Multivariate Discrete Distributions Research Articles

Semiparametric multinomial mixed-effects models: A university students profiling tool

Many applicative studies deal with multinomial responses and hierarchical data. Performing clustering at the highest level of grouping, in multilevel multinomial regression, is also often of interest. In this study we analyse Politecnico di Milano data with the aim of profiling students, modelling their probabilities of belonging to different categories and considering their nested structure within engineering degree programmes. In particular, we are interested in clustering degree programmes standing on their effects on different types of student career. To this end, we propose an EM algorithm for implementing semiparametric mixed-effects models dealing with a multinomial response. The novel semiparametric approach assumes the random effects to follow a multivariate discrete distribution with an a priori unknown number of support points, that is, allowed to differ across response categories. The advantage of this modelling is twofold: the discrete distribution on random effects allows, first, to express the marginal density as a weighted sum, avoiding numerical problems in the integration step, typical of the parametric approach, and, second, to identify a latent structure at the highest level of the hierarchy where groups are clustered into subpopulations.

State-Variable Modeling for a Class of Two-Stage Stochastic Optimization Problems

This paper considers a class of two-stage stochastic mixed-integer optimization problems where, for a given first-stage solution, we can determine the optimal values of recourse variables sequentially. This class of problems arises in a wide variety of applications. In the case of multivariate discrete distributions for uncertain parameters, a standard stochastic programming formulation of these problems involves an exponential number of scenarios, therefore an exponential number of variables and constraints. We propose a new mixed-integer programming modeling approach where the number of variables and constraints is independent of the number of scenarios and scales at most pseudopolynomially with the problem size. The proposed modeling approach relies on state variables that track the system’s state as the uncertainty realizes sequentially. We demonstrate the advantages of the proposed approach in two applications arising in project scheduling and operating room allocation. Summary of Contribution: This paper proposes a new modeling approach for a class of two-stage stochastic optimization problems that is computationally more efficient than the traditional scenario-based stochastic integer programming models. The proposed modeling approach relies on state variables that track the system's state as the uncertainty realizes sequentially. We demonstrated the efficiency of the proposed approach by computational results on two applications in project scheduling and operating room allocation.

An entropy divergence, D-distribution and its selected application in multivariate financial time series

This paper proposes an Entropy Divergence (ED) which is the difference between the two Shannon's differential entropies for multivariate discrete and continuous distributions. This is illustrated based on the multivariate version of the central limit theorem to evaluate the divergence between the multivariate scaled normal distribution (f) and scaled student's t distribution (g). This helps to firmly establish the exact distribution of ED, and its density is visualised in terms of Meijer G function. Moreover, the authors derived the distribution of the difference (D) between two EDs in terms of the sum of p-variate independent Fisher's-Z variable plus a normalising constant. Similarly, the moments of the two cases are derived which are visualised through digamma and poly-gamma functions. The percentage points of ED distribution of p-variates are computed at 5% and 1% significance level for respective degrees of freedom. Plots show the percentage points (discrepancy distance) between the multivariate distribution (normal) of financial time series and the distribution of cross-section data (student's t) graphically. The proposed ED is applied to test the sphericity of Nifty 50 stock returns in National Stock Exchange. Similarly, D-statistic is used to check the equality of two Shannon's differential entropy, where returns are grouped based on the demonetisation crisis as before and after in India.

A family of multivariate discrete distributions

A family of multivariate discrete distributions

Fuzzy Hypercubes and their time-like evolution

The concept of Fuzzy Hypercube is defined as a simple trigonometric extension of the binary structure of an N-dimensional Boolean hypercube. Moreover, in the present study, there is a discussion on the possibility of defining Fuzzy Hypercubes as a set of $$ 2^{N} $$ vertices, which remain Stationary or might undergo Synchronous or Asynchronous evolution. Finally, the connection of Fuzzy Hypercubes with multivariate discrete probability distributions is also considered.

Finding the modes of some multivariate discrete probability distributions: Application of the resource allocation problem

Determination of the modes of discrete multivariate distributions are of significant interest in many application contexts. Thus far, most papers in the literature have focused on developing specialized algorithms to find the modes of some specific distributions, e.g., the multinomial distribution. In contrast, this paper proposes a unified and efficient solution approach for the mode-finding problem for many well-known multivariate discrete distributions. In particular, we reformulate and solve the mode-finding problem as a discrete resource allocation problem using a greedy algorithm. Numerical examples show the effectiveness of our new solution approach.

A novel scale-space approach for multinormality testing and the k-sample problem in the high dimension low sample size scenario.

Two classical multivariate statistical problems, testing of multivariate normality and the k-sample problem, are explored by a novel analysis on several resolutions simultaneously. The presented methods do not invert any estimated covariance matrix. Thereby, the methods work in the High Dimension Low Sample Size situation, i.e. when n ≤ p. The output, a significance map, is produced by doing a one-dimensional test for all possible resolution/position pairs. The significance map shows for which resolution/position pairs the null hypothesis is rejected. For the testing of multinormality, the Anderson-Darling test is utilized to detect potential departures from multinormality at different combinations of resolutions and positions. In the k-sample case, it is tested whether k data sets can be said to originate from the same unspecified discrete or continuous multivariate distribution. This is done by testing the k vectors corresponding to the same resolution/position pair of the k different data sets through the k-sample Anderson-Darling test. Successful demonstrations of the new methodology on artificial and real data sets are presented, and a feature selection scheme is demonstrated.

Multivariate discrete distributions via sums and shares

In this article, we develop a sum and share decomposition to model multivariate discrete distributions, and more specifically multivariate count data that can be divided into a number of distinct categories. From a Poisson mixture model for the sum and a multinomial mixture model for the shares, a rich ensemble of properties, examples and relationships arises. As a main example, a seemingly new multivariate model involving a negative binomial sum and Pólya shares is considered, previously seen only in the bivariate case, for which we present two contrasting applications. For other choices of the distribution of the sum, natural but novel discrete multivariate Liouville distributions emerge; an important special case of these is that of Schur constant distributions. Analogies and interactions with related continuous distributions are to the fore throughout.

Multivariate Poisson hidden Markov models with a case study of modelling seismicity

SummaryWhile the literature on multivariate models for continuous data flourishes, there is a lack of models for multivariate counts. We aim to contribute to this framework by extending the well known class of univariate hidden Markov models to the multidimensional case, by introducing multivariate Poisson hidden Markov models. Each state of the extended model is associated with a different multivariate discrete distribution. We consider different distributions with Poisson marginals, starting from the multivariate Poisson distribution and then extending to copula based distributions to allow flexible dependence structures. An EM type algorithm is developed for maximum likelihood estimation. A real data application is presented to illustrate the usefulness of the proposed models. In particular, we apply the models to the occurrence of strong earthquakes (surface wave magnitude ≥5), in three seismogenic subregions in the broad region of the North Aegean Sea for the time period from 1 January 1981 to 31 December 2008. Earthquakes occurring in one subregion may trigger events in adjacent ones and hence the observed time series of events are cross‐correlated. It is evident from the results that the three subregions interact with each other at times differing by up to a few months. This migration of seismic activity is captured by the model as a transition to a state of higher seismicity.

On the evaluation of some multivariate compound distributions with Sarmanov’s counting distribution

In this paper, we consider Sarmanov’s multivariate discrete distribution as counting distribution in two multivariate compound models: the first model assumes different types of independent claim sizes (corresponding to, e.g., different types of insurance policies), while in the second model, we introduce some dependency between the claims (motivated by the events that can simultaneously affect several types of policies). Since Sarmanov’s distribution can join different types of marginals, we also assume that these marginals belong to Panjer’s class of distributions and discuss the evaluation of the resulting compound distribution based on recursions. Alternatively, the evaluation of the same distribution using the Fast Fourier Transform method is also presented, with the purpose to significantly reduce the computing time, especially in the dependency case. Both methods are numerically illustrated and compared from the point of view of speed and accuracy.

Sampling Designs on Finite Populations with Spreading Control Parameters

We present new sampling methods in finite population that allow to control the joint inclusion probabilities of units and especially the spreading of sampled units in the population. They are based on the use of renewal chains and multivariate discrete distributions to generate the difference of population ranks between two successive selected units. With a Bernoulli sampling design, these differences follow a geometric distribution, and with a simple random sampling design they follow a negative hypergeometric distribution. We propose to use other distributions and introduce a large class of sampling designs with and without fixed sample size. The choice of the rank-difference distribution allows us to control units joint inclusion probabilities with a relatively simple method and closed form formula. Joint inclusion probabilities of neighboring units can be chosen to be larger, or smaller, compared to those of Bernoulli or simple random sampling, thus allowing to more or less spread the sample on the population. This can be useful when neighboring units have similar characteristics or, on the contrary, are very different. A set of simulations illustrates the qualities of this method.

New sufficient conditions for strong unimodality of multivariate discrete distributions

New sufficient conditions that ensure the strong unimodality of multivariate discrete distributions are obtained by the use of a special simplicial subdivision of multidimensional space. Strong unimodality of multivariate Pólya–Eggenberger distribution is shown.

Morfizmy markowskie: zespolone podejście do wielowymiarowych kwantyli oparte o funkcje łącznikowe i zagadnienie transportowe

Our purpose is both conceptual and practical. On the one hand, we discuss the question which properties are basic ingredients of a general conceptual notion of a multivariate quantile. We propose and argue that the object “quantile” should be defined as a Markov morphism which carries over similar algebraic, ordering and topological properties as known for quantile functions on the real line. On the other hand, we also propose a practical quantile Markov morphism which combines a copula standardization and the recent optimal mass transportation method of Chernozhukov et al. (2015). Its empirical counterpart has the advantages of being a bandwidth-free, monotone invariant, a.s. consistent transformation. The proposed the approach gives a general and unified framework to quantiles and their corresponding depth areas, for both a continuous or a discrete multivariate distribution.

A note on inconsistent families of discrete multivariate distributions

We construct a d-dimensional discrete multivariate distribution for which any proper subset of its components belongs to a specific family of distributions. However, the joint d-dimensional distribution fails to belong to that family and in other words, it is ‘inconsistent’ with the distribution of these subsets. We also address preservation of this ‘inconsistency’ property for the symmetric Binomial distribution, and some discrete distributions arising from the multivariate discrete normal distribution.

Discrete random vectors generated by Taylor expansions

ABSTRACTWe demonstrate how univariate discrete and multivariate discrete distributions can be generated using Taylor expansions. Some of the results involve use of Bell polynomials.

Getting More Out of the Exposed Structure in Constraint Programming Models of Combinatorial Problems

To solve combinatorial problems, Constraint Programming builds high-level models that expose much of the structure of the problem. The distinctive driving force of Constraint Programming has been this direct access to problem structure. This has been key to the design of powerful filtering algorihms but we could do much more. Considering the set of solutions to each constraint as a multivariate discrete distribution opens the door to more structure-revealing computations that may significantly change this solving paradigm. As a result we could improve our ability to solve combinatorial problems and our understanding of the structure of practical problems.

On the Evaluation of Some Multivariate Compound Distributions with Sarmanov's Counting Distribution

In this paper, we consider Sarmanov's multivariate discrete distribution as counting distribution in two multivariate compound models: the First model assumes different types of independent claim sizes (corresponding to, e.g., different types of insurance policies), while in the second model, we introduce some dependency between the claims (motivated by the events that can simultaneously affect several types of policies). Since Sarmanov's distribution can join different types of marginals, we also assume that these marginals belong to Panjer's class of distributions and discuss the evaluation of the resulting compound distribution based on recursions. Alternatively, the evaluation of the same distribution using the Fast Fourier Transform method is also presented, with the purpose to significantly reduce the computing time, especially in the dependency case. Both methods are numerically illustrated and compared from the point of view of speed and accuracy.

Applying Multivariate Discrete Distributions to Genetically Informative Count Data.

We present a novel method of conducting biometric analysis of twin data when the phenotypes are integer-valued counts, which often show an L-shaped distribution. Monte Carlo simulation is used to compare five likelihood-based approaches to modeling: our multivariate discrete method, when its distributional assumptions are correct, when they are incorrect, and three other methods in common use. With data simulated from a skewed discrete distribution, recovery of twin correlations and proportions of additive genetic and common environment variance was generally poor for the Normal, Lognormal and Ordinal models, but good for the two discrete models. Sex-separate applications to substance-use data from twins in the Minnesota Twin Family Study showed superior performance of two discrete models. The new methods are implemented using R and OpenMx and are freely available.

Parameter Estimation of Discrete Multivariate Phase-Type Distributions

This paper considers parameter estimation of a class of discrete multi-variate phase-type distributions (DMPH). Such discrete phase-type distributions are based on discrete Markov chains with marked transitions introduced by He and Neuts (Stoch Process Appl 74(1):37–52, 1998) and is a generalization of the discrete univariate phase–type distributions. Properties of the DMPHs are presented. An EM-algorithm is developed for estimating the parameters for DMPHs. A number of numerical examples are provided to address some interesting parameter selection issues and to show possible applications of DMPHs.

Maximal coupling of empirical copulas for discrete vectors

For a vector X with a purely discrete multivariate distribution, we give simple short proofs of uniform a.s. convergence on their whole domain of two versions of genuine empirical copula functions, obtained either via probabilistic continuation, i.e. kernel smoothing, or via the distributional transform. These results give a positive answer to some delicate issues related to the convergence of copula functions in the discrete case. They are obtained under the very weak hypothesis of ergodicity of the sample, a framework which encompasses most types of serial dependence encountered in practice. Moreover, they allow to derive, as simple corollaries, almost sure consistency results for some recent extensions of concordance measures attached to discrete vectors. The proofs are based on a maximal coupling construction of the empirical cdf, a result of independent interest.