Characterization and classification of multiple stable Tweedie models

Extending normal stable Tweedie models, the multiple stable Tweedie (MST) models are recently introduced as a huge class of multivariate distributions. They are composed by a fixed univariate stable Tweedie variable having a positive mean domain and random variables that, given the fixed one, are real independent stable Tweedie variables, possibly different, with the same dispersion parameter equal to the fixed component.Within the framework of exponential dispersion models, we completely prove the characterization of the MST models through their variance functions under steepness property. Thereforewe deduce a new classification of the Poisson-MST, gamma-MST, noncentral-gamma-MST, and inverse-Gaussian-MST families, where each of them contains one element of the normal stable Tweedie models, namely normal-Poisson, normal-gamma (or gamma-Gaussian), normal-noncentral-gamma, and normal-inverse-Gaussian distributions, respectively.     

Multivariate Tweedie distributions and some related capital-at-risk analyses

We study a multivariate extension of the univariate exponential dispersion Tweedie family of distributions. The class, referred to as the multivariate Tweedie family (MTwF), on the one hand includes multivariate Poisson, gamma, inverse Gaussian, stable and compound Poisson distributions and on the other hand introduces a high variety of new dependent probabilistic models unstudied so far. We investigate various properties of MTwF and discuss its possible applications to financial risk management.     

Simulations of full multivariate Tweedie with flexible dependence structure

We employ a variables-in-common method for constructing multivariate Tweedie distributions, based on linear combinations of independent univariate Tweedie variables. The method lies on the convolution and scaling properties of the Tweedie laws, using the cumulant generating function for characterization of the distributions and correlation structure. The routine allows the equivalence between independence and zero correlation and gives a parametrization through given values of the mean vector and dispersion matrix, similarly to the Gaussian vector. Our approach leads to a matrix representation of multivariate Tweedie models, which permits the simulations of many known distributions, including Gaussian, Poisson, non-central gamma, gamma, and inverse Gaussian, both positively or negatively correlated.     

Domains of attraction to Tweedie distributions

We derive new Tauberian theorems for natural exponential families, characterizing regularity properties of the family’s variance function in terms of those of an extreme generating measure. This provides normal and general domains of attraction to Tweedie distributions and leads to new results on weak convergence of natural exponential families to Tweedie distributions, parallel to weak convergence toward stable laws. In particular, we give the domains of attraction to the gamma and compound Poisson-gamma distributions.     

The covariance structure and likelihood function for multivariate dyadic data

This paper extends the results in Li and Loken [A unified theory of statistical analysis and inference for variance component models for dyadic data, Statist. Sinica 12 (2002) 519-535] on the statistical analysis of measurements taken on dyads to the situations in which more than one attribute are measured on each dyad. Starting from the covariance structure for the univariate case obtained in Li and Loken (2002), the covariance structure for the multivariate case is derived based on the group symmetry induced by the assumed exchangeability in the units. Our primary objective is to document the Gaussian likelihood and the sufficient statistics for multivariate dyadic data in closed form, so that they can be referenced by researchers as they analyze those data. The derivation carried out can also serve as an example of multivariate extension of univariate models based on exchangeability.     

A multiple window scan statistic for time series models

In this article we extend the results derived for scan statistics in Wang and Glaz (2014) for independent normal observations. We investigate the performance of two approximations for the distribution of fixed window scan statistics for time series models. An R algorithm for computing multivariate normal probabilities established in Genz and Bretz (2009) can be used along with proposed approximations to implement fixed window scan statistics for ARMA models. The accuracy of these approximations is investigated via simulation. Moreover, a multiple window scan statistic is defined for detecting a local change in the mean of a Gaussian white noise component in ARMA models, when the appropriate length of the scanning window is unknown. Based on the numerical results, for power comparisons of the scan statistics, we can conclude that when the window size of a local change is unknown, the multiple window scan statistic outperforms the fixed window scan statistics.     

Generalized variance estimators in the multivariate gamma models

It is already known that the uniformly minimum variance unbiased (UMVU) estimator of the generalized variance always exists for any natural exponential family. However, in practice, this estimator is often difficult to obtain. This paper provides explicit forms of the UMVU estimators for the bivariate and symmetric multivariate gamma models, which are diagonal quadratic exponential families. For the non-independent multivariate gamma models, it is shown that the UMVU and the maximum likelihood estimators are not proportional.      

A non-negative matrix factorization model based on the zero-inflated Tweedie distribution

Non-negative matrix factorization (NMF) is a technique of multivariate analysis used to approximate a given matrix containing non-negative data using two non-negative factor matrices that has been applied to a number of fields. However, when a matrix containing non-negative data has many zeroes, NMF encounters an approximation difficulty. This zero-inflated situation occurs often when a data matrix is given as count data, and becomes more challenging with matrices of increasing size. To solve this problem, we propose a new NMF model for zero-inflated non-negative matrices. Our model is based on the zero-inflated Tweedie distribution. The Tweedie distribution is a generalization of the normal, the Poisson, and the gamma distributions, and differs from each of the other distributions in the degree of robustness of its estimated parameters. In this paper, we show through numerical examples that the proposed model is superior to the basic NMF model in terms of approximation of zero-inflated data. Furthermore, we show the differences between the estimated basis vectors found using the basic and the proposed NMF models for \(\beta \) divergence by applying it to real purchasing data.     

Invariant Measures and Asymptotic Gaussian Bounds for Normal Forms of Stochastic Climate Model

The systematic development of reduced low-dimensional stochastic climate models from observations or comprehensive high dimensional climate models is an important topic for atmospheric low-frequency variability, climate sensitivity, and improved extended range forecasting. Recently, techniques from applied mathematics have been utilized to systematically derive normal forms for reduced stochastic climate models for low-frequency variables. It was shown that dyad and multiplicative triad interactions combine with the climatological linear operator interactions to produce a normal form with both strong nonlinear cubic dissipation and Correlated Additive and Multiplicative (CAM) stochastic noise. The probability distribution functions (PDFs) of low frequency climate variables exhibit small but significant departure from Gaussianity but have asymptotic tails which decay at most like a Gaussian. Here, rigorous upper bounds with Gaussian decay are proved for the invariant measure of general normal form stochastic models. Asymptotic Gaussian lower bounds are also established under suitable hypotheses.     

Offline and online weighted least squares estimation of nonstationary power ARCH processes

This paper proposes two estimation methods based on a weighted least squares criterion for non-(strictly) stationary power ARCH models. The weights are the squared volatilities evaluated at a known value in the parameter space. The first method is adapted for fixed sample size data while the second one allows for online data available in real time. It will be shown that these methods provide consistent and asymptotically Gaussian estimates having asymptotic variance equal to that of the quasi-maximum likelihood estimate (QMLE) regardless of the value of the weighting parameter. Finite-sample performances of the proposed WLS estimates are shown via a simulation study for various sub-classes of power ARCH models.     

Dependence Properties of Conditional Distributions of some Copula Models

For multivariate data from an observational study, inferences of interest can include conditional probabilities or quantiles for one variable given other variables. For statistical modeling, one could fit a parametric multivariate model, such as a vine copula, to the data and then use the model-based conditional distributions for further inference. Some results are derived for properties of conditional distributions under different positive dependence assumptions for some copula-based models. The multivariate version of the stochastically increasing ordering of conditional distributions is introduced for this purpose. Results are explained in the context of multivariate Gaussian distributions, as properties for Gaussian distributions can help to understand the properties of copula extensions based on vines.     

Conditional independence models for seemingly unrelated regressions with incomplete data

We consider normal ≡ Gaussian seemingly unrelated regressions (SUR) with incomplete data (ID). Imposing a natural minimal set of conditional independence constraints, we find a restricted SUR/ID model whose likelihood function and parameter space factor into the product of the likelihood functions and the parameter spaces of standard complete data multivariate analysis of variance models. Hence, the restricted model has a unimodal likelihood and permits explicit likelihood inference. In the development of our methodology, we review and extend existing results for complete data SUR models and the multivariate ID problem.     

Testing for Varying Dispersion in Exponential Family Nonlinear Models

A diagnostic model and several new diagnostic statistics are proposed for testing for varying dispersion in exponential family nonlinear models. A score statistic and an adjusted score statistic based on Cox and Reid (1987, J. Roy. Statist. Soc. Ser. B, 55, 467-471) are derived in normal, inverse Gaussian, and gamma nonlinear models. An adjusted likelihood ratio statistic is also given for normal and inverse Gaussian nonlinear models. The results of simulation studies are presented, which show that the adjusted tests keep their sizes better and are more powerful than the ordinary tests.     

Small and Large Scale Asymptotics of some Lévy Stochastic Integrals

We provide general conditions for normalized, time-scaled stochastic integrals of independently scattered, Lévy random measures to converge to a limit. These integrals appear in many applied problems, for example, in connection to models for Internet traffic, where both large scale and small scale asymptotics are considered. Our result is a handy tool for checking such convergence. Numerous examples are provided as illustration. Somewhat surprisingly, there are examples where rescaling towards large times scales yields a Gaussian limit and where rescaling towards small time scales yields an infinite variance stable limit, and there are examples where the opposite occurs: a Gaussian limit appears when one converges towards small time scales and an infinite variance stable limit occurs when one converges towards large time scales.      

A differential harmony search based hybrid interval type2 fuzzy EGARCH model for stock market volatility prediction

In this paper a new hybrid model integrating an interval type2 fuzzy logic system (IT2FLS) with a computationally efficient functional link artificial neural network (CEFLANN) and an Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH) model has been proposed for accurate forecasting and modeling of financial data with changing variance over time. The proposed model denoted as IT2F-CE-EGARCH helps to enhance the ability of EGARCH model through a joint estimation of the important features of EGARCH like leverage effect, asymmetric shock by leverage effect with the secondary membership functions of interval type2 TSK FLS and the functional expansion and learning component of a CEFLANN. The secondary membership functions with upper and lower limits of IT2FLS provide a forecasting interval for handling more complicated uncertainties involved in volatility forecasting compared to type1 FLS. The performance of the proposed model has been observed with two membership functions i.e. Gaussian with fixed mean, uncertain variance and Gaussian with fixed variance and uncertain mean. The proposed model has also been compared with a few other fuzzy time series models and GARCH family models based on four performance metrics: MSFE, RMSFE, MAFE and Rel MAE. Again a differential harmony search (DHS) algorithm has been suggested for optimizing the parameters of all the fuzzy time series models. The results indicate that the proposed IT2F-CE-EGARCH model offers significant improvements in volatility forecasting performance in comparison with all other specified models over BSE Sensex and CNX Nifty dataset.     

Robust inference in linear mixed model with skew normal-symmetric error

Linear mixed effects models with general skew normal-symmetric (SNS) error are considered and several properties of the SNS distributions are obtained. Under the SNS settings, ANOVA-type estimates of variance components in the model are unbiased, the ANOVA-type F-tests are exact F-tests in SNS setting, and the exact confidence intervals for fixed effects are constructed. Also the power of ANOVA-type F-tests for components are free of the skewing function if the random effects normally distributed. For illustration of the main results, simulation studies on the robustness of the models are given by comparisons of multivariate skew-normal, multivariate skew normal-Laplace, multivariate skew normal-uniform, multivariate skew normal-symmetric, and multivariate normal distributed errors. A real example is provided for the illustration of the proposed method.     

误差服从多元t分布的一类线性模型下参数估计的若干注记

研究一类线性模型下参数估计的若干问题．这类模型包含了多个因变量线性模型、增长曲线模型、扩充的增长曲线模型、似乎不相关回归方程组、方差分量模型等常用模型．在这类线性模型下,证明了当误差服从多元t分布时与误差服从多元正态分布时,具有相同的完全统计量和无偏估计,且在后一种情况下的充分统计量必为前一种情况下的充分统计量．对于带有多种协方差结构的前述几种模型,把在误差服从多元正态分布下,相应的协方差阵及有关参数的一致最小风险无偏(UMRU)估计存在性的结论推广到了相应的误差服从多元t分布情形．此外,对于误差服从多元t分布的这类统一的线性模型,给出了回归系数的线性可估函数的无偏估计的协方差阵的C-R下界．     

More on the inadmissibility of step-up

Cohen and Sackrowitz [Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure, Ann. Statist. 33 (2005) 145-158] proved that the step-up multiple testing procedure is inadmissible for a multivariate normal model with unknown mean vector and known intraclass covariance matrix. The hypotheses tested are each mean is zero vs. each mean is positive. The risk function is a 2×1 vector where one component is average size and the other component is one minus average power. In this paper, we extend the inadmissibility result to several different models, to two-sided alternatives, and to other risk functions. The models include one-parameter exponential families, independent t-variables, independent χ²-variables, t-tests arising from the analysis of variance, and t-tests arising from testing treatments against a control. The additional risk functions are linear combinations where one component is the false discovery rate (FDR).     

Characterizations of natural exponential families with power variance functions by zero regression properties

Summary Series of new characterizations by zero regression properties are derived for the distributions in the class of natural exponential families with power variance functions. Such a class of distributions has been introduced in Bar-Lev and Enis (1986) in the context of an investigation of reproductible exponential families. This class is broad and includes the following families: normal, Poisson-type, gamma, all families generated by stable distributions with characteristic exponent an element of the unit interval (among these are the inverse Gaussian, Modified Bessel-type, and Whittaker-type distributions), and families of compound Poisson distributions generated by gamma variates. The characterizations by zero regression properties are obtained in a unified approach and are based on certain relations which hold among the cumulants of the distributions in this class. Some remarks are made indicating how the techniques used here can be extended to obtain characterizations of general exponential families.The work of this author was performed while he was a visitor in the Department of Statistics, State University of New York at Buffalo     

Local asymptotic normality in a stationary model for spatial extremes

De Haan and Pereira (2006) [6] provided models for spatial extremes in the case of stationarity, which depend on just one parameter β>0 measuring tail dependence, and they proposed different estimators for this parameter. We supplement this framework by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold. Standard arguments from LAN theory then provide the asymptotic minimum variance within the class of regular estimators of β. It turns out that the relative frequency of exceedances is a regular estimator sequence with asymptotic minimum variance, if the underlying observations follow a multivariate extreme value distribution or a multivariate generalized Pareto distribution.