applications are given to tests for multivariate slippage and peakedness and to the analysis of stochastic dependency.

Keywords: Multivariate stochastic orders, upper and lower orthant ordering, multivariate slippage, multivariate peakedness, stochastic dependence.

# Abstract: An Interactive Decision Procedure with Multiple Attributes under Risk (Hartmut Holz and Karl Mosler)

Consider a finite set of alternatives under risk which have multiple attributes. MARPI is an interactive computer-based procedure to find an efficient choice in the sense of linear expected utility. The choice is based on incomplete information about the decision maker's preferences which is elicited and processed in a sequential way. The information includes qualitative properties of the multivariate utility function such as monotoneity, risk aversion, and separability. Further, in case of an additively separable utility function, bounds on the scaling constants are elicited, and preferences (not necessarily indifferences) between sure amounts and lotteries are asked from the decision maker. The lotteries are Bernoulli lotteries generated by MARPI using special strategies. At every stage of the procedure the efficient set of alternatives is determined with respect to the information elicited so far.

Keywords: Partial information, linear expected utility, MAUT, bounds on expected utility, multivariate stochastic dominance (first and second degree, weak and strong), computer-based procedure.

# Abstract: Majorization in Economic Disparity Measures (Karl Mosler)

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 curve is introduced.

# Abstract: Multidimensional Welfarisms (Karl Mosler)

The problem of comparing social states with respect to several attributes is investigated in a purely ordinal framework. Axioms are given for an additive (in individual welfare levels) representation of the social welfare function. Then a general class of social welfare orderings is introduced which can be characterized by comparing the numbers of individuals (headcounts) in "critical sets" of individual states. A new approach to multivariate welfare orderings is proposed: Starting with one or a few critical sets, the headcounts in these sets are compared.

# Abstract: Stochastic Dominance with Nonadditive Probabilities (Rainer Dyckerhoff and Karl Mosler)

Choquet expected utility which uses capacities (i.e. nonadditive probability measures) in place of $\sigma$-additive probability measures has been introduced to decision making under uncertainty to cope with observed effects of ambiguity aversion like the Ellsberg paradox. In this paper we present necessary and sufficient conditions for stochastic dominance between capacities (i.e. the expected utility with respect to one capacity exceeds that with respect to the other one for a given class of utility functions). One wide class of conditions refers to probability inequalities on certain families of sets. To yield another general class of conditions we present sufficient conditions for the existence of a probability measure P with $\int f dC=\int f dP$ for all increasing functions f when C is a given capacity. Examples include n-th degree stochastic dominance on the reals and many cases of so-called set dominance. Finally, applications to decision making are given including anticipated utility with unknown distortion function.

# Abstract: Location of a Spatially Extended Facility with Respect to a Point (Karl Mosler)

In many planning situations a facility has to be located in space so that it cannot be reasonably modelled as a geometric point. Examples in public planning are the location of a recreational area (as a park) and the location of a large industrial plant and the location of a shopping area. In situations like these the location of a facility should be considered as a subset rather than a point in two-space. Besides the facility's extension in space often also the degree of agglomeration or the dispersion of activities in the area is of interest. In this paper, an area location of a facility is considered as a distribution of the facility's activities in the plane. Given a subset of the plane, the planning region, every point of this subset may be used for activities of the new facility. There is an existing facility whose location is given, and the only inhomogeneity of the plane is introduced by distances to this location. To give several examples, a new residential area may be located at an existing railway station, a worker's camp at a given mining shaft, or a commerical area at a bridge, a gate, or a custom's point. Also, an area location may be chosen with respect to an obnoxious facility like an incineration plant or a nuclear centre. The evaluation of an area location is based here on the local benefits per unit activity which depend solely on distances to the given facility, and on the local costs of agglomeration which depend on the density of activities. Criteria are maxisum benefit (i.e. minisum cost) and maximin benefit (i.e. minimax cost). E.g., when the facility is a residential area to be located near an existing railway station and every future resident has the same individual utility function depending on her distance to the station only, the criterion may be either utilitarian or Rawlsian, which corresponds to maxisum or maximin residents' utility.

# Abstract: Multiattribute utility functions, partial information on coefficients and efficient choice (Karl Mosler)

The expected utility analysis of decision under risk needs information on the alternatives and on the decision maker's preferences which in many practical situations are difficult to obtain. This paper presents a procedure for choosing between multiattribute risky alternatives when the probabilities of outcome are known, the utility function is general multilinear (i.e. can be decomposed into sums and products of univariate utility functions), and there is some partial information on univariate attributes (viz. increasingness) and arbitrary partial information on the scaling coefficients. Pairwise comparisons in the set of alternatives yield a subset which is efficient under the given partial information. Additive and multiplicative utility functions are special cases of the multilinear one. The paper gives particular attention to linear partial information (LPI) on coefficients, which is obtained by standard assessment procedures. The approach can be combined with dominance procedures which use other partial information as LPI on probabilities.

Key words: Decision theory, multiattribute choice, stochastic dominance, multilinear utility decomposition, linear partial information

# Abstract: Location of Physical Distribution Networks: A Continuous Approach (Karl Mosler)

This paper is about freight networks for distributing materials from one or several origins to a large number of destinations. Origins are manufacturing plants or suppliers, while destinations are sites of consumption or further manufacturing. A distributional system includes depots, trunks leading from plants to depots, and delivery lines from depots to consumption sites which together make up a distributional network. Our research focuses on the task: Determine the number and site of depots and network lines as well as the size of shipments. Our goal is to minimize the total costs of trunking and delivery as well as depots which include costs of production, of transportation and trunking and delivery, as well as costs of storage at plants, depots and customers.