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In this chapter we extend the theory from univariate distributions to multivariate formulations for a select few distributions: the Gaussian, lognormal, and gamma distributions. We shall also introduce mixed distribution, which is a combination of Gaussian and lognormal random variables. We shall highlight that not all of the desired properties of the univariate descriptive statistics carry over to the multivariate formulation, but start to lay the foundations for the derivation of many of the data assimilation schemes in later chapters.

To improve the adaptability for renewable energy sources (RES), a multi-time scale optimal dispatch method, which considers the multiple correlation of RES, is proposed in the active distribution network (ADN). The proposed method includes a multiple correlation model of RES and a scheduling framework. Multiple correlation model of RES is formulated by t Copula function, Sklar's theorem and conditional distribution of multivariate t distribution. The multiple correlation model can estimate the forecast error of the future period according to the historical data and the latest prediction. The scheduling framework further employs the revised forecast to coordinate the fast/slow-response regulation resources in the ADN under three timescales. In the day-ahead stage, the operation state of slow-response resources is optimized and determined. In the intraday stage, the output of the fast-response resource, which is transferred to the real-time stage as reference trajectory, is optimized based on the short-term prediction. In the real-time stage, the stochastic model predictive control is adopted to adjust the output of fast-response resources based on ultra-short-term prediction and reference trajectory. Case analyses indicate that the proposed method can effectively reduce the voltage fluctuation, promote the utilization of demand-side resources, and improve the system economic performance by introducing the multiple correlation model.

In this contribution, a vector-autoregressive (VAR) process with multivariate t-distributed random deviations is incorporated into the Gauss-Helmert model (GHM), resulting in an innovative adjustment model. This model is versatile since it allows for a wide range of functional models, unknown forms of auto- and cross-correlations, and outlier patterns. Subsequently, a computationally convenient iteratively reweighted least squares method based on an expectation maximization algorithm is derived in order to estimate the parameters of the functional model, the unknown coefficients of the VAR process, the cofactor matrix, and the degree of freedom of the t-distribution. The proposed method is validated in terms of its estimation bias and convergence behavior by means of a Monte Carlo simulation based on a GHM of a circle in two dimensions. The methodology is applied in two different fields of application within engineering geodesy: In the first scenario, the offset and linear drift of a noisy accelerometer are estimated based on a Gauss-Markov model with VAR and multivariate t-distributed errors, as a special case of the proposed GHM. In the second scenario real laser tracker measurements with outliers are adjusted to estimate the parameters of a sphere employing the proposed GHM with VAR and multivariate t-distributed errors. For both scenarios the estimated parameters of the fitted VAR model and multivariate t-distribution are analyzed for evidence of auto- or cross-correlations and deviation from a normal distribution regarding the measurement noise.

In this paper, we propose a new kind of multivariate t distribution by allowing different degrees of freedom for each univariate component. Compared with the classical multivariate t distribution, it is more flexible in the model specification that can be used to deal with the variant amounts of tail weights on marginals in multivariate data modeling. In particular, it could include components following the multivariate normal distribution, and it contains the product of independent t-distributions as a special case. Subsequently, it is extended to the regression model as the joint distribution of the error terms. Important distributional properties are explored and useful statistical methods are developed. The flexibility of the specified structure in better capturing the characteristic of data is exemplified by both simulation studies and real data analyses.

Network meta-analysis (NMA) is gaining popularity in evidence synthesis and network meta-regression (NMR) allows us to incorporate potentially important covariates into network meta-analysis. In this paper, we propose a Bayesian network meta-regression hierarchical model and assume a general multivariate t distribution for the random treatment effects. The multivariate t distribution is desired for heavy-tailed random effects and converges to the multivariate normal distribution when the degrees of freedom go to infinity. Moreover, in NMA, some treatments are compared only in a single study. To overcome such sparsity, we propose a log-linear regression model for the variances of the random effects and incorporate aggregate covariates into modeling the variance components. We develop a Markov chain Monte Carlo sampling algorithm to sample from the posterior distribution via the collapsed Gibbs technique. We further use the Deviance Information Criterion (DIC) and the logarithm of the Pseudo-marginal likelihood (LPML) for model comparison. A simulation study is conducted and a detailed analysis from our motivating case study is carried out to further demonstrate the proposed methodology.

Skewed t-copulas recently became popular as a modeling tool of non-linear dependence in statistics. In this paper we consider three different versions of skewed t-copulas introduced by Demarta and McNeill; Smith, Gan and Kohn; and Azzalini and Capitanio. Each of these versions represents a generalization of the symmetric t-copula model, allowing for a different treatment of lower and upper tails. Each of them has certain advantages in mathematical construction, inferential tools and interpretability. Our objective is to apply models based on different types of skewed t-copulas to the same financial and insurance applications. We consider comovements of stock index returns and times-to-failure of related vehicle parts under the warranty period. In both cases the treatment of both lower and upper tails of the joint distributions is of a special importance. Skewed t-copula model performance is compared to the benchmark cases of Gaussian and symmetric Student t-copulas. Instruments of comparison include information criteria, goodness-of-fit and tail dependence. A special attention is paid to methods of estimation of copula parameters. Some technical problems with the implementation of maximum likelihood method and the method of moments suggest the use of Bayesian estimation. We discuss the accuracy and computational efficiency of Bayesian estimation versus MLE. Metropolis-Hastings algorithm with block updates was suggested to deal with the problem of intractability of conditionals.

Due to uncertainty in the nature of indirect remotely sensed precipitation products, bias adjustment is a crucial step. In this research, bias adjustment of Integrated Multi-satellitE Retrievals for GPM (Global Precipitation Measurement) (IMERG) snowfall estimates with 0.1° spatial resolution and half-hour temporal resolution was performed over a basin located in the Western United States based on Snowpack Telemetry (SNOTEL) ground station snow accumulation (SA) observations using the Gridded based Bias Adjustment using Copula (GBAC) approach. After dividing the study area into several boxes, usual and unusual pairs of cumulative distribution functions (CDFs) were separated in station pixels. Then, simulated biases were obtained using multivariate copula functions in station boxes. Next, based on the T copula function, bias adjustment bands were generated during the calibration period in station pixels. This step was followed by the validation stage. Also, bias adjustment bands were generated in pixels with no station. Based on calibration results, in boxes located west of 120° W, the range between the lower bound and the 10th percentile of the band had the most impact on satellite SA adjustment. Conversely, in boxes located east of 120° W, the middle range of the band (between 10th and 90th percentiles) was more effective. Thus, longitude was an effective factor in satellite SA bias adjustment. The validation results showed that in station boxes with a longitude over 120° west, the adjustment band was narrower. This is while in station boxes with longitude less than 120° west, the bias adjustment band better encompassed the ground SA data.

Semiparametric mixed-effects models (SMM) have received increasing attention in recent years because of the greater flexibility in analysing longitudinal trajectories. However, the normality assump...

In this paper, we consider a system of seemingly unrelated regression equations: yi=Xiβi+ei,i=1,…,m,m≥2, with μ{X1}⊇⋯⊇μ{Xm}, where the symbol μ{Xi} denotes the linear space spanned by all the colum...

Denoising of natural images is a basic problem in image processing. The present paper proposes a new algorithm for image denoising based on the maximum a-posteriori (MAP) estimator in undecimated dual-tree complex wavelet transform. The undecimated dual-tree complex wavelet transform (UDT-CWT), along with the directional selectivity of the dual-tree complex wavelet transform (DT-CWT), offers exact translational invariance property through removing the down-sampling of filter outputs together with the up-sampling of the complex filter pairs of DT-CWT. These properties are very important in image denoising. The performance of the MAP estimator depends strongly on the probability of noise-free wavelet coefficients. In our proposed denoising method, multivariate t-distribution is applied as the prior probability of noise-free coefficients. The t-distribution can accurately model the statistics of wavelet coefficients, which have peaky and heavy-tailed characteristics. On the other hand, the multivariate model makes it possible to take into account the dependencies of wavelet coefficients and their neighbors. Also, in our work, the necessary parameters of the multivariate distribution will be estimated in a locally-adaptive way to improve the denoising results via using the correlations among the amplitudes of neighbor coefficients. Simulation results delineate that the proposed algorithm outperforms state-of-the-art denoising algorithms in the literature.

This paper considers the vector autoregressive model of order p, VAR(p), with multivariate t error distributions, the latter being more prevalent in real life than the usual multivariate normal distribution. It is believed that the maximum-likelihood equations for the multivariate t distribution have convergence problem, hence we develop estimation procedures for VAR(p) model using the normal mean–variance mixture representation of multivariate t distribution. The procedure relies on the computational ease available in Expectation Maximization-based algorithms. The estimators obtained are explicit functions of sample observations and therefore are easy to compute. Extensive simulation experiments show that the estimators have negligible bias and are considerably more efficient than an existing method that uses the least-squares error approach. It is shown that the proposed estimators are robust to plausible deviations from an assumed distribution and hence are more advantageous when compared with the other estimator. One real-life example is given for illustration purposes.

The objective of the paper is to introduce a copula methodology of economic capital modeling, which is practically applicable for life insurance companies. Copula methods make it possible to address multiple dependent risk factors including both investment and underwriting risks in the framework of a portfolio approach. We identify a relevant set of asset and liability variables, and suggest a copula model for the joint distribution of these variables. Estimates of economic capital are constructed via VaR and TVaR calculations based on the tails of this joint distribution. This approach requires ARIMA and copula model selection followed by Monte Carlo simulation of the time series of the joint asset/liability portfolio. Models are implemented in open source software (R and MS Excel) and tested using historical and simulated asset/liability data. The results are applied to the construction of a software tool which can be utilized for customization and direct user application. The novelty of the approach consists in estimating interdependent underwriting and investment risks in one multivariate model taking into account short-term (daily or monthly) fluctuations of the market. In particular, we address the challenges that life insurance companies face in the low interest environment, using the market data for the 15-year period 2003–2018.

Chapter 6 is generally concerned with issues related to multivariate data. The chapter begins with basic techniques for measuring the overall dispersion of a data cloud and how to measure the extent a particular point is nested within the cloud. These methods have practical value when testing hypotheses. Next, robust multivariate measures of location and scatter are introduced, followed by multivariate outlier detection techniques. Included are recent results related to situations where the number of variables is large. Confidence regions for robust multivariate measures of location, which take into account the overall structure of the data when dealing with outliers, are described. Inferential methods in the one-sample case are described, including an extension of Hotelling's method to trimmed means. Multivariate extensions of the Wilcoxon–Mann–Whitney test are covered, one of which is based on a projection of the data that helps characterize the extent two groups differ. The notion of data depth can be used to compare groups, which is based in part on a multivariate notion of ranks. This is followed by methods for comparing dependent groups. The chapter concludes with recent advances relevant to robust principal components, multivariate discriminate analysis, as well as some recent results related to functional data. R functions are described and illustrated.

This paper examines the impact of the novel coronavirus (COVID-19) on the interdependence of stock markets in China, Japan and U. S. Data is divided into two sub-periods: pre-crisis period (Jul. 1st, 2019-Dec.31st, 2019) and post-crisis period (Jan.1st, 2020-Jun. 30th, 2020). Method of Conditional Maximum Likelihood Estimation is used. The results from copula methods indicate 1) following the COVID-19 outbreak, degree of dependence among stocks have increased; 2) the outbreak has converted left-tailed dependence into a right-tailed dependence; 3) Gumbel copula has better in-sample fitness before crisis while for post-crisis period, Student's T copula is better.

Symmetric elliptical distributions have been intensively used in data modeling and robustness studies. The area of applications was considerably widened after transforming elliptical distributions into the skew elliptical ones that preserve several good properties of the corresponding symmetric distributions and increase possibilities of data modeling. We consider three-parameter p-variate skew t-distribution where p-vector μ is the location parameter, Σ:p×p is the positive definite scale parameter, p-vector α is the skewness or shape parameter, and the number of degrees of freedom ν is fixed. Special attention is paid to the two-parameter distribution when μ=0 that is useful for construction of the skew t-copula. Expressions of the parameters are presented through the moments and parameter estimates are found by the method of moments. Asymptotic normality is established for the estimators of Σ and α. Convergence to the asymptotic distributions is examined in simulation experiments.

It has been argued that fitting a t-copula to financial data is superior to a normal copula. To overcome the shortcoming that a t-copula only has one parameter for the degrees of freedom, the t-copula with multiple parameters of degrees of freedom has been proposed in the literature, which generalizes both the t-copulas and the grouped t-copulas. Like the inference for a t-copula, a computationally efficient inference procedure is to first estimate the correlation matrix via Kendalls $\tau$ and then to estimate the parameters of degrees of freedom via pseudo maximum likelihood estimation. This paper proposes a jackknife empirical likelihood test for testing the equality of some parameters of degrees of freedom based on this two-step inference procedure, and shows that the Wilks theorem holds.

This study is investigating the predictability of the five Fama–French factors and explores their optimal portfolio allocation for factor investing during 2000–2017. Firstly, we forecast each factor with a pool of linear and nonlinear models. Next, the individual forecasts are combined through dynamic model averaging, and their performance is benchmarked by the best performing individual predictor and other forecast combination techniques. Finally, we use the generalized autoregressive score model and the skewed t copula method to estimate the correlation of assets. The generalized autoregressive score performance is also compared with other traditional approaches such as dynamic conditional correlation model and asymmetric dynamic conditional correlation. The performance of the constructed portfolios is assessed through traditional metrics and ratios accounting for the conditional value‐at‐risk and the conditional diversification benefits approach. Our results show that combining Bayesian forecast combinations with copulas is leading to significant improvements in the portfolio optimization process, and forecasting covariance accounting for asymmetric dependence between the factors adds diversification benefits to the obtained portfolios.

Water accessibility is an important factor in water use assessments for water users. Water accessibility is affected by climate and socio-economic changes, and notably large regional differences exist. Therefore, understanding water accessibility is essential for effective water management. Based on the length, runoff and viewshed value of each water consumption grid, this paper characterized the corresponding water accessibility using the standardized accumulative probability distribution of the three variables. The Copula function was used to connect the accumulative probability distributions of the three variables and construct a joint distribution. The joint probability distribution was applied to characterize the water accessibility of all of the water consumption grids, and a grid-scale water accessibility evaluation model-LRV (Length-Runoff-Viewshed) was constructed. This paper chose the Yongdinghe River Basin as a case study to evaluate water accessibility using the LRV model. The T Copula function was chosen as the joint distribution function to obtain the water accessibility distribution pattern of the Yongdinghe River Basin. This model obtained the water accessibility of every water consumption grid and provided technical support for refined water resources management.

The recent financial crises have motivated the researchers to study the cross-market linkage, and how shocks are transmitted across countries. Several scholars have studied financial contagion and its different impacts on markets and international portfolios using static models. This paper employs a regular vine copula approach to model the dependence dynamics between major American and European stock markets by distinguishing the effects during crisis periods and tranquility periods. Empirical results show a significant change in the connectedness and shock transmissions during both periods, indicating strong evidence of financial contagion with the Eurozone at its origin. Moreover, the regular vine copula surpasses the multivariate t copula. Hence, the shock transmission path is as important as the dependence itself.

In applications tail dependence is an important property of a copula. Bivariate tail dependence is investigated in many papers, but multivariate tail dependence has not been studied widely. We define multivariate upper and lower tail dependence coefficients as limits of the probability that values of one marginal will be large if at least one of other marginals will be as large also. Further we derive some relations between introduced tail dependence and bivariate tail dependence coefficients. Applications have shown that the multivariate t-copula has been successfully used in practice because of its tail dependence property. Therefore, t-copula can be used as an alternative method for risk assessment under Solvency II for insurance models. We have paid attention to the properties of the introduced multivariate tail dependence coefficient for t-copula and examine it in the simulation experiment.

When doing multivariate data analysis, one common obstacle is the presence of incomplete observations, i.e., observations for which one or more key fields are missing data. Missing data is often countered by deleting entire observations that contain missing data. The negative effects of deleting entire observations are multiple: deleting observations reduces sample size and can also result in biased inferences even if data is missing at random. In addition, knowledge contained within incomplete observations is knowledge lost when they are deleted- and the effort spent collecting that knowledge is effort wasted. Data imputation methods, or methods of statistically “filling-in” missing data, can help combat small sample sizes by using the existing information in partially complete observations with the end goal of producing less biased and higher confidence inferences. When a sample from a multivariate normal population is only partially complete, and the missing data meets appropriate assumptions (missing at random), robust data imputation of the missing data can be implemented with monotone data augmentation (MDA) using the multivariate t distribution. Missing data imputation is applied to data from the NASA Instrument Cost Model (NICM) using the MDA algorithm under the assumption of having a multivariate t distribution with fixed degrees of freedom. A sensitivity analysis to the degrees of freedom parameter is presented to demonstrate robustness of the multivariate t distribution when dealing with small samples as compared to the multivariate normal distribution.

Best integer equivariant (BIE) estimators provide minimum mean squared error (MMSE) solutions to the problem of GNSS carrier-phase ambiguity resolution for a wide range of distributions. The associated BIE estimators are universally optimal in the sense that they have an accuracy which is never poorer than that of any integer estimator and any linear unbiased estimator. Their accuracy is therefore always better or the same as that of Integer Least-Squares (ILS) estimators and Best Linear Unbiased Estimators (BLUEs).Current theory is based on using BIE for the multivariate normal distribution. In this contribution this will be generalized to the contaminated normal distribution and the multivariate t-distribution, both of which have heavier tails than the normal. Their computational formulae are presented and discussed in relation to that of the normal distribution. In addition a GNSS real-data based analysis is carried out to demonstrate the universal MMSE properties of the BIE estimators for GNSS-baselines and associated parameters. Keywords: Integer equivariant (IE) estimation · Best integer equivariant (BIE) · Integer Least-Squares (ILS) . Best linear unbiased estimation (BLUE) · Multivariate contaminated normal · Multivariate t-distribution . Global Navigation Satellite Systems (GNSSs)

International audience; The multivariate t (MVT)-distribution is a widely used statistical model in various application domains, mainly due to its adaptability to heavy-tailed data. However, estimating the degree of freedom (d.o.f) parameter, that controls the shape of the distribution, remains a challenging problem. In this work, we develop a novel methodology and design various algorithms for estimating the d.o.f parameter. More precisely, based on a key relationship between scatter and covariance matrices for the t-distribution, the estimator is derived from the expectation of a particular quadratic form and is proved to converge although the classical independence assumption is not fulfilled. finally, some preliminary simulations show the improvement of the proposed approach with respect to state-of-the-art methods.