Pairwise likelihood estimation for multivariate mixed Poisson models generated by Gamma intensities

Estimating the parameters of multivariate mixed Poisson models is an important problem in image processing applications, especially for active imaging or astronomy. The classical maximum likelihood approach cannot be used for these models since the corresponding masses cannot be expressed in a simple closed form. This paper studies a maximum pairwise likelihood approach to estimate the parameters of multivariate mixed Poisson models when the mixing distribution is a multivariate Gamma distribution. The consistency and asymptotic normality of this estimator are derived. Simulations conducted on synthetic data illustrate these results and show that the proposed estimator outperforms classical estimators based on the method of moments. An application to change detection in low-flux images is also investigated.     

Multivariate Control Chart Based on Multivariate Smirnov Test

Robust control charts are useful in statistical process control (SPC) when there is limited knowledge about the underlying process distribution, especially for multivariate observations. This article develops a new robust and self-starting multivariate procedure based on multivariate Smirnov test (MST), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on multivariate empirical distribution function (MEDF) and the change-point model. As expected, simulation results show that our proposed control chart is robust to nonnormally distributed data, and moreover, it is efficient in detecting process shifts, especially large shifts, which is one of the main drawbacks of most robust control charts in the literature. As it avoids the need for a lengthy data-gathering step, the proposed chart is particularly useful in start-up or short-run situations. Comparison results and a real data example show that our proposed chart has great potential for application.     

Vector correlation for elliptical distributions

A measure of multivariate correlation between two sets of vectors is considered when the underlying joint distribution is a member of the class of elliptical distributions. Its asymptotic distribution is derived under different situations and these results are used to test hypotheses on vector correlation when the underlying joint distribution is non-normal.     

Estimates of regression coefficients based on the sign covariance matrix

Summary. A new estimator of the regression parameters is introduced in a multivariate multiple-regression model in which both the vector of explanatory variables and the vector of response variables are assumed to be random. The affine equivariant estimate matrix is constructed using the sign covariance matrix (SCM) where the sign concept is based on Oja's criterion function. The influence function and asymptotic theory are developed to consider robustness and limiting efficiencies of the SCM regression estimate. The estimate is shown to be consistent with a limiting multinormal distribution. The influence function, as a function of the length of the contamination vector, is shown to be linear in elliptic cases; for the least squares (LS) estimate it is quadratic. The asymptotic relative efficiencies with respect to the LS estimate are given in the multivariate normal as well as the t -distribution cases. The SCM regression estimate is highly efficient in the multivariate normal case and, for heavy-tailed distributions, it performs better than the LS estimate. Simulations are used to consider finite sample efficiencies with similar results. The theory is illustrated with an example.     

On the Multivariate Upcrossings Index

The multivariate extremal index function is a measure of the clustering among the extreme values of a multivariate stationary sequence. In this article, we introduce a measure of the degree of clustering of upcrossings in a multivariate stationary sequence, called multivariate upcrossings index, which is a multivariate generalization of the concept of upcrossings index. We derive the main properties of this function, namely the relations with the multivariate extremal index and the clustering of upcrossings.

Imposing general local and asymptotic dependence restrictions on the sequence or on its marginals we compute the multivariate upcrossings index from the marginal upcrossings indices and from the joint distribution of a finite number of variables. A couple of illustrative examples are exploited.     

Likelihood Inference for Multivariate Extreme Value Distributions Whose Spectral Vectors have known Conditional Distributions

Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail dependence. The main approaches to inference for multivariate extremes consist in approximating either the distribution of block component‐wise maxima or the distribution of the exceedances over a high threshold. Although the expressions of the asymptotic density functions of these distributions may be characterized, they cannot be computed in general. In this paper, we study the case where the spectral random vector of the multivariate max‐stable distribution has known conditional distributions. The asymptotic density functions of the multivariate extreme value distributions may then be written through univariate integrals that are easily computed or simulated. The asymptotic properties of two likelihood estimators are presented, and the utility of the method is examined via simulation.     

On multivariate separating Hill estimator under estimated location and scatter

We consider estimating the tail-index of a distribution under the assumption of multivariate ellipticity. Recently, a separating Hill estimator for multivariate elliptical distributions was proposed. This estimator is an affine invariant alternative to using the marginal observations in tail-index estimation and is hence unaffected by, e.g. change of units of measurement. However, the separating Hill estimator depends on the location and scatter of the elliptical distribution, which, in practice, have to be estimated. The effect of replacing the true location and scatter of the distribution by estimates has previously been only examined through simulations. In this article we show that the error caused by replacing the location and scatter of the distribution by estimates indeed is asymptotically negligible. This fact is essential for the practicality of the separating Hill estimator. In addition to providing the theoretical results, we present simulation results on the asymptotic behaviour of the estimators.     

Analyzing Mean Profiles of Nonnormal Populations

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 SO₄ concentrations from a forestry experiment is used to illustrate the methods.     

A new class of models for bivariate joint tails

Summary.  A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.     

Semiparametric Regression Estimation in Copula Models

We consider some methods of semiparametric regression estimation in multivariate models when the common distribution function is represented using a copula and the marginals satisfy a generalized regression model using a transfer functional. Sufficient conditions for consistency and joint asymptotic normality of the finite-dimensional parameters are obtained.     

Multivariate trimmed means based on the Tukey depth

In univariate statistics, the trimmed mean has long been regarded as a robust and efficient alternative to the sample mean. A multivariate analogue calls for a notion of trimmed region around the center of the sample. Using Tukey's depth to achieve this goal, this paper investigates two types of multivariate trimmed means obtained by averaging over the trimmed region in two different ways. For both trimmed means, conditions ensuring asymptotic normality are obtained; in this respect, one of the main features of the paper is the systematic use of Hadamard derivatives and empirical processes methods to derive the central limit theorems. Asymptotic efficiency relative to the sample mean as well as breakdown point are also studied. The results provide convincing evidence that these location estimators have nice asymptotic behavior and possess highly desirable finite-sample robustness properties; furthermore, relative to the sample mean, both of them can in some situations be highly efficient for dimensions between 2 and 10.     

Case-deletion Influence Measures for the Data from Multivariate t Distributions

For the data from multivariate t distributions, it is very hard to make an influence analysis based on the probability density function since its expression is intractable. In this paper, we present a technique for influence analysis based on the mixture distribution and EM algorithm. In fact, the multivariate t distribution can be considered as a particular Gaussian mixture by introducing the weights from the Gamma distribution. We treat the weights as the missing data and develop the influence analysis for the data from multivariate t distributions based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. Several case-deletion measures are proposed for detecting influential observations from multivariate t distributions. Two numerical examples are given to illustrate our methodology.     

Simulation-assisted saddlepoint approximation

A general saddlepoint/Monte Carlo method to approximate (conditional) multivariate probabilities is presented. This method requires a tractable joint moment generating function (m.g.f.), but does not require a tractable distribution or density. The method is easy to program and has a third-order accuracy with respect to increasing sample size in contrast to standard asymptotic approximations which are typically only accurate to the first order.

The method is most easily described in the context of a continuous regular exponential family. Here, inferences can be formulated as probabilities with respect to the joint density of the sufficient statistics or the conditional density of some sufficient statistics given the others. Analytical expressions for these densities are not generally available, and it is often not possible to simulate exactly from the conditional distributions to obtain a direct Monte Carlo approximation of the required integral. A solution to the first of these problems is to replace the intractable density by a highly accurate saddlepoint approximation. The second problem can be addressed via importance sampling, that is, an indirect Monte Carlo approximation involving simulation from a crude approximation to the true density. Asymptotic normality of the sufficient statistics suggests an obvious candidate for an importance distribution.

The more general problem considers the computation of a joint probability for a subvector of random T, given its complementary subvector, when its distribution is intractable, but its joint m.g.f. is computable. For such settings, the distribution may be tilted, maintaining T as the sufficient statistic. Within this tilted family, the computation of such multivariate probabilities proceeds as described for the exponential family setting.     

An Algorithm to Simulate VMA Processes Having a Spectrum with Fixed Condition Number

This article proposes an algorithm to generate vector moving average (VMA) processes with a variable spectrum having a fixed condition number across frequencies. This method is based on the theory of multivariate linear spectrum for VMA processes, and is developed in a two-step procedure. Specific examples are provided, and the precision of generated time series is discussed. Such an algorithm is a useful tool to assess the performance of selected multivariate spectral estimators, and it turns out to be particularly appropriated in the Kolmogorov asymptotic estimation framework.     

Multivariate CUSUM and EWMA Control Charts for Skewed Populations Using Weighted Standard Deviations

This article proposes a heuristic method of constructing multivariate cumulative sum and exponentially weighted moving average control charts for skewed populations based on the weighted standard deviation method which adjusts the variance–covariance matrix of quality characteristics and approximates the probability density function using several multivariate normal distributions. These control charts, however, reduce to the conventional control charts when the underlying distribution is symmetric. In-control and out-of-control average run lengths of the proposed control charts are compared with those of the conventional control charts for multivariate lognormal and Weibull distributions. Simulation results show that considerable improvements over the standard method can be achieved when the underlying distribution is skewed.     

A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship

Abstract

We propose a new multivariate extension of the inverse Gaussian distribution derived from a certain multivariate inverse relationship. First we define a multivariate extension of the inverse relationship between two sets of multivariate distributions, then define a reduced inverse relationship between two multivariate distributions. We derive the multivariate continuous distribution that has the reduced multivariate inverse relationship with a multivariate normal distribution and call it a multivariate inverse Gaussian distribution. This distribution is also characterized as the distribution of the location of a multivariate Brownian motion at some stopping time. The marginal distribution in one direction is the inverse Gaussian distribution, and the conditional distribution in the space perpendicular to this direction is a multivariate normal distribution. Mean, variance, and higher order cumulants are derived from the multivariate inverse relationship with a multivariate normal distribution. Other properties such as reproductivity and infinite divisibility are also given.     

Characterizations of Bivariate and Multivariate Life Distributions Based on Reciprocal Subtangent

The subtangent is the projection of the tangent upon the axis of abscissa. The usefulness of the reciprocal subtangent as a measure of the survival and density curves has earlier been reported in the literature for univariate distributions. This measure was generalized for bivariate and multivariate setups and related characterization problems were examined. The conditionally specified bivariate exponential distribution has been uniquely determined from the local constancy of the bivariate reciprocal subtangents. The case of global constancy and other related results have been studied.

Conditionally specified bivariate Lomax distribution and normal distribution were also studied. Further, the conditionally specified multivariate exponential distribution was uniquely determined from the local constancy of the multivariate reciprocal subtangents.     

Estimation and inference of the joint conditional distribution for multivariate longitudinal data using nonparametric copulas

In this paper we study estimating the joint conditional distributions of multivariate longitudinal outcomes using regression models and copulas. For the estimation of marginal models, we consider a class of time-varying transformation models and combine the two marginal models using nonparametric empirical copulas. Our models and estimation method can be applied in many situations where the conditional mean-based models are not good enough. Empirical copulas combined with time-varying transformation models may allow quite flexible modelling for the joint conditional distributions for multivariate longitudinal data. We derive the asymptotic properties for the copula-based estimators of the joint conditional distribution functions. For illustration we apply our estimation method to an epidemiological study of childhood growth and blood pressure.     

On a multivariate log-gamma distribution and the use of the distribution in the Bayesian analysis

In this article, we propose a new generalized multivariate log-gamma distribution. We consider the usage of the proposed multivariate distribution as the prior distribution in the Bayesian analysis. The generalized multivariate log-gamma distribution allows for the inclusion of prior knowledge on correlations between model parameters when likelihood is not in the form of a normal distribution. Use of the proposed distribution in the Bayesian analysis of log-linear models is also discussed.     

A Multivariate Synthetic Control Chart for Monitoring the Process Mean Vector of Skewed Populations Using Weighted Standard Deviations

This article proposes a multivariate synthetic control chart for skewed populations based on the weighted standard deviation method. The proposed chart incorporates the weighted standard deviation method into the standard multivariate synthetic control chart. The standard multivariate synthetic chart consists of the Hotelling's T ² chart and the conforming run length chart. The weighted standard deviation method adjusts the variance–covariance matrix of the quality characteristics and approximates the probability density function using several multivariate normal distributions. The proposed chart reduces to the standard multivariate synthetic chart when the underlying distribution is symmetric. In general, the simulation results show that the proposed chart performs better than the existing multivariate charts for skewed populations and the standard T ² chart, in terms of false alarm rates as well as moderate and large mean shift detection rates based on the various degrees of skewnesses.