Estimation of a parameter of Morgenstern type bivariate exponential distribution by ranked set sampling

Ranked set sampling is applicable whenever ranking of a set of sampling units can be done easily by a judgement method or based on the measurement of an auxiliary variable on the units selected. In this work, we consider ranked set sampling, in which ranking of units are done based on measurements made on an easily and exactly measurable auxiliary variable X which is correlated with the study variable Y. We then estimate the mean of the study variate Y by the BLUE based on the measurements made on the units of the ranked set sampling regarding the study variable Y, when (X ,Y) follows a Morgenstern type bivariate exponential distribution. We then consider unbalanced multistage ranked set sampling and estimate the mean of the study variate Y by the BLUE based on the observations made on the units of multistage ranked set sample regarding the study variable Y. Efficiency comparison is also made on all estimators considered in this work.     

Bootstrap Algorithms for Risk Models with Auxiliary Variable and Complex Samples

Resampling methods are often invoked in risk modelling when the stability of estimators of model parameters has to be assessed. The accuracy of variance estimates is crucial since the operational risk management affects strategies, decisions and policies. However, auxiliary variables and the complexity of the sampling design are seldom taken into proper account in variance estimation. In this paper bootstrap algorithms for finite population sampling are proposed in presence of an auxiliary variable and of complex samples. Results from a simulation study exploring the empirical performance of some bootstrap algorithms are presented.      

Multivariate logistic regression with incomplete covariate and auxiliary information

In this article, we propose and explore a multivariate logistic regression model for analyzing multiple binary outcomes with incomplete covariate data where auxiliary information is available. The auxiliary data are extraneous to the regression model of interest but predictive of the covariate with missing data. Horton and Laird [N.J. Horton, N.M. Laird, Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information, Biometrics 57 (2001) 34–42] describe how the auxiliary information can be incorporated into a regression model for a single binary outcome with missing covariates, and hence the efficiency of the regression estimators can be improved. We consider extending the method of [9] to the case of a multivariate logistic regression model for multiple correlated outcomes, and with missing covariates and completely observed auxiliary information. We demonstrate that in the case of moderate to strong associations among the multiple outcomes, one can achieve considerable gains in efficiency from estimators in a multivariate model as compared to the marginal estimators of the same parameters.     

Comparison of Methods for the Computation of Multivariate t Probabilities

This article compares methods for the numerical computation of multivariate t probabilities for hyper-rectangular integration regions. Methods based on acceptance-rejection, spherical-radial transformations, and separation-of-variables transformations are considered. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz for multivariate normal probabilities. These methods allow moderately accurate multivariate t probabilities to be quickly computed for problems with as many as 20 variables. Methods for the noncentral multivariate t distribution are also described.     

On generating multivariate Poisson data in management science applications

Generating multivariate Poisson random variables is essential in many applications, such as multi echelon supply chain systems, multi‐item/multi‐period pricing models, accident monitoring systems, etc. Current simulation methods suffer from limitations ranging from computational complexity to restrictions on the structure of the correlation matrix, and therefore are rarely used in management science. Instead, multivariate Poisson data are commonly approximated by either univariate Poisson or multivariate Normal data. However, these approximations are often not adequate in practice. In this paper, we propose a conceptually appealing correction for NORTA (NORmal To Anything) for generating multivariate Poisson data with a flexible correlation structure and rates. NORTA is based on simulating data from a multivariate Normal distribution and converting it into an arbitrary continuous distribution with a specific correlation matrix. We show that our method is both highly accurate and computationally efficient. We also show the managerial advantages of generating multivariate Poisson data over univariate Poisson or multivariate Normal data. Copyright © 2011 John Wiley & Sons, Ltd.     

A New Algorithm for Simulating a Correlation Matrix Based on Parameter Expansion and Reparameterization

The correlation matrix (denoted by R) plays an important role in many statistical models. Unfortunately, sampling the correlation matrix in Markov chain Monte Carlo (MCMC) algorithms can be problematic. In addition to the positive definite constraint of covariance matrices, correlation matrices have diagonal elements fixed at one. In this article, we propose an efficient two-stage parameter expanded reparameterization and Metropolis-Hastings (PX-RPMH) algorithm for simulating R. Using this algorithm, we draw all elements of R simultaneously by first drawing a covariance matrix from an inverse Wishart distribution, and then translating it back to a correlation matrix through a reduction function and accepting it based on a Metropolis-Hastings acceptance probability. This algorithm is illustrated using multivariate probit (MVP) models and multivariate regression (MVR) models with a common correlation matrix across groups. Via both a simulation study and a real data example, the performance of the PX-RPMH algorithm is compared with those of other common algorithms. The results show that the PX-RPMH algorithm is more efficient than other methods for sampling a correlation matrix.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

On the exact distribution of linear combinations of order statistics from dependent random variables

We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions.     

Approximating Unknown Mappings: An Experimental Evaluation

Different methodologies have been introduced in recent years with the aim of approximating unknown functions. Basically, these methodologies are general frameworks for representing non-linear mappings from several input variables to several output variables. Research into this problem occurs in applied mathematics (multivariate function approximation), statistics (nonparametric multiple regression) and computer science (neural networks). However, since these methodologies have been proposed in different fields, most of the previous papers treat them in isolation, ignoring contributions in the other areas. In this paper we consider five well known approaches for function approximation. Specifically we target polynomial approximation, general additive models (Gam), local regression (Loess), multivariate additive regression splines (Mars) and artificial neural networks (Ann).Neural networks can be viewed as models of real systems, built by tuning parameters known as weights. In training the net, the problem is to find the weights that optimize its performance (i.e. to minimize the error over the training set). Although the most popular method for Ann training is back propagation, other optimization methods based on metaheuristics have recently been adapted to this problem, outperforming classical approaches. In this paper we propose a short term memory tabu search method, coupled with path relinking and BFGS (a gradient-based local NLP solver) to provide high quality solutions to this problem. The experimentation with 15 functions previously reported shows that a feed-forward neural network with one hidden layer, trained with our procedure, can compete with the best-known approximating methods. The experimental results also show the effectiveness of a new mechanism to avoid overfitting in neural network training.     

Automated Factor Slice Sampling

Markov chain Monte Carlo (MCMC) algorithms offer a very general approach for sampling from arbitrary distributions. However, designing and tuning MCMC algorithms for each new distribution can be challenging and time consuming. It is particularly difficult to create an efficient sampler when there is strong dependence among the variables in a multivariate distribution. We describe a two-pronged approach for constructing efficient, automated MCMC algorithms: (1) we propose the “factor slice sampler,” a generalization of the univariate slice sampler where we treat the selection of a coordinate basis (factors) as an additional tuning parameter, and (2) we develop an approach for automatically selecting tuning parameters to construct an efficient factor slice sampler. In addition to automating the factor slice sampler, our tuning approach also applies to the standard univariate slice samplers. We demonstrate the efficiency and general applicability of our automated MCMC algorithm with a number of illustrative examples. This article has online supplementary materials.     

A class of multivariate skew-normal models

The existing model for multivariate skew normal data does not cohere with the joint distribution of a random sample from a univariate skew normal distribution. This incoherence causes awkward interpretation for data analysis in practice, especially in the development of the sampling distribution theory. In this paper, we propose a refined model that is coherent with the joint distribution of the univariate skew normal random sample, for multivariate skew normal data. The proposed model extends and strengthens the multivariate skew model described in Azzalini (1985,Scandinavian Journal of Statistics,12, 171–178). We present a stochastic representation for the newly proposed model, and discuss a bivariate setting, which confirms that the newly proposed model is more plausible than the one given by Azzalini and Dalla Valle (1996,Biometrika,83, 715–726).     

Some multivariate linear regression testing problems with additional observations

In an earlier paper, the present author ([6], Calcutta Statist. Assoc. Bull.28, 47–56) proposed a similar test for a mean testing problem with additional observations on a set of correlated auxiliary variables. This idea has been extended here to cover some multivariate linear regression testing problems with the same type of additional observations on a set of correlated auxiliary variables.     

Multivariate Polynomial Minimization and Its Application in Signal Processing

We make a conjecture that the number of isolated local minimum points of a 2n-degree or (2n+1)-degree r-variable polynomial is not greater than n ^r when n 2. We show that this conjecture is the minimal estimate, and is true in several cases. In particular, we show that a cubic polynomial of r variables may have at most one local minimum point though it may have 2^r critical points. We then study the global minimization problem of an even-degree multivariate polynomial whose leading order coefficient tensor is positive definite. We call such a multivariate polynomial a normal multivariate polynomial. By giving a one-variable polynomial majored below a normal multivariate polynomial, we show the existence of a global minimum of a normal multivariate polynomial, and give an upper bound of the norm of the global minimum and a lower bound of the global minimization value. We show that the quartic multivariate polynomial arising from broad-band antenna array signal processing, is a normal polynomial, and give a computable upper bound of the norm of the global minimum and a computable lower bound of the global minimization value of this normal quartic multivariate polynomial. We give some sufficient and necessary conditions for an even order tensor to be positive definite. Several challenging questions remain open.     

Invariance and efficiency of convex representations

We consider two notions for the representations of convex cones G-representation and lifted-G-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one notion of representation but not the other. In particular, we prove that lifted-G-representation is closed under duality when the representing cone is self-dual. We also prove that strict complementarity of a convex optimization problem in conic form is preserved under G-representations. Then we move to study efficiency measures for representations. We evaluate the representations of homogeneous convex cones based on the “smoothness” of the transformations mapping the central path of the representation to the central path of the represented optimization problem. Research of the first author was supported in part by a grant from the Faculty of Mathematics, University of Waterloo and by a Discovery Grant from NSERC. Research of the second author was supported in part by a Discovery Grant from NSERC and a PREA from Ontario, Canada.     

Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error

Efficient estimation of tail probabilities involving heavy tailed random variables is amongst the most challenging problems in Monte-Carlo simulation. In the last few years, applied probabilists have achieved considerable success in developing efficient algorithms for some such simple but fundamental tail probabilities. Usually, unbiased importance sampling estimators of such tail probabilities are developed and it is proved that these estimators are asymptotically efficient or even possess the desirable bounded relative error property. In this paper, as an illustration, we consider a simple tail probability involving geometric sums of heavy tailed random variables. This is useful in estimating the probability of large delays in M/G/1 queues. In this setting we develop an unbiased estimator whose relative error decreases to zero asymptotically. The key idea is to decompose the probability of interest into a known dominant component and an unknown small component. Simulation then focuses on estimating the latter ‘residual’ probability. Here we show that the existing conditioning methods or importance sampling methods are not effective in estimating the residual probability while an appropriate combination of the two estimates it with bounded relative error. As a further illustration of the proposed ideas, we apply them to develop an estimator for the probability of large delays in stochastic activity networks that has an asymptotically zero relative error.      

Binary clustering with missing data

A clustering method is presented for analysing multivariate binary data with missing values. When not all values are observed, Govaert³ has studied the relations between clustering methods and statistical models. The author has shown how the identification of a mixture of Bernoulli distributions with the same parameter for all clusters and for all variables corresponds to a clustering criterion which uses L₁ distance characterizing the MNDBIN method (Marchetti⁸). He first generalized this model by selecting parameters which can depend on variables and finally by selecting parameters which can depend both on variables and on clusters. We use the previous models to derive a clustering method adapted to missing data. This method optimizes a criterion by a standard iterative partitioning algorithm which removes the necessity either to ignore objects or to substitute the missing data. We study several versions of this algorithm and, finally, a brief account is given of the application of this method to some simulated data.     

Bayesian Copulae Distributions, with Application to Operational Risk Management

The aim of this paper is to introduce a new methodology for operational risk management, based on Bayesian copulae. One of the main problems related to operational risk management is understanding the complex dependence structure of the associated variables. In order to model this structure in a flexible way, we construct a method based on copulae. This allows us to split the joint multivariate probability distribution of a random vector of losses into individual components characterized by univariate marginals. Thus, copula functions embody all the information about the correlation between variables and provide a useful technique for modelling the dependency of a high number of marginals. Another important problem in operational risk modelling is the lack of loss data. This suggests the use of Bayesian models, computed via simulation methods and, in particular, Markov chain Monte Carlo. We propose a new methodology for modelling operational risk and for estimating the required capital. This methodology combines the use of copulae and Bayesian models.      

Average case complexity of linear multivariate problems I. Theory

We study the average case complexity of linear multivariate problems, that is, the approximation of continuous linear operators on functions of d variables. The function spaces are equipped with Gaussian measures. We consider two classes of information. The first class Λ^std consists of function values, and the second class Λ^all consists of all continuous linear functionals. Tractability of a linear multivariate problem means that the average case complexity of computing an ε-approximation is O((1/)^p) with p independent of d. The smallest such p is called the exponent of the problem. Under mild assumptions, we prove that tractability in Λ^all is equivalent to tractability in Λ^std and that the difference of the exponents is at most 2. The proof of this result is not constructive. We provide a simple condition to check tractability in Λ^all. We also address the issue of how to construct optimal (or nearly optimal) sample points for linear multivariate problems. We use relations between average case and worst case settings. These relations reduce the study of the average case to the worst case for a different class of functions. In this way we show how optimal sample points from the worst case setting can be used in the average case. In Part II we shall apply the theoretical results to obtain optimal or almost optimal sample points, optimal algorithms, and average case complexity functions for linear multivariate problems equipped with the folded Wiener sheet measure. Of particular interest will be the multivariate function approximation problem.     

Choosing the best set of variables in regression analysis using integer programming

This paper is concerned with an algorithm for selecting the best set of s variables out of k(> s) candidate variables in a multiple linear regression model. We employ absolute deviation as the measure of deviation and solve the resulting optimization problem by using 0-1 integer programming methodologies. In addition, we will propose a heuristic algorithm to obtain a close to optimal set of variables in terms of squared deviation. Computational results show that this method is practical and reliable for determining the best set of variables.     

Covariance Decompositions for Accurate Computation in Bayesian Scale-Usage Models

Analyses of multivariate ordinal probit models typically use data augmentation to link the observed (discrete) data to latent (continuous) data via a censoring mechanism defined by a collection of “cutpoints.” Most standard models, for which effective Markov chain Monte Carlo (MCMC) sampling algorithms have been developed, use a separate (and independent) set of cutpoints for each element of the multivariate response. Motivated by the analysis of ratings data, we describe a particular class of multivariate ordinal probit models where it is desirable to use a common set of cutpoints. While this approach is attractive from a data-analytic perspective, we show that the existing efficient MCMC algorithms can no longer be accurately applied. Moreover, we show that attempts to implement these algorithms by numerically approximating required multivariate normal integrals over high-dimensional rectangular regions can result in severely degraded estimates of the posterior distribution. We propose a new data augmentation that is based on a covariance decomposition and that admits a simple and accurate MCMC algorithm. Our data augmentation requires only that univariate normal integrals be evaluated, which can be done quickly and with high accuracy. We provide theoretical results that suggest optimal decompositions within this class of data augmentations, and, based on the theory, recommend default decompositions that we demonstrate work well in practice. This article has supplementary material online.