Multivariate Normal Distribution

Let Z t = (Z1,. .. , Zp) be a p-variate Gaussian complex random variable. Let α = (n1, m1,. .. , np, mp) be a vector in N 2p and let ν(α) be the correspondent moment: ν(α) = 1 π p det(Σz) C p z n 1 1 z m 1 1 z n 2 2 z m 2 2 • • • z np p z mp p exp(−z t Σ −1 z) dz1 • • • dzp. We present conditions for nullity of the moment ν(α). Furthermore, we show the value of the non null moments for p = 1 and p = 2 and we conjecture the value for p > 2.

Transformed empirical processes (TEPs) have been used by the authors in a previous paper to construct consistent and selectively efficient goodness-of-fit tests of the Kolmogorov-Smirnov type. A straightforward application of the same ideas to the construction of tests of the Cramér-von Mises type with the same properties leads to cumbersome computations. This short note exhibits some of the inconvenients encountered, and introduces a new family of quadratic statistics of the Cramér-von Mises type, in order to circumvent the difficulties.

We now turn attention to statistical models in which the family F of possible pdfs for the observable X ∈ X are a k-dimensional parametric family F = {f(x | θ) : θ ∈ Θ} for some parameter space Θ ⊆ Rk and function f : X × Θ → R+. Examples include the Poisson distribution Po(θ) with Θ = R+ and the Be(α, β) distribution with θ = (α, β) ∈ Θ ⊂ R 2 +. Other examples include the univariate normal distribution No(μ, σ), with k = 2 and Θ = R×R+ with θ = (μ, σ ), and the pdimensional multivariate normal distribution No(μ,Σ) with k = p(p+ 3)/2-dimensional parameter θ = (μ,Σ), with mean vector μ ∈ Rp and p× p positive-definite covariance matrix Σ ∈ S +.

We now turn attention to statistical models in which the family F of possible pdfs for the observable X ∈ X are a k-dimensional parametric family F = {f(x | θ) : θ ∈ Θ} for some parameter space Θ ⊆ Rk and function f : X × Θ → R+. Examples include the Poisson distribution Po(θ) with Θ = R+ and the Be(α, β) distribution with θ = (α, β) ∈ Θ ⊂ R 2 +. Other examples include the univariate normal distribution No(μ, σ), with k = 2 and Θ = R×R+ with θ = (μ, σ ), and the pdimensional multivariate normal distribution No(μ,Σ) with k = p(p+ 3)/2-dimensional parameter θ = (μ,Σ), with mean vector μ ∈ Rp and p× p positive-definite covariance matrix Σ ∈ S +.

Supervised classification or pattern recognition is a method to solve decision problems in Social Sciences. It is organized on the basis of specific sets of predictor variables and the existence of classes known a priori. Based on a training sample, its main objective is to construct a classification rule in order to predict the class to which a new object

We consider all two-times iterated Itô integrals obtained by pairing m independent standard Brownian motions. First we calculate the conditional joint characteristic function of these integrals, given the Brownian increments over the integration interval, and show that it has a form entirely similar to what is obtained in the univariate case. Then we propose an algorithm for the simultaneous simulation of the m 2 integrals conditioned on the Brownian increments that achieves a mean square error of order 1/n 2 , where n is the number of terms in a truncated sum. The algorithm is based on approximation of the tail-sum distribution, which is a multivariate normal variance mixture, by a multivariate normal distribution.

In this paper, from Frèchet's metric, diagnostic tools are constructed for the detection of influential observations in Profile Analysis with elliptically distributed random errors. This distributional hypothesis allows the application of the proposed diagnostics to a wide variety of random experiences, not only for data from a multivariate normal distribution but also from other symmetric distributions, commonly used in studies of several sciences. The diagnostics are based on Frèchet's distance between the distributions of the basic statistics in Profile Analysis, in the postulated model and in the perturbed model obtained by deleting an observation from the sample data. This metric is highly useful since it enables the analysis of the influence on the point estimation and the estimation error. Applications on two data sets are provided.

With the widespread use of image processing technologies, objective image quality metrics are a fundamental and challenging problem. In this paper, we present a new No-Reference Image Quality Assessment (NR-IQA) algorithm based on visual attention modeling and a multivariate Gaussian distribution to predict the final quality score from the extracted features. Computational modeling of visual attention is performed to compute saliency maps at three resolution levels. At each level, distortions of the input image are extracted and weighted by the saliency maps in order to highlight degradations of visually attracting regions. The generated features are used by a probabilistic model to predict the final quality score. Experimental results demonstrate the effectiveness of the metric and show better performance when compared to well known NR-IQA algorithms.

Statistical approaches tailored to analyzing longitudinal data that have multiple outcomes with different distributions are scarce. This paucity is due to the non-availability of multivariate distributions that jointly model outcomes with different distributions other than the multivariate normal. A plethora of research has been done on the specific combination of binary-Gaussian bivariate outcomes but a more general approach that allows other mixtures of distributions for multiple longitudinal outcomes has not been thoroughly demonstrated and examined. Here we study a multivariate generalized linear mixed models approach that jointly models multiple longitudinal outcomes with different combinations of distributions and incorporates the correlations between the various outcomes through separate yet correlated random intercepts. Every outcome is linked to the set of covariates through a proper link function that allows the incorporation and joint modelling of different distributions. A novel application was demonstrated on a cohort study of Type 1 diabetic patients to jointly model a mix of longitudinal cardiovascular outcomes and to explore for the first time the effect of glycemic control treatment, plasma prekallikrein biomarker, gender and age on cardiovascular risk factors collectively.

The severity of type II errors is frequently ignored when deriving a multiple testing procedure, even though utilizing it properly can greatly help in making correct decisions. This paper puts forward a theory behind developing a multiple testing procedure that can incorporate the type II error severity and is optimal in the sense of minimizing a measure of false non-discoveries among all procedures controlling a measure of false discoveries. The theory is developed under a general model allowing arbitrary dependence by taking a compound decision theoretic approach to multiple testing with a loss function incorporating the type II error severity. We present this optimal procedure in its oracle form and offer numerical evidence of its superior performance over relevant competitors.

Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: (1) the vector of critical simplex counts attained by a lexicographical Morse matching, (2) the vector of simplex counts in the link of a fixed simplex, and (3) the vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein's method.

Invariance in Random Utility (RU) Models is the property that the distribution of achieved utility is invariant across the alternatives chosen. In this note we study a generalization termed Power Invariance (PI). It generalizes the property, showed by Mattsson and Weibull, that invariance holds for an independent RU Model, where the cdf's of the random terms are positive powers of a given base cdf. The power invariant copulas turn out to be the copulas of Multivariate Extreme Value distributions. We further show that, under natural regularity conditions, the copulas of the marginal distributions of a PI copula are power invariant.

Spatial statistics deals with statistical methods in which spatial locations play an explicit role in the analysis of data. Spatial data are often modeled as a realization of a Gaussian process or a function of a Gaussian process. However there are events, such as rain, snow, storms, hurricanes, earthquakes, where extremes are of the main interest. Here multivariate normal distributions are inappropriate for modeling tail behavior. The natural class of processes for dealing with extremes is the max-stable processes. This work briefly reviews the approaches to the statistical modeling of spatial extremes. Models for max-stable processes will be considered in an application to the annual maxima of daily precipitation over the North of Portugal.

Spatial statistics deals with statistical methods in which spatial locations play an explicit role in the analysis of data. Spatial data are often modeled as a realization of a Gaussian process or a function of a Gaussian process. However there are events, such as rain, snow, storms, hurricanes, earthquakes, where extremes are of the main interest. Here multivariate normal distributions are inappropriate for modeling tail behavior. The natural class of processes for dealing with extremes is the max-stable processes. This work briefly reviews the approaches to the statistical modeling of spatial extremes. Models for max-stable processes will be considered in an application to the annual maxima of daily precipitation over the North of Portugal.

This paper explores the use of robust location estimators such as AverageExcluding-High-And-Low and Huber’s M-estimators in loss reserving. Standard order statistics results are used to investigate the nite-sample properties of Average-Excluding-High-And-Low for positively skewed distributions including bias and eciency, based on the criterion of mean squared error. The paper concludes that Averages-Excluding-High-And-Low, although biased with respect to the population mean for positively skewed distributions, is more ecient than the sample average in small samples. The paper also shows that the use of Huber’s M-estimators can enhance the consistency in loss development factor selections by identifying the implied risk preference.

An international portfolio allows simultaneous investment in both domestic and foreign markets. It hence has the potential for improved performance by exploiting a wider range of returns, and diversification benefits, than portfolios investing in just one market. However, to obtain the most efficient portfolios (along with the usual management of assets) the risks from currency fluctuations need good management, such as by using appropriate hedging. In this paper, we present a two-stage stochastic international portfolio optimisation model to find an optimal allocation for the combination of both assets and currency hedging positions. Our optimisation model allows a "currency overlay", or a deviation of currency exposure from asset exposure, to provide flexibility in hedging against, or in speculation using, currency exposure. The transaction costs associated with both trading and hedging are also included. To model the realistic dependence structure of the multivariate return distributions, a new scenario generation method, employing a regular-vine copula is developed. The use of vine copulas allows a better representation of the characteristics of returns, specifically, their non-normality and asymmetric dependencies. It hence improves the representation of the uncertainty underlying decisions needed for international portfolio optimisation problems. Efficient portfolios optimised with scenarios generated from the new vine-copula method are compared with the portfolios from a standard scenario generation method. Experimental results show that the proposed method, using realistic non-normal uncertainty, produces portfolios that give better risk-return reward than those from a standard scenario generation approach, using normal distributions. The difference in risk-return compensation is largest when the portfolios are constrained to require higher returns. The paper shows that it can be important to model the nonnormality in uncertainty, and not just assume normal distributions.

A new distribution is introduced, which we call the twin-t distribution. This distribution is heavy-tailed like the t distribution, but closer to normality in the central part of the curve. Its properties are described, e.g. the pdf, the distribution function, moments, and random number generation. This distribution could have many applications, but here we focus on its use as an aid to robustness. We give examples of its application in robust regression and in curve fitting. Extensions such as skew and multivariate twin-t distributions, and a twin of the McDonald-Newey generalised t distribution are also discussed.

Maximum likelihood estimation of a log-concave density has certain advantages over other nonparametric approaches, such as kernel density estimation, which requires a bandwidth selection. Furthermore, finding the optimal bandwidth gets more difficult as a dimension increases. On the other hand, the shape-constrained approach is automatic and does not need any tuning parameters. However, for both the kernel and log-concave estimators, the rate of convergence slows down as the dimension d increases. To handle this “curse of dimensionality”, we study an intermediate semiparametric copula approach and we estimate the marginals using the log-concave shape-constrained MLE and use a parametric approach to fit the copula parameters. We prove √ n rate of convergence for the parametric estimator and that the joint density converges at a rate of n−2/5 regardless of dimension. This is faster than the conjectured rate of n−2/(d+4) for the multivariate log-concave estimators [Cule, 2009]. We exam...

One of the most important 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. We propose a general Monte Carlo simulation approach in order to solve an asset allocation problem with shortfall constraint, and to evaluate the exact portfolio risk-level when managers assume a misspecified tails behaviour. In particular, in a stochastic optimisation problem, we assume that returns are generated by a multivariate Student-t distribution, while in reality returns come from a multivariate distribution where each marginal is a Student-t but with different degrees of freedom. Stochastic optimisation allows us to value the effective risk for managers. In the specific case analysed, it is also interesting to observe that a multivariate density resulting from different Student-t marginal distributions produces a shortfall probability and a shortfall return level that can be approximated adequately by assuming a multivariate Student-t with a common degree of freedom in the optimisation problem. The present simulation based approach could be an important instrument for investors who need a qualitative assessment of the reliability and sensitivity of their investment strategies when their models are potentially misspecified.

This paper derives the prediction distribution of future responses from the linear model with errors having an elliptical distribution with known covariance parameters. For unknown covariance parameters, the marginal likelihood function of the parameters has been obtained and the prediction distribution has been modified by replacing the covariance parameters by their estimates obtained from the marginal likelihood function. It is observed that the prediction distribution with elliptical error has a multivariate Student's t-distribution with appropriate degrees of freedom. The results for some special cases such as the Intra-class correlation model, AR(1), MA(1), and ARMA(1,1) models have been obtained from the general results. As an application, the ;-expectation tolerance region has been constructed. An example has been added.

We develop parametric and nonparametric bootstrap methods for multi-factor multivariate data, without assuming normality, and allowing for covariance matrices that are heterogeneous between groups. The newly proposed, general procedure includes several situations as special cases, such as the multivariate Behrens-Fisher problem, the multivariate one-way layout, as well as crossed and hierarchically nested two-way layouts. We derive the asymptotic distribution of the bootstrap tests for general factorial designs and evaluate their performance in an extensive comparative simulation study. For moderate sample sizes, the bootstrap approach provides an improvement to existing methods in particular for situations with nonnormal data and heterogeneous covariance matrices in unbalanced designs. For balanced designs, less computationally intensive alternatives based on approximate sampling distributions of multivariate tests can be recommended.

We consider testing whether the mean vectors of two or more populations have parallel, coincident, or flat profiles when the validity of normality is not known, and the sample sizes are moderate. Using some properties of multivariate moments and matrix manipulations, we obtain the asymptotic expansions for the null distribution of the Lawley-Hotelling statistics. We also derive the corresponding results in the situation where interest lies in coincidence and flatness alone. Accuracy of all the asymptotic expansions in approximating the exact null distributions is examined via simulation. Profile analysis of SO 4 concentrations from a forestry experiment is used to illustrate the methods.

This paper addresses a parameter estimation problem for Markov-modulated compound Poisson process (MMCPP) and compound Markovian arrival process (CMAP). MM-CPP and CMAP are extended from Markov-modulated Poisson process (MMPP) and Markovian arrival process (MAP) by combining compound Poisson process (CPP). The EM (expectation-maximization) algorithm is well known as an effective method in order to perform the statistical estimation for the family of MAPs. In this paper, we develop the EM algorithm for MMCPP and CMAP by using the similar technique to the forward-backward algorithm of hidden Markov model (HMM). In particular, we derive concrete estimation algorithms for MMCPP and CMAP whose outputs are given by exponential distributions or multivariate normal distributions.

Temperature and precipitation are two critical climate parameters that influence agricultural productivity and various extreme hydrological and meteorological phenomena. Both temperature and precipitation have non-normal marginal distribution and have varying correlation over time. In many cases, while the marginal distributions of these two variables are known, their joint distributions remain unknown. Modelling the potential dependence under varying correlation and non-normal distribution can be achieved using Copula. In this study, we analysed the relationship between total precipitation and temperatures within the Bafra Plain using the Copula method considering maximum, minimum and average temperature, and total precipitation. First, the assumption of autocorrelation was tested using Ljung-Box unit root, Mann-Kendall trend, and Ollech-Webel seasonality tests. Then, the presence of autocorrelation was verified through autocorrelation functions (ACF). To mitigate autocorrelation, appropriate SARIMA and NNAR models were determined based on ACF. A multivariate analysis was conducted on residuals by examining the marginals distributions and copula dependency. Parameters of the marginal distributions and copula families were estimated by maximizing log-likelihood. The suitable copula families were determined based on Bayesian information criteria (BIC). Copula Kendall correlations () together with Spearman () and Pearson correlation coefficient () calculated to show the effect of copula in revealing correct relationship. As a result, the Copula method demonstrated moderate negative correlation of minimum and maximum temperature with precipitation which is higher compared to low negative correlation of and. For average temperature and precipitation, all three methods showed similar low negative correlation. The outcomes contribute to establishing more robust foundations for implementing measures to preserve and strengthen the region's agricultural sustainability.

In this paper we give an extension of the theory of local minimax property of Giri and Kiefer (1964, Ann. Math. Statist., 35, 21-35) to the family of elliptically symmetric distributions which contains the multivariate normal distribution as a member.

In this paper, we study, within a modeling framework, the joint treatment of nonignorable dropout and informative sampling for longitudinal survey data, by specifying the probability distribution of the observed measurements when the sampling design is informative. The sample distribution of the observed measurements model is extracted from the population distribution model, assumed to be multivariate normal. The sample distribution is derived first by identifying and estimating the conditional expectations of first order sample inclusion probabilities, (assuming complete response at the first time period), given the study variable, based on a variety of models, such as linear, exponential, logit and probit. Next, we consider a logistic model for the informative dropout process. The proposed method combines two methodologies used in the analysis of sample surveys: for the treatment of informative sampling and informative dropout. One incorporates the dependence of the first order inclusion probabilities at the initial time period on the study variable, see Eideh and Nathan (2006), while the other incorporates the dependence of the probability of nonresponse on unobserved or missing observations, see Diggle and Kenward (1994). An empirical example based on British Labour Force Survey data illustrates the methods proposed.

An estimation of parameters of a multivariate Gaussian Mixture Model is usually based on a criterion (e.g. Maximum Likelihood) that is focused mostly on training data. Therefore, testing data, which were not seen during the training procedure, may cause problems. Moreover, numerical instabilities can occur (e.g. for low-occupied Gaussians especially when working with full-covariance matrices in high-dimensional spaces). Another question concerns the number of Gaussians to be trained for a specific data set. The approach proposed in this paper can handle all these issues. It is based on an assumption that the training and testing data were generated from the same source distribution. The key part of the approach is to use a criterion based on the source distribution rather than using the training data itself. It is shown how to modify an estimation procedure in order to fit the source distribution better (despite the fact that it is unknown), and subsequently new estimation algorithm for diagonal-as well as full-covariance matrices is derived and tested.

This mathematical research involves formulating mathematical explanations and solving discrete probabilistic problems related to the numerical estimation of target hits. These estimation formulas enable the cost estimation of operating and support funding for weapons payloads. Two of the three analytical problems involve estimating the probability that a target is hit after a tactical munitions is dispensed from an aircraft or missile launcher and the submunitions are ejected to clear a targeted area. The first problem entails determining the probability of hitting a target if a munitions may fall within a circle of radius r with a landing spread resembling a Gaussian or bivariate normal distribution, i.e., a two-dimensional probability distribution given jointly by (X,Y) ~N(0, � 2); the second problem generalizes the first and assumes that a mine's true position and a munitions' targeting are subject to random errors whose distributions are circular Gaussian; this problem is solved using discrete convolution techniques in the spatial domain and multiple integrals for the dispersing distribution. A third Poisson model is offered under some standard assumptions for multiple target hits.

The Malaysian women population frequently face the problem of finding the best fitting shoes. This problem is created by the absence of a Malaysian women shoe sizing system. Standard statistical methods involving the Multivariate Normal distribution are used in a novel process of addressing issues related to the creation of a shoe sizing system, in particular, the problem of defining categories of shoe sizes. This study focused on the use of five-foot measurement namely, foot length (FL), foot breadth (FB), foot's ball girth (BG), instep length (IL), and fibulare instep length (FIL). Univariate hypothesis testing was performed taking advantage of the existence of normal probability distribution. For brevity, details for FL, FB, and BG are shown in this paper, followed by a comparison of performance results between (FL, FB, BG) and (FL, FB, BG, IL, FIL). Our results were compared to a similar study showing almost the same aggregate loss and coverage percentage. The result shows that a modest sample size of 160 was sufficient to define categories of shoe sizes to help develop a prototype shoe sizing system using the proposed novel approach. The proposed prototype shoe sizing system provides information for the planning, design, and manufacturing of Malaysian women's footwear with implications for better fitness and comfort.

The dispersion control charts monitor the variability of a process that may increase or decrease. An increase in dispersion parameter implies deterioration in the process for an assignable cause, while a decrease in dispersion indicates an improvement in the process. Multivariate variability control charts are used to monitor the shifts in the process variance-covariance matrix. Although multivariate EWMA and CUSUM dispersion control charts are designed to detect the small amount of change in the covariance matrix but to gain more efficiency, we have developed a Mixed Multivariate EWMA-CUSUM (MMECD) chart. The proposed MMECD chart is compared with its existing counterparts by using some important performance run length-based properties such as ARL, SDRL, EQL, SEQL, and different quantile of run length distribution. A real application related to carbon fiber tubing process is presented for practical considerations. INDEX TERMS Control charts, dispersion parameter, mixed EWMA-CUSUM, memory type, multivariate normality.

In this article, we present an approach which allows taking into account the effect of extreme values in the modeling of financial asset returns and in the valorisation of associated options. Specifically, the marginal distribution of asset returns is modelled by a mixture of two Gaussian distributions. Moreover, we model the joint dependence structure of the returns using a copula function, the extremal one, which is suitable for our financial data, particularly the extreme values copulas. Applications are made on the Atos and Dassault Systems actions of the CAC40 index. Monte Carlo method is used to compute the values of some equity options such as the call on maximum, the call on minimum, the digital option, and the spreads option with the basket (Atos, Dassault systems) as underlying.

The statistical bibliography frequently refers to omnibus tests intended to be sensitive to all or at least a wide variety of alternatives, and focused or directional tests directed to detect efficiently some specific alternatives. In fact, the apparent opposition between omnibus and focused is artificial, and, for instance, K-S test is focused on changes in position of Double Exponential distribution, as well as Cramér-von Mises is focused on changes in position of the distribution with density f (t) = 1/(2 cosh(πt/2)). We provide in this article a simple proof of this latter fact.

A general way of constructing classes of goodness-of-fit tests for multivariate samples is presented. These tests are based on a random signed measure that plays the same role as the empirical process in the construction of the classical Kolmogorov-Smirnov tests. The resulting tests are consistent against any fixed alternative, and, for each sequence of contiguous alternatives, a test in each class can be chosen so as to optimize the discrimination of those alternatives. Résumé Processus empiriques transformés et tests de Kolmogorov-Smirnov modifiés, pour des distributions multivariantes. On considere ici une méthode générale pour construire certaines classes de tests d'ajustement pour des échantillons multivariantes. Ces tests sont basés sur des mesures aléatories signées qui jouent le meme role que le proces empirique pour la construction du test de Kolmogorov-Smirnov classique. Les tests resultants sont efficients et consistents sous toute alternative, et, pour chaque suite d'alternatives contigües, on peut choisir dans chaque classe un test que maximise la discrimination de ces alternatives.

The complete characterization of the Gorbunov and Pinsker [1], [2] nonanticipatory epsilon entropy of multivariate Gauss-Markov sources with square-error fidelity is derived, which remained an open problem since 1974. Specifically, it is shown that the optimal matrices of the stochastic realization of the optimal test channel or reproduction distribution, admit spectral representations with respect to the same unitary matrices, and that the optimal reproduction process is generated, subject to pre-processing and post-processing by memoryless parallel additive Gaussian noise channels. The derivations and analyses are new and bring out several properties of such optimization problems over the space of conditional distributions and their realizations.

For any probability model M ≡ {p(x | θ, ω), θ ∈ Θ, ω ∈ Ω} assumed to describe the probabilistic behaviour of data x ∈ X, it is argued that testing whether or not the available data are compatible with the hypothesis H 0 ≡ {θ = θ 0 } is best considered as a formal decision problem on whether to use (a 0), or not to use (a 1), the simpler probability model (or null model) M 0 ≡ {p(x | θ 0 , ω), ω ∈ Ω}, where the loss difference L(a 0 , θ, ω) − L(a 1 , θ, ω) is proportional to the amount of information δ(θ 0 , θ, ω) which would be lost if the simplified model M 0 were used as a proxy for the assumed model M. For any prior distribution π(θ, ω), the appropriate normative solution is obtained by rejecting the null model M 0 whenever the corresponding posterior expectation δ(θ 0 , θ, ω) π(θ, ω | x) dθ dω is sufficiently large. Specification of a subjective prior is always difficult, and often polemical, in scientific communication. Information theory may be used to specify a prior, the reference prior, which only depends on the assumed model M , and mathematically describes a situation where no prior information is available about the quantity of interest. The reference posterior expectation, d(θ 0 , x) = δ π(δ | x) dδ, of the amount of information δ(θ 0 , θ, ω) which could be lost if the null model were used, provides an attractive non-negative test function, the intrinsic statistic, which is invariant under reparametrization. The intrinsic statistic d(θ 0 , x) is measured in units of information, and it is easily calibrated (for any sample size and any dimensionality) in terms of some average log-likelihood ratios. The corresponding Bayes decision rule, the Bayesian reference criterion (BRC), indicates that the null model M 0 should only be rejected if the posterior expected loss of information from using the simplified model M 0 is too large or, equivalently, if the associated expected average log-likelihood ratio is large enough. The BRC criterion provides a general reference Bayesian solution to hypothesis testing which does not assume a probability mass concentrated on M 0 and, hence, it is immune to Lindley's paradox. The theory is illustrated within the context of multivariate normal data, where it is shown to avoid Rao's paradox on the inconsistency between univariate and multivariate frequentist hypothesis testing.

The multivariate normality assumption is used in many multivariate statistical analyses. It is, therefore, important to assess the validity of this assumption. The main aim of this study is to develop a JAVA program for applying the recently developed Z p and C p test statistics. The application and results of the program are illustrated on two real data sets.