Multivariate Normal Distribution

One possible approach to cluster analysis is the mixture maximum likelihood method, in which the data to be clustered are assumed to come from a finite mixture of populations. The method has been well developed, and much used, for the case of multivariate normal populations. Practical applications, however, often involve mixtures of categorical and continuous variables. Everitt (1988) and Everitt and Merette (1990) recently extended the normal model to deal with such data by incorporating the use of thresholds for the categorical variables. The computations involved in this model are so extensive, however, that it is only feasible for data containing very few categorical variables. In the present paper we consider an alternative model, known as the homogeneous Conditional Gaussian model in graphical modelling and as the location model in discriminant analysis. We extend this model to the finite mixture situation, obtain maximum likelihood estimates for the population parameters, and show that computation is feasible for an arbitrary number of variables. Some data sets are clustered by this method, and a small simulation study demonstrates characteristics of its performance.

A new quasi-imputation strategy for correlated ordinal responses is proposed by borrowing ideas from random number generation. The essential idea is collapsing ordinal levels to binary ones, and converting correlated binary outcomes to multivariate normal outcomes in a sensible way so that re-conversion to the binary and then ordinal scale, after conducting multiple imputation, yields the original marginal distributions and correlations. This conversion process ensures that the correlations are transformed reasonably which in turn allows us to take advantage of well-developed imputation techniques for Gaussian outcomes. We use the phrase "quasi" because the original observations are not guaranteed to be preserved. We present an application using a data set from psychiatric research. We conclude that the proposed method may be a promising tool for handling incomplete longitudinal or clustered ordinal outcomes.

This paper considers the estimation of the mean vector of a p-variate normal distribution with unknown covariance matrix when it is suspected that for a p × r known matrix B the hypothesis = B , ∈ R r may hold. We consider empirical Bayes estimators which includes (i) the unrestricted unbiased (UE) estimator, namely, the sample mean vector (ii) the restricted estimator (RE) which is obtained when the hypothesis = B holds (iii) the preliminary test estimator (PTE), (iv) the James-Stein estimator (JSE), and (v) the positive-rule Stein estimator (PRSE). The biases and the risks under the squared loss function are evaluated for all the five estimators and compared. The numerical computations show that PRSE is the best among all the five estimators even when the hypothesis = B is true.

Class-modeling techniques, classic and recent, are studied with special reference with the new applications to data sets characterized by many variables, frequently noisy variables without importance in the characterization of the studied class. UNEQ (based on the hypothesis of multivariate normal distribution and on the Hotelling T 2 statistics), SIMCA (with a model built on the class principal components), POTFUN (Potential Functions Modeling, where the probability distribution is estimated by means of the potential functions), MRM (Multivariate Range Modeling, where the model is obtained with the range of the original variables and of discriminant functions) are compared by means of the sensitivities and specificities of the models evaluated both by means of cross validation and with the model forced to accept all the objects of the modeled category. The parameters used to evaluate the performance of class-modeling techniques are critically reviewed. The performances of class-modeling techniques, both in classification and in modeling, have been evaluated on real data sets, with the original variables and on subsets of variables obtained after elimination of nondiscriminant variables. The effect of noisy variables and of deviation from the underlying hypotheses are discussed. Therefore, a "perfect" class model has 100% sensitivity and 100% specificity.

We present an adaptive method for the automatic scaling of Random-Walk Metropolis-Hastings algorithms, which quickly and robustly identifies the scaling factor that yields a specified overall sampler acceptance probability. Our method relies on the use of the Robbins-Monro search process, whose performance is determined by an unknown steplength constant. We give a very simple estimator of this constant for proposal distributions that are univariate or multivariate normal, together with a sampling algorithm for automating the method. The effectiveness of the algorithm is demonstrated with both simulated and real data examples. This approach could be implemented as a useful component in more complex adaptive Markov chain Monte Carlo algorithms, or as part of automated software packages.

A statistical framework is presented for examining cost and effect data on competing interventions obtained from an RCT or from an observational study. Parameters of the joint distribution of costs and effects or a regression function linking costs and effects are used to define cost-effectiveness (c-e) measures. Several new c-e measures are proposed that utilize the linkage between costs and effects on the patient level. These measures reflect perspectives that are different from those of the commonly used measures, such as the ratio of expected cost to expected effect, and they can lead to different relative rankings of the interventions. The cost-effectiveness of interventions are assessed statistically in a two stage procedure that first eliminates clearly inferior interventions. Members of the remaining admissible set are then rank ordered according to a c-e preference measure. Statistical techniques, particularly in the multivariate normal case, are given for several commonly used c-e measures. These techniques provide methods for obtaining confidence intervals, for testing the hypothesis of admissibility and for the equality of interventions, and for ranking interventions. The ideas are illustrated for a hypothetical clinical trial of antipsychotic agents for community-based persons with mental illness. 0

Many cost-effectiveness analyses (CEAs) use data from observational studies. Statistical methods can only address selection bias if they make plausible assumptions. No quality assessment tool is available for appraising CEAs that use observational studies. We developed a new checklist to assess statistical methods for addressing selection bias in CEAs that use observational data. The checklist criteria were informed by a conceptual review and applied in a systematic review of economic evaluations. Criteria included whether the study assessed the ‘no unobserved confounding’ assumption, overlap of baseline covariates between the treatment groups and the specification of the regression models. The checklist also considered structural uncertainty from the choice of statistical approach. We found 81 studies that met the inclusion criteria: studies tended to use regression (51%), matching on individual covariates (25%) or matching on the propensity score (22%). Most studies (77%) did not ...

In this work parametric and non-parametric statistical methods are proposed to analyze Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) data. A Multivariate Normal Distribution is proposed as a parametric statistical model of diffusion tensor data when magnitude MR images contain no artifacts other than Johnson noise. We test this model using Monte Carlo (MC) simulations of DT-MRI experiments. The non-parametric approach proposed here is an implementation of bootstrap methodology that we call the DT-MRI bootstrap. It is used to estimate an empirical probability distribution of experimental DT-MRI data, and to perform hypothesis tests on them. The DT-MRI bootstrap is also used to obtain various statistics of DT-MRI parameters within a single voxel, and within a region of interest (ROI); we also use the bootstrap to study the intrinsic variability of these parameters in the ROI, independent of background noise. We evaluate the DT-MRI bootstrap using MC simulations and apply it to DT-MRI data acquired on human brain in vivo, and on a phantom with uniform diffusion properties. Published by Elsevier Science (USA).

Experimental conjoint choice analysis is among the most frequently used methods for measuring and analyzing consumer preferences. The data from such experiments have been typically analyzed with the Multinomial Logit (MNL) model. However, there are several problems associated with the standard MNL model because it is based on the assumption that the error terms of the underlying random utilities are independent across alternatives, choice sets, and subjects. The Multinomial Probit model (MNP) is well known to alleviate this assumption of independence of the error terms. Accounting for covariances in utilities in modeling choice experiments with the MNP is important because variation of the coefficients in the choice model may occur due to context effects. Previous research has shown that subjects' utilities for alternatives depend on the choice context, that is, the particular set of alternatives evaluated. Simonson and Tversky's tradeoff contrast principle describes the effect of the choice context on attribute importance and patterns of choice. They distinguish local contrast effects, which are caused by the alternatives in the offered set only, and background contrast effects, which are due to the influence of alternatives previously considered in choice experiments. These effects are hypothesized to cause correlations in the utilities of alternatives within and across choice sets, respectively.

We analyse the Generalised Hyperbolic distribution as a model for fat tails and asymmetries in multivariate conditionally heteroskedastic dynamic regression models. We provide a standardised version of this distribution, obtain analytical expressions for the log-likelihood score, and explain how to evaluate the information matrix. In addition, we derive tests for the null hypotheses of multivariate normal and Student t innovations, and decompose them into skewness and kurtosis components, from which we obtain more powerful one-sided versions. Finally, we present an empirical illustration with UK sectorial stock returns, which suggests that their conditional distribution is asymmetric and leptokurtic.

In a regression setting, the partial correlation coef®cient is often used as a measure of standardized' partial association between the outcome y and each of the covariates in x 9 (x 1 , F F F, x K ). In a linear regression model estimated by using ordinary least squares, with y as the response, the estimated partial correlation coef®cient between y and x k can be shown to be a monotone function of the z-statistic for testing whether the regression coef®cient of x k is 0; it can also be shown to be a monotone function of the likelihood ratio statistic. When y is normal but the data are clustered so that y and x are obtained from each member of a cluster, maximum likelihood using a multivariate normal distribution is often used to estimate the regression parameters of the model for y given x. For clustered data, we propose two measures of partial association: the exact same monotone function of the repeated measures' z-statistic that is used to form the partial correlation coef®cient in the normal linear regression setting and the exact same monotone function of the likelihood ratio statistic that is used to form the partial correlation coef®cient in the normal linear regression setting. We also propose an R 2 -like measure which can be used to measure the overall predictive ability of the model. To illustrate the method, we use a longitudinal study concerning the stress on the wall of the heart from chemotherapy in children with leukaemia.

Dalam perkembangan sistem ekonomi Indonesia, persaingan antar perusahaan pada era globalisasi semakin ketat dan menjadi hal yang wajar. Daya saing perusahaan sendiri tidak hanya terjadi dalam keunggulan produk yang ditawarkan, melainkan kualitas dan keamanan produk bagi kesehatan juga turut dipertimbangkan. Perusahaan “X” merupakan salah satu pabrik yang bergerak di industri kosmetik, utamanya produksi parfum remaja. Dengan target pasar remaja, dimana remaja lebih sensitif terhadap zat-zat kimia, perusahaan “X” menjaga kualitas produknya dengan menetapkan spesifikasi untuk karakteristik kualitas dalam produk parfum yang diproduksi. Pengendalian kualitas statistik merupakan teknik penyelesaian masalah yang digunakan untuk memonitor, mengendalikan, menganalisis, mengelola, dan memperbaiki produk dan proses menggunakan metode-metode statistik. Pada praktikum ini, akan dibahas mengenai pengendalian kualitas terhadap karakteristik kualitas pada parfum remaja yang diproduksi oleh Perusahaan “X” yang diukur dengan pH dan massa jenis menggunakan peta kendali Generalized Variance dan T2 Hotelling. Data yang dianalisis harus memenuhi asumsi dependen dan berdistribusi normal multivariat, dan selanjutnya akan dilakukan analisis menggunakan peta kendali Generalized Variance dan T2 Hotelling yang kemudian dilanjutkan dengan analisis kapabilitas proses. Dari pengujian yang dilakukan, diketahui bahwa data telah memenuhi asumsi dependen dan berdistribusi normal multivariat. Peta kendali Generalized Variance menunjukkan bahwa variabilitas proses produksi parfum remaja produksi perusahaan “X” telah terkendali secara statistik atau sudah stabil sehingga dapat dilanjutkan menganalisis dengan peta kendali T2 Hotelling. Pada peta kendali T2 Hotelling terdapat satu pengamatan yang out of control sehingga rata-rata proses produksi parfum remaja oleh perusahaan “X” belum terkendali secara statistik atau belum stabil. Pengamatan yang out of control tersebut adalah pengamatan ke-3 dan setelah diidentifikasi, diketahui penyebabnya adalah kedua karakteristik kualitas, yaitu pH dan massa jenis. Namun, pada praktikum ini diasumsikan telah in control. Pada keadaan in control diperoleh MCp sebesar 1,74 dan MCpk sebesar 1,08 sehingga dapat disimpulkan bahwa produk parfum remaja memiliki variabilitas rendah dan telah sesuai spesifikasi perusahaan yang artinya kualitas produk parfum remaja produksi perusahaan “X” telah baik.

The views expressed in this Working Paper are those of the author(s) and do not necessarily represent those of the IMF or IMF policy. Working Papers describe research in progress by the author(s) and are published to elicit comments and to further debate.

Robust estimators of location and dispersion are often used in the elliptical model to obtain an uncontaminated and highly representative subsample by trimming the data outside an ellipsoid based in the associated Mahalanobis distance. Here we analyze some one (or $k$)-step Maximum Likelihood Estimators computed on a subsample obtained with such a procedure. We introduce different models which arise naturally from the ways in which the discarded data can be treated, leading to truncated or censored likelihoods, as well as to a likelihood based on an only outliers gross errors model. Results on existence, uniqueness, robustness and asymptotic properties of the proposed estimators are included. A remarkable fact is that the proposed estimators generally keep the breakdown point of the initial (robust) estimators, but they could improve the rate of convergence of the initial estimator because our estimators always converge at rate $n^{1/2}$, independently of the rate of convergence of the initial estimator.

Seiring dengan semakin diminatinya minuman wine, kebutuhan konsumen akan minuman ini meningkat dengan banyak negara yang juga mendukung pertumbuhan industri ini. Sertifikasi guna meyakinkan konsumen akan kualitas serta pencegahan terhadap pemalsuan wine juga diperlukan. Oleh sebab itu, diperlukan penilaian kualitas anggur, di mana variabel yang masuk dalam penelitian ini adalah alkohol, asam malat dan ash. Dalam penelitian ini, akan dibahas mengenai statistika deskriptif serta analisis diskriminan menggunakan metode canonical pada data penilaian kualitas anggur Portugis "Vinho Verde" di tahun 2009. Dalam melakukan analisis, asumsi yang harus dipenuhi adalah bahwa data mengikuti distribusi normal multivariat dan asumsi homogenitas. Setelah dilakukan analisis, didapatkan hasil bahwa data mengikuti distribusi normal multivariat dan diasumsikan homogen sehingga memenuhi asumsi dalam melakukan analisis diskriminan. Berdasarkan hasil analisis diskriminan menggunakan metode canonical didapatkan nilai akurasi sebesar 92,38%, sehingga pengelompokan kualitas wine dengan analisis diskriminan menggunakan metode canonical dapat mengklasifikasikan kualitas wine sebesar 92,38%.

I n this research, we investigate the behavior of Cronbach's coefficient alpha and its new standard error. We systematically analyze the effects of sample size, scale length, strength of item intercorrelations, and scale dimensionality. We demonstrate the beneficial effects of sample size on alpha's standard error and of scale length and the strengths of item intercorrelations (effects that are substitutes in their benefits) on both alpha and its standard error. Our findings also speak to this adage: Heterogeneity within the item covariance matrix (e.g., through multidimensionality or poor items) negatively impacts reliability by decreasing the precision of the estimation. We also examined the question of "equilibrium" scale length, showing the conditions for which it is optimal to add no items, or one, or multiple items to a scale. In terms of "best practices," we recommend that researchers report a confidence interval or standard error along with the coefficient alpha point estimate.

The heavy-tailed Multivariate Normal Inverse Gaussian (MNIG) distribution is a recent variance-mean mixture of a multivariate Gaussian with a univariate inverse Gaussian distribution. Due to the complexity of the likelihood function, parameter estimation by direct maximization is exceedingly difficult. To overcome this problem, we propose a fast and accurate multivariate Expectation-Maximization (EM) algorithm for maximum likelihood estimation of the scalar, vector, and matrix parameters of the MNIG distribution. Important fundamental and attractive properties of the MNIG as a modeling tool for multivariate heavy-tailed processes are discussed. The modeling strength of the MNIG, and the feasibility of the proposed EM parameter estimation algorithm, are demonstrated by fitting the MNIG to real world wideband synthetic aperture sonar data.

In this article, an improved method of computing tolerance factors for multivariate normal distributions is proposed. The method involves an approximation and simulation, and is more accurate than the several approximate methods considered in Krishnamoorthy and Mathew (1999). The accuracies of the tolerance regions are evaluated using Monte Carlo simulation. Simulation study shows that the new approach is very satisfactory

In this paper we compare some modern algorithms i.e. Direct Maximization of the Likelihood (DML), the EM algorithm, and Multiple Imputation (MI) for analyzing multivariate normal data with missing responses. We also compare two approaches for modeling incomplete data (1) ignoring missing data and (2) joint modeling of response and non-response mechanisms. Several types of Software which can be used

Understanding the dynamics of high dimensional non-normal dependency structure is a challenging task. This research aims at attacking this problem by building up a hidden Markov model (HMM) for Hierarchical Archimedean Copulae (HAC), where the HAC represent a wide class of models for high dimensional dependency, and HMM is a statistical technique to describe time varying dynamics. HMM applied to HAC provide flexible modeling for high dimensional non Gaussian time series. Consistency results for both parameters and HAC structures are established in an HMM framework. The model is calibrated to exchange rate data with a VaR application, where the model’s performance is compared with other dynamic models, and in the second application we simulate rainfall process.

Multivariate GARCH models are in principle able to accommodate the features of the dynamic conditional correlations processes, although with the drawback, when the number of financial returns series considered increases, that the parameterizations entail too many parameters.In general, the interaction between model parametrization of the second conditional moment and the conditional density of asset returns adopted in the estimation determines the fitting of such models to the observed dynamics of the data. This paper aims to evaluate the interactions between conditional second moment specifications and probability distributions adopted in the likelihood computation, in forecasting volatilities and covolatilities. We measure the relative performances of alternative conditional second moment and probability distributions specifications by means of Monte Carlo simulations, using both statistical and financial forecasting loss functions.

This paper analyzes discrete time portfolio selection models with Lévy processes. We first implement portfolio models under the hypotheses the vector of log-returns follow or a multivariate Variance Gamma model or a Multivariate Normal Inverse Gaussian model or a Brownian Motion. In particular, we propose an ex-ante and an ex-post empirical comparisons by the point of view of different investors. Thus, we compare portfolio strategies considering different term structure scenarios and different distributional assumptions when unlimited short sales are allowed.

This paper presents Natural Evolution Strategies (NES), a recent family of algorithms that constitute a more principled approach to black-box optimization than established evolutionary algorithms. NES maintains a parameterized distribution on the set of solution candidates, and the natural gradient is used to update the distribution's parameters in the direction of higher expected fitness. We introduce a collection of techniques that address issues of convergence, robustness, sample complexity, computational complexity and sensitivity to hyperparameters. This paper explores a number of implementations of the NES family, ranging from general-purpose multi-variate normal distributions to heavy-tailed and separable distributions tailored towards global optimization and search in high dimensional spaces, respectively. Experimental results show best published performance on various standard benchmarks, as well as competitive performance on others.

An asymmetric multivariate generalization of the recently proposed class of normal mixture GARCH models is developed. Issues of parametrization and estimation are discussed. Conditions for covariance stationarity and the existence of the fourth moment are derived, and expressions for the dynamic correlation structure of the process are provided. In an application to stock market returns, it is shown that the disaggregation of the conditional (co)variance process generated by the model provides substantial intuition. Moreover, the model exhibits a strong performance in calculating out-of-sample Value-at-Risk measures.

In this paper we present a new multi-asset pricing model, which is built upon newly developed families of solvable multi-parameter single-asset diffusions with a nonlinear smile-shaped volatility and an affine drift. Our multi-asset pricing model arises by employing copula methods. In particular, all discounted single-asset price processes are modeled as martingale diffusions under a risk-neutral measure. The price processes are so-called UOU diffusions and they are each generated by combining a variable (Ito) transformation with a measure change performed on an underlying Ornstein-Uhlenbeck (Gaussian) process. Consequently, we exploit the use of a normal bridge copula for coupling the single-asset dynamics while reducing the distribution of the multi-asset price process to a multivariate normal distribution. Such an approach allows us to simulate multidimensional price paths in a precise and fast manner and hence to price path-dependent financial derivatives such as Asian-style and...

The main purpose of this paper is the study of the multivariate Behrens-Fisher distribution. It is defined as the convolution of two independent multivariate Student t distributions. Some representations of this distribution as the mixture of known distributions are shown. An important result presented in the paper is the elliptical condition of this distribution in the special case of proportional scale matrices of the Student t distributions in the defining convolution. For the bivariate Behrens-Fisher problem, the authors propose a non-informative prior distribution leading to highest posterior density (H.P.D.) regions for the difference of the mean vectors whose coverage probability matches the frequentist coverage probability more accurately than that obtained using the independence-Jeffreys prior distribution, even with small samples.

We have compared statistical properties of the interior-branch and bootstrap tests of phylogenetic trees when the neighbor-joining tree-building method is used. For each interior branch of a predetermined topology, the interiorbranch and bootstrap tests provide the confidence values, PC and PB, respectively, that indicate the extent of statistical support of the sequence cluster generated by the branch. In phylogenetic analysis these two values are often interpreted in the same way, and if PC, and Ps are high (say, 20.95 ), the sequence cluster is regarded as reliable. We have shown that PC. is in fact the complement of the P-value used in the standard statistical test, but PB is not. Actually, the bootstrap test usually underestimates the extent of statistical support of species clusters. The relationship between the confidence values obtained by the two tests varies with both the topology and expected branch lengths of the true (model) tree. The most conspicuous difference between PC and PB is observed when the true tree is starlike, and there is a tendency for the difference to increase as the number of sequences in the tree increases. The reason for this is that the bootstrap test tends to become progressively more conservative as the number of sequences in the tree increases. Unlike the bootstrap, the interior-branch test has the same statistical properties irrespective of the number of sequences used when a predetermined tree is considered. Therefore, the interior-branch test appears to be preferable to the bootstrap test as long as unbiased estimators of evolutionary distances are used. However, when the interior-branch is applied to a tree estimated from a given data set, PC may give an overestimate of statistical confidence. For this case, we developed a method for computing a modified version (Pk.) of the PC, value and showed that this P& tends to give a conservative estimate of statistical confidence, though it is not as conservative as P B. In this paper we have introduced a model in which evolutionary distances between sequences follow a multivariate normal distribution. This model allowed us to study the relationships between the two tests analytically.

We propose inference tools to analyse the ordering of concordance of random vectors. The analysis in the bivariate case relies on tests for upper and lower quadrant dominance of the true distribution by a parametric or semiparametric model, i.e. for a parametric or semiparametric model to give a probability that two variables are simultaneously small or large at least as great as it would be were they left unspecified. Tests for its generalisation in higher dimensions, namely joint lower and upper orthant dominance, are also analysed. The parametric and semiparamettric setting are based on the copula representation for multivariate distribution, which allows for disentangling behaviour of margins and dependence structure. We propose two types of testing procedures for each setting. The first procedure is based on a formulation of the dominance concepts in terms of values taken by random variables, while the second procedure is based on a formulation in terms of probability levels. For each formulation a distance test and an intersection-union test for inequality constraints are developed depending on the definition of null and alternative hypotheses. An empirical illustration is given for US insurance claim data.

The fan plot of the score statistic for transformation during the forward search is a powerful tool for detecting masked outliers that indicate an incorrect transformation. We use simulation to investigate the distribution of this statistic during the search. Data structures where the normal approximation is, and is not, satisfactory are characterised.

One of the crucial aspects in asset allocation problems is the assumption concerning the probability distribution of asset returns. Financial managers generally suppose normal distribution, even if extreme realizations usually have an higher frequency than in the Gaussian case. The aim of this paper is to propose a general Monte Carlo simulation approach able to solve an asset allocation problem with shortfall constraint, and to evaluate the exact portfolio risk-level when managers assume a misspecified return behaviour. We assume that returns are generated by a multivariate skewed Student-t distribution where each marginal can have different degrees of freedom. The stochastic optimization allows us to value the effective risk for managers. In the empirical application we consider a symmetric and heterogeneous case, and interestingly note that a multivariate Student-t with heterogeneous marginal distributions produces in the optimization problem a shortfall probability and a shortfall return level that can be adequately approximated by assuming a multivariate Student-t with common degrees of freedom. Thus, the proposed simulation-based approach could be an important instrument for investors who require a qualitative assessment of the reliability and sensitivity of their investment strategies in the case their models could be potentially misspecified. Copyright © 2007 John Wiley & Sons, Ltd.

In the last decade, substantial progress has been made on methods for imputation of missing data. Modern imputation methods have become widely available for practitioners through software products such as S-Plus 6.0 (Schimert, Schafer, Hesterberg, Fraley, and Clarkson 2000), SAS PROC MI (SAS 2001), and SOLAS (2001) . The key idea underlying most of these methods is to impute missing

The main objective of this work is to calculate and compare different measures of multivariate skewness for the skew-normal family of distributions. For this purpose, we consider the Mardia (1970) [10], Malkovich and Afifi (1973) [9], Isogai (1982) [17], Srivastava (1984) [15], Song (2001) [14], Móri et al. (1993) [11], Balakrishnan et al. (2007) [3] and Kollo (2008) [7] measures of skewness. The exact expressions of all measures of skewness, except for Song’s, are derived for the family of skew-normal distributions, while Song’s measure of shape is approximated by the use of delta method. The behavior of these measures, their similarities and differences, possible interpretations, and their practical use in testing for multivariate normal are studied by evaluating their power in the case of some specific members of the multivariate skew-normal family of distributions.► We calculate different measures of multivariate skewness for skew-normal distributions. ► Exact expressions of 7 measures of skewness are derived for multivariate skew-normal distributions. ► Behavior, differences, interpretations and tests for normality against skew-normal on all these measures are studied and their power properties are assessed.

Generating multivariate Poisson data is essential in many applications. Current simulation methods suffer from limitations ranging from computational complexity to restrictions on the structure of the correlation matrix. We propose a computationally efficient and conceptually appealing method for generating multivariate Poisson data. The method is based on simulating multivariate Normal data and converting them to achieve a specific correlation matrix and Poisson rate vector. This allows for generating data that have positive or negative correlations as well as different rates.

In this paper we introduce a new flexible mixed model for multinomial discrete choice where the key individual-and alternative-specific parameters of interest are allowed to follow an assumptionfree nonparametric density specification while other alternative-specific coefficients are assumed to be drawn from a multivariate normal distribution which eliminates the independence of irrelevant alternatives assumption at the individual level. A hierarchical specification of our model allows us to break down a complex data structure into a set of submodels with the desired features that are naturally assembled in the original system. We estimate the model using a Bayesian Markov Chain Monte Carlo technique with a multivariate Dirichlet Process (DP) prior on the coefficients with nonparametrically estimated density. We employ a "latent class" sampling algorithm which is applicable to a general class of models including non-conjugate DP base priors. The model is applied to supermarket choices of a panel of Houston households whose shopping behavior was observed over a 24-month period in years 2004-2005. We estimate the nonparametric density of two key variables of interest: the price of a basket of goods based on scanner data, and driving distance to the supermarket based on their respective locations. Our semi-parametric approach allows us to identify a complex multi-modal preference distribution which distinguishes between inframarginal consumers and consumers who strongly value either lower prices or shopping convenience.

This paper reviews some important results dealing with the norms of distributions of several members of spherical distributions in an accessible manner. Moments of the norms of some spherical distributions are discussed. Then they have been used to derive covariance matrices and higher order standardized moments.