On the hyper-dirichlet type 1 and hyper-liouville distributions

This paper concerns the characterization of a new family of multivariate beta distribution functions - the hyper-Dirichlet type 1 distribution. This family describes the joint density function of the terminal variates of an arbitrary tree constructed from finite sequences of probability vectors having independent Dirichlet type 1 distributions. Expressions for the general properties of the hyper-Dirichlet type 1 distribution are presented. In addition, the hyper-Liouville distribution is described and its properties are discussed as well as a generalization of the Liouville integral identity.     

Eliciting expert judgements about a set of proportions

Eliciting expert knowledge about several uncertain quantities is a complex task when those quantities exhibit associations. A well-known example of such a problem is eliciting knowledge about a set of uncertain proportions which must sum to 1. The usual approach is to assume that the expert's knowledge can be adequately represented by a Dirichlet distribution, since this is by far the simplest multivariate distribution that is appropriate for such a set of proportions. It is also the most convenient, particularly when the expert's prior knowledge is to be combined with a multinomial sample since then the Dirichlet is the conjugate prior family. Several methods have been described in the literature for eliciting beliefs in the form of a Dirichlet distribution, which typically involve eliciting from the expert enough judgements to identify uniquely the Dirichlet hyperparameters. We describe here a new method which employs the device of over-fitting, i.e. eliciting more than the minimal number of judgements, in order to (a) produce a more carefully considered Dirichlet distribution and (b) ensure that the Dirichlet distribution is indeed a reasonable fit to the expert's knowledge. The method has been implemented in a software extension of the Sheffield elicitation framework (SHELF) to facilitate the multivariate elicitation process.     

On the Dirichlet distributions on symmetric matrices

In this paper, we introduce a generalization of the Dirichlet distribution on symmetric matrices which represents the multivariate version of the Connor and Mosimann generalized real Dirichlet distribution. We establish some properties concerning this generalized distribution. We also extend to the matrix Dirichlet distribution a remarkable characterization established in the real case by Darroch and Ratcliff.     

Two multivariate generalized beta families

A multivariate generalized beta distribution is introduced that extends the univariate generalized beta distribution and includes many multivariate distributions, such as the multivariate beta of the first and second kind, the generalized gamma, and the Burr and Dirichlet distributions as special and limiting cases. These interrelationships can be illustrated using a distributional family tree. The corresponding marginal distributions are univariate generalized beta distributions and their special cases. Selected expressions for the moments are reported, and an application to the joint distribution of income and wealth is presented. A simple transformation of the multivariate generalized beta distribution leads to what will be referred to as a multivariate exponential generalized beta distribution, which includes a multivariate form of the logistics and Burr distributions as special cases.     

Nonparametric Mixtures Based on Skew-normal Distributions: An Application to Density Estimation

This article addresses the density estimation problem using nonparametric Bayesian approach. It is considered hierarchical mixture models where the uncertainty about the mixing measure is modeled using the Dirichlet process. The main goal is to build more flexible models for density estimation. Extensions of the normal mixture model via Dirichlet process previously introduced in the literature are twofold. First, Dirichlet mixtures of skew-normal distributions are considered, say, in the first stage of the hierarchical model, the normal distribution is replaced by the skew-normal one. We also assume a skew-normal distribution as the center measure in the Dirichlet mixture of normal distributions. Some important results related to Bayesian inference in the location-scale skew-normal family are introduced. In particular, we obtain the stochastic representations for the full conditional distributions of the location and skewness parameters. The algorithm introduced by MacEachern and Müller in 1998 MacEachern, S.N., Müller, P. (1998). Estimating mixture of Dirichlet Process models. J. Computat. Graph. Statist. 7(2):223–238.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar] is used to sample from the posterior distributions. The models are compared considering simulated data sets. Finally, the well-known Old Faithful Geyser data set is analyzed using the proposed models and the Dirichlet mixture of normal distributions. The model based on Dirichlet mixture of skew-normal distributions captured the data bimodality and skewness shown in the empirical distribution.     

Wilks’ factorization of the matrix-variate Dirichlet–Riesz distributions

The Riesz distributions on positive definite symmetric matrices are used to introduce a class of Dirichlet–Riesz distributions. In addition, several distributional properties are stated. Essentially, we show the relationship between the Dirichlet–Riesz distributions of the first kind and the second kind, respectively. We derive Wilks’ factorization of the matrix-variate Dirichlet–Riesz. Further, several results on the product of Riesz and beta–Riesz matrices with a set of Dirichlet–Riesz matrices of the first kind have been derived.     

Estimations of the parameter of a Dirichlet distribution using residual allocation model representations and sampling properties

Estimating the parameter of a Dirichlet distribution is an interesting question since this distribution arises in many situations of applied probability. Classical procedures are based on sample of Dirichlet distribution. In this paper we exhibit five different estimators from only one observation. They are based either on residual allocation model decompositions or on sampling properties of Dirichlet distributions. Two ways are investigated: the first one uses fragments’ size and the second one uses size-biased permutations of a partition. Numerical computations based on simulations are supplied. The estimators are finally used to estimate birth probabilities per month.     

Analysing the interevent time distribution to identify seismicity phases: a Bayesian nonparametric approach to the multiple-changepoint problem

In the study of earthquakes, several aspects of the underlying physical process, such as the time non-stationarity of the process, are not yet well understood, because we lack clear indications about its evolution in time. Taking as our point of departure the theory that the seismic process evolves in phases with different activity patterns, we have attempted to identify these phases through the variations in the interevent time probability distribution within the framework of the multiple-changepoint problem. In a nonparametric Bayesian setting, the distribution under examination has been considered a random realization from a mixture of Dirichlet processes, the parameter of which is proportional to a generalized gamma distribution. In this way we could avoid making precise assumptions about the functional form of the distribution. The number and location in time of the phases are unknown and are estimated at the same time as the interevent time distributions. We have analysed the sequence of main shocks that occurred in Irpinia, a particularly active area in southern Italy: the method consistently identifies changepoints at times when strong stress releases were recorded. The estimation problem can be solved by stochastic simulation methods based on Markov chains, the implementation of which is improved, in this case, by the good analytical properties of the Dirichlet process.     

On a class of distributions on the simplex

In the present paper we define and investigate a novel class of distributions on the simplex, termed normalized infinitely divisible distributions, which includes the Dirichlet distribution. Distributional properties and general moment formulae are derived. Particular attention is devoted to special cases of normalized infinitely divisible distributions which lead to explicit expressions. As a by-product new distributions over the unit interval and a generalization of the Bessel function distribution are obtained.     

On upper bounds for the characteristic values of the covariance matrix for multinomial,dirichlet and multivariate hypergeometric distributions

For the characteristic values T1 of the matrix V:=Diag(p)-pp^T with p=(p1,...,pk), p1≥p2≥...≥pk≥pk+1>0 and p1+p2+...+pk+pk+1=1 the inequalities p1≥τ1≥p2≥τ2≥...≥pk≥τk>0 are given by RONNING (1982). These inequalities give, if p and pk+1 are unknown, the upper bound 1≥T1. However, in this note the bound 1/2≥T1 is derived. V is proportional to the covariance matrix for multinomial, Dirichlet and multivariate hypergeometric distributions. A statistical application for the multinomial distribution is given.     

Bayesian nonparametric survival analysis using mixture of Burr XII distributions

ABSTRACT

Recently, the Bayesian nonparametric approaches in survival studies attract much more attentions. Because of multimodality in survival data, the mixture models are very common. We introduce a Bayesian nonparametric mixture model with Burr distribution (Burr type XII) as the kernel. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and real failure time datasets, we show the preference of this model and compare it with Dirichlet process mixture models with different kernels. The Markov chain Monte Carlo (MCMC) simulation methods to calculate the posterior distribution are used.     

On several estimates to the precision parameter of Dirichlet process prior

The article presents careful comparisons among several empirical Bayes estimates to the precision parameter of Dirichlet process prior, with the setup of univariate observations and multigroup data. Specifically, the data are equipped with a two-stage compound sampling model, where the prior is assumed as a Dirichlet process that follows within a Bayesian nonparametric framework. The precision parameter α measures the strength of the prior belief and kinds of estimates are generated on the basis of observations, including the naive estimate, two calibrated naive estimates, and two different types of maximum likelihood estimates stemming from distinct distributions. We explore some theoretical properties and provide explicitly detailed comparisons among these estimates, in the perspectives of bias, variance, and mean squared error. Besides, we further present the corresponding calculation algorithms and numerical simulations to illustrate our theoretical achievements.     

Posterior Distributions for the Gini Coefficient Using Grouped Data

When available data comprise a number of sampled households in each of a number of income classes, the likelihood function is obtained from a multinomial distribution with the income class population proportions as the unknown parameters. Two methods for going from this likelihood function to a posterior distribution on the Gini coefficient are investigated. In the first method, two alternative assumptions about the underlying income distribution are considered, namely a lognormal distribution and the Singh–Maddala (1976) income distribution. In these cases the likelihood function is reparameterized and the Gini coefficient is a nonlinear function of the income distribution parameters. The Metropolis algorithm is used to find the corresponding posterior distributions of the Gini coefficient from a sample of Bangkok households. The second method does not require an assumption about the nature of the income distribution, but uses (a) triangular prior distributions, and (b) beta prior distributions, on the location of mean income within each income class. By sampling from these distributions, and the Dirichlet posterior distribution of the income class proportions, alternative posterior distributions of the Gini coefficient are calculated.     

A SEMIPARAMETRIC BAYESIAN APPROACH TO NETWORK MODELLING USING DIRICHLET PROCESS PRIOR DISTRIBUTIONS

This paper considers the use of Dirichlet process prior distributions in the statistical analysis of network data. Dirichlet process prior distributions have the advantages of avoiding the parametric specifications for distributions, which are rarely known, and of facilitating a clustering effect, which is often applicable to network nodes. The approach is highlighted for two network models and is conveniently implemented using WinBUGS software.     

PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE

This paper investigates the roles of partial correlation and conditional correlation as measures of the conditional independence of two random variables. It first establishes a sufficient condition for the coincidence of the partial correlation with the conditional correlation. The condition is satisfied not only for multivariate normal but also for elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial and Dirichlet distributions. Such families of distributions are characterized by a semigroup property as a parametric family of distributions. A necessary and sufficient condition for the coincidence of the partial covariance with the conditional covariance is also derived. However, a known family of multivariate distributions which satisfies this condition cannot be found, except for the multivariate normal. The paper also shows that conditional independence has no close ties with zero partial correlation except in the case of the multivariate normal distribution; it has rather close ties to the zero conditional correlation. It shows that the equivalence between zero conditional covariance and conditional independence for normal variables is retained by any monotone transformation of each variable. The results suggest that care must be taken when using such correlations as measures of conditional independence unless the joint distribution is known to be normal. Otherwise a new concept of conditional independence may need to be introduced in place of conditional independence through zero conditional correlation or other statistics.     

Topics on Dynamic Panel Data Models with Random Effects Using Semi-Parametric Bayesian Approach

A common assumption in fitting panel data models is normality of stochastic subject effects. This can be extremely restrictive, making vague most potential features of true distributions. The objective of this article is to propose a modeling strategy, from a semi-parametric Bayesian perspective, to specify a flexible distribution for the random effects in dynamic panel data models. This is addressed here by assuming the Dirichlet process mixture model to introduce Dirichlet process prior for the random-effects distribution. We address the role of initial conditions in dynamic processes, emphasizing on joint modeling of start-up and subsequent responses. We adopt Gibbs sampling techniques to approximate posterior estimates. These important topics are illustrated by a simulation study and also by testing hypothetical models in two empirical contexts drawn from economic studies. We use modified versions of information criteria to compare the fitted models.     

A Bayesian semi-parametric approach to the ordinal calibration problem

ABSTRACT

We introduce a semi-parametric Bayesian approach based on skewed Dirichlet processes priors for location parameters in the ordinal calibration problem. This approach allows the modeling of asymmetrical error distributions. Conditional posterior distributions are implemented, thus allowing the use of Markov chains Monte Carlo to generate the posterior distributions. The methodology is applied to both simulated and real data.     

Household budget-share distributions and welfare implications: an application of multivariate distributional statistics

In this paper the consequences of considering the household ‘food share’ distribution as a welfare measure, in isolation from the joint distribution of itemized budget shares, is examined through the unconditional and conditional distribution of ‘food share’ both parametrically and nonparametrically. The parametric framework uses Dirichlet and Beta distributions, while the nonparametric framework uses kernel smoothing methods. The analysis, in a three commodity setup (‘food’, ‘durables’, ‘others’), based on household level rural data for West Bengal, India, for the year 2009–2010 shows significant underrepresentation of households by the conventional unconditional ‘food share’ distribution in the higher range of food budget shares that correspond to the lower end of the income profile. This may have serious consequences for welfare measurement.     

Saddlepoint Approximations to the Density and the Distribution Functions of Linear Combinations of Ratios of Partial Sums

A class of ratios of partial sums, including Normal, Weibull, Gamma, and Exponential distributions, is considered. The distribution of a linear combination of ratios of partial sums from this class is characterized by the distribution of a linear combination of Dirichlet components. This article presents two saddlepoint approaches to calculate the density and the distribution function for such a class of linear combinations. A simulation study is conducted to assess the performance of the saddlepoint methods and shows the great accuracy of the approximations over the usual asymptotic approximation. Applications of the presented approximations in statistical inferences are discussed.     

Modeling a minimal spanning tree

We model the distribution of the normalized interpoint distances (IDs) on the minimal spanning tree (MST) using multivariate beta vectors. We use a multivariate normal copula with beta marginals and a Dirichlet distribution to obtain beta vectors. Based on the normalized ordered IDs of the MST, we define a multivariate Gini index to measure the scatter of a data cloud. An example considers the MST of numerals in 11 European languages and obtains their Gini index. A simulation study compares the Gini index, the maximum and the range of the IDs for multivariate normal and log-normal data, with the results of modeling the distances on the MST.