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This article introduces co-quantiles, a new concept in rank statistics that generalizes quantiles, order statistics, and concomitants. In contrast to conventional order statistics that rank and record the same attribute of a population, or concomitants that consider different attributes observed over the same time period, co-quantiles allow for the ranking and recording of different attributes, and more generally their functions, across different time periods. In the special case of affine co-quantiles (defined formally in the article), we derive explicit expressions for their probability density functions and moments under the assumption that the distribution of the underlying data is multivariate normal. In particular, we establish that the former is a sum of unified skew-normal distributions introduced in Arellano-Valle and Azzalini (2006). The co-quantile results naturally reduce to those for order statistics and concomitants, and generalize those on the distributions of linear combinations and the maxima of vector-valued random variables obtained in Arellano-Valle and Genton (2007, 2008) and on cross-sectional momentum returns obtained in Kwon and Satchell (2018). As an example that has not been analyzed in the financial literature, we consider momentum spill-over returns and establish theoretically that these returns are susceptible to sudden changes in skewness and kurtosis during periods of market uncertainty. [ABSTRACT FROM AUTHOR]

We introduce a novel probability density function for modeling the distribution of points around an ellipsoidal surface. This density is part of the family of elliptical distributions. We establish the theoretical convergence properties of the parameter estimators and validate them using simulated data. Furthermore, we propose a mixture model utilizing this density, and we estimate its parameters using the Expectation-Maximization (EM) algorithm. To assess its performance, we compare the algorithm to a state-of-the-art ellipse fitting method and conduct experiments on 3D real data obtained from depth cameras. [ABSTRACT FROM AUTHOR]

The distribution of the sample correlation matrix when the sample comes from a real multivariate normal population and when the population covariance matrix is diagonal or when the population variables are independently distributed is available in the literature. In the general case, that is, when the covariance matrix in the normal population is a general real positive definite matrix, the distribution of the sample correlation matrix is not available in a compact computable form in the literature. Some scattered results on some aspects of the general case are available, but the representations are not easily tractable. This article deals with a covariance structure coming from a real matrix-variate gamma when the scale parameter matrix is a general real positive definite matrix. General procedures for deriving asymptotic chi-square and asymptotic normal for some general moment structures are also presented in this article. The density in some particular cases are given explicitly. Graphs are also provided in the particular cases. [ABSTRACT FROM AUTHOR]

Hüsler–Reiss vectors and Brown–Resnick fields are popular models in multivariate and spatial extreme-value theory, respectively, and are widely used in applications. We provide analytical formulas for the correlation between powers of the components of the bivariate Hüsler–Reiss vector, extend these to the case of the Brown–Resnick field, and thoroughly study the properties of the resulting dependence measure. The use of correlation is justified by spatial risk theory, while power transforms are insightful when taking correlation as dependence measure, and are moreover very suited damage functions for weather events such as wind extremes or floods. This makes our theoretical results worthwhile for, e.g., actuarial applications. We finally perform a case study involving insured losses from extreme wind speeds in Germany, and obtain valuable conclusions for the insurance industry. [ABSTRACT FROM AUTHOR]

We derive an equation system for finding Maximum Likelihood Estimators (MLEs) for the parameters of a p-dimensional t-distribution with ν degrees of freedom, t p , ν , and use the MLEs for testing covariance structures for the t p , ν -distributed population. The likelihood ratio test (LRT), Rao score test (RST) and Wald test (WT) statistics are derived under the general null-hypothesis H 0 : Σ = Σ 0 , using a matrix derivative technique. Here the p × p -matrix Σ is a dispersion/scale parameter. Convergence to the asymptotic chi-square distribution under the null hypothesis is examined in extensive simulation experiments. Also the convergence to the chi-square distribution is studied empirically in the situation when the MLEs of a t p , ν -distribution are changed to the corresponding estimators for a normal population. Type I errors and the power of the tests are also examined by simulation. In the simulation study the RST behaved more adequately than all remaining statistics in the situation when the dimensionality p was growing. [ABSTRACT FROM AUTHOR]

Correlation coefficients and linear regression values computed from group averages can differ from correlation coefficients and linear regression values computed using individual scores. This observation known as the ecological fallacy often assumes that all the individual scores are available from a population. In many situations, one must use a sample from the larger population. In such cases, the computed correlation coefficient and linear regression values will depend on the sample that is chosen and the underlying sampling distribution. The sampling distribution of correlation coefficients and linear regression values for group averages will be identical to the sampling distribution for individuals for normally distributed variables for random samples drawn from infinitely large continuous distributions. However, data that is acquired in practice is often acquired when sampling without replacement from a finite population. Our objective is to demonstrate through Monte Carlo simulations that the sampling distributions for correlation and linear regression will also be similar for individuals and group averages when sampling without replacement from normally distributed variables. These simulations suggest that when a random sample from a population is selected, the correlation coefficients and linear regression values computed from individual scores will not be more accurate in estimating the entire population values compared to samples when group averages are used as long as the sample size is the same. [ABSTRACT FROM AUTHOR]

The multivariate inverse hypergeometric (MIH) distribution is an extension of the negative multinomial (NM) model that accounts for sampling without replacement in a finite population. Even though most studies on longitudinal count data with a specific number of 'failures' occur in a finite setting, the NM model is typically chosen over the more accurate MIH model. This raises the question: How much information is lost when inferring with the approximate NM model instead of the true MIH model? The loss is quantified by a measure called deficiency in statistics. In this paper, asymptotic bounds for the deficiencies between MIH and NM experiments are derived, as well as between MIH and the corresponding multivariate normal experiments with the same mean-covariance structure. The findings are supported by a local approximation for the log-ratio of the MIH and NM probability mass functions, and by Hellinger distance bounds. [ABSTRACT FROM AUTHOR]

AbstractIn this study, we consider the problems of testing for a mean vector and testing the equality of two mean vectors with monotone missing data. We propose new test statistics similar to the simplified Hotelling’s T2-type test statistic for one-sample and two-sample problems under general-step monotone missing data. Approximate upper percentiles of this new statistics are provided by asymptotic expansion along with transformed test statistics based on Bartlett adjustment. Approximate simultaneous confidence intervals for pairwise comparisons among mean vectors are also presented. Furthermore, we investigated the asymptotic behavior of the proposed statistics using a Monte Carlo simulation, and the approximate accuracy of the proposed approximations are provided and discussed. Finally, the numerical power for several statistics is given. [ABSTRACT FROM AUTHOR]

Dried grapes (or Raisins) are among the most frequently grown and consumed cereal crops worldwide. They are also an important source of nutrition and nourishment in a variety of countries including Türkiye, the United States, Greece, etc. In addition to that, raisins consist of 15% water, 79% carbs (including 4% fiber), 3% protein, and very little fat. In our study, there were a total of 900 raisin grains used, with 450 pieces from each type: Kecimen and Besni raisin. Seven morphological features were taken from these images after going through several steps of pre-processing. Since machine learning algorithms can analyze large datasets quickly, automatic classification is made possible. With enough training and testing, machine learning models can attain a high degree of precision in classifying raisin grains. They are able to detect variations in size, shape, color, and texture that would be difficult for humans to detect consistently. Eleven machine learning and five different types of artificial intelligence have been used to classify these features. As part of this study, we look into different machine learning and deep learning methods: GaussianNB, Decision Tree, K-Nearest Neighbor, Random Forest, Support vector machine (SVM), XGBoost, LightGBM, and AdaBoost, Logistic Regression, Artificial Neural Network and Deep Learning Network. Study efficacy is evaluated using standard metrics as F1 score and ROC area under the curve (AUC). Using the caret, H2O, neuralnet, and keras packages, AdaBoost and LightGBM, two of the fourteen models, achieve an accuracy of 90.30% and 98.40%, respectively, and a ROC curve score of around 90%. [ABSTRACT FROM AUTHOR]

In this paper, we derive the analytical behavior of the limiting spectral distribution of non-central covariance matrices of the "general information-plus-noise" type, as studied in Zhou (JTP 36:1203–1226, 2023). Through the equation defining its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and we show the determination criterion for its support. We also extend the result in Zhou (JTP 36:1203-1226, 2023) to allow for all possible ratios of row to column of the underlying random matrix. [ABSTRACT FROM AUTHOR]

Simplices are the fundamental domain when integrating over convex polytopes. The aim of this work is to establish a novel framework of Monte Carlo integration over simplices, throughout from sampling to variance reduction. Namely, we develop a uniform sampling method on the standard simplex consisting of two independent procedures and construct theories on change of measure on each of the two independent elements in the developed sampling technique with a view towards variance reduction by importance sampling. We provide illustrative figures and numerical results to support our theoretical findings and demonstrate the strong potential of the developed framework for effective implementation and acceleration of Monte Carlo integration over simplices. [ABSTRACT FROM AUTHOR]

In this paper, the Bayesian inference of the model of dependent lives is considered. We use the bivariate Gompertz (BGP) distribution. This new bivariate survival model and its extension have recently been proposed by Shoaee and Khorram. This model is more flexible and can be applied in the actuarial science of life insurance, and reliability theory in competing risks and shock models. As we know, the maximum likelihood estimates do not always exist and therefore cannot always be calculated. So, the Bayesian estimations are considered using the squared error loss function and a priori distributions that create a dependency between the hyper-parameters. But given the assumptions, one can see that explicit expressions cannot be obtained for Bayesian estimations. The importance sampling method is proposed to calculate the Bayes estimations and also to create the corresponding HPD credible intervals of the unknown parameters. The Monte Carlo simulation studies and analysis of a real data set are also performed to evaluate the proposed method for estimating parameters. Finally, a generalization of the dependent lives models is presented and the Bayesian estimate for this new bivariate family distribution is introduced and a comparison is made between the models of this new family. [ABSTRACT FROM AUTHOR]

In recent years, the study of Receiver Operating Characteristic (ROC) curve analysis has gained significant attention as a means of accurately assessing test performance and determining optimal cutoff points. Traditionally, ROC models have been developed for bi-distributional univariate and multivariate data, such as Bi-normal, Bi-Exponential, Multivariate ROC models, and so forth. However, in current practical scenarios, the prevalence of high-dimensional matrix variate data poses a challenge for accurate test evaluation. To address this issue, this paper presents a novel ROC model that incorporates matrix variate normal distribution to effectively explain the accuracy of a test in the context of matrix data. Further, the accuracy measure, Area under the Curve (AUC) is derived, which helps in explaining the variability of the curve and provides the sensitivity at a particular value of specificity and vice versa. The proposed methodology is supported by a real data set and simulation studies. [ABSTRACT FROM AUTHOR]

This paper addresses certain distributional aspects of a scale-free scatter matrix denoted by R that is stemming from a matrix-variate gamma distribution having a positive definite scale parameter matrix B. Under the assumption that B is a diagonal matrix, a structural representation of the determinant of R is derived; the exact density functions of products and ratios of determinants of matrices possessing such a structure are obtained; a closed form expression is given for the density function of R. Moreover, a novel procedure is utilized to establish that certain functions of the determinant of the sample scatter matrix are asymptotically distributed as chi-square or normal random variables. Then, representations of the density function of R that respectively involve multiple integrals, multiple series and Gauss' hypergeometric function are provided for the general case of a positive definite scale parameter matrix, and an illustrative numerical example is presented. Cutting-edge mathematical techniques have been employed to derive the results. Naturally, they also apply to the conventional sample correlation matrix which is encountered in various multivariate inference contexts. [ABSTRACT FROM AUTHOR]

We provide a formal definition of an M-state multivariate Markov switching (MS) trend, describe its asymptotic distribution, and consider vector autoregressive processes with MS trends which contain either unit roots or a stationary part. Then, we estimate the coefficients of such models via ordinary least squares (OLS) , and determine the asymptotic distributions of OLS estimators in terms of functionals on a multivariate Brownian motion. [ABSTRACT FROM AUTHOR]

Polarimetric synthetic aperture radar (PolSAR) systems are powerful remote sensing instruments. Despite their ability to acquire images in all weather conditions, day and night, and to detect dielectric properties and structures, PolSAR images are corrupted by multidimensional speckle. Such contamination, resulting from the use of coherent illumination, does not obey the classical hypothesis of additive Gaussian noise and therefore require special treatment. Moreover, the data are structured as positive-definite Hermitian matrices. The Wishart and G m 0 laws, both defined on the set of all positive definite Hermitian matrices, successfully describe fully PolSAR returns. The Shannon entropy is a scalar that measures the contrast between two PolSAR observations in this context. We obtain the Shannon entropy of these models and, using the h - ϕ entropy formalism, its asymptotic distribution. With these two elements, and using the Stochastic Expectation-Maximisation algorithm, we obtain better segmentations than those provided by the k-means and k-medoids with nine-dimensional inputs. We show applications to actual PolSAR data. Our results show that G m 0 entropy-based segmentation is better suited for images where the class features are not well separated, while its Wishart counterpart gives the best results in scenarios with more separable regions. [ABSTRACT FROM AUTHOR]

It is now known that early government interventions in pandemic management helps in slowing down the pandemic in the initial phase, during which a conservative basic reproduction number can be maintained. There have been several ways to evaluate these early response strategies for COVID-19 during its outbreak globally in 2020. As a novelty, we evaluate them through the lens of patient recovery logistics. Here, we use a data-driven approach of recovery analysis in a case study of Singapore during January 22–April 01, 2020, which is effectively the analysis of length-of-stay in the government healthcare facility, National Center for Infectious Diseases. We propose the use of a data-driven method involving periodization, statistical analysis, regression models, and epidemiological models. We demonstrate that the estimates of reproduction number in Singapore shows variation in different age groups and periods, indicating the success of early intervention strategy in the initial transmission stages of the pandemic.

In order to better realize the innovation and entrepreneurship model of college students, the author proposed a statistical method based on probability theory. Based on the data obtained from the questionnaire on "university students in the community of innovation and entrepreneurship", this paper analyzes the existing problems already in the innovation and entrepreneurship of college students in the community. Second, on the basis of pattern analysis and data reliability, the analysis method based on principal analysis is used to determine the eight factors that influence the innovation and entrepreneurship among college students in the community. In addition, innovative and effective business opportunities for community college students based on statistical results have been developed. The test results show that: from the prediction of the last three sets of models, the average relative error of y1 and y2 is 0.0013 and 0.0984 respectively, and the error is less 0.1, indicating that the relative error of the model is within the acceptable range. The results calculated by the model are accurate according to the actual situation from the question. It can be seen that the author uses the analysis method as the principal analysis to determine the eight important factors that affect the innovation and entrepreneurship of the students. High society is necessary, which proves that the design model created by the author is scientific and practical.

In order to solve the problem of Poisson equation, the author puts forward a research on the character image setting of 3D animation works. 3D animation refers to the three-dimensional virtual image produced by using computer software, also known as 3D animation, is a new technology produced with the development of computer software and hardware technology in recent years. It is the art of photography, set design, and stage lighting reasonable arrangement of various arts and techniques. At the same time, the design and production of 3D animation need more artistic foundation and creativity. A good 3D animation, it requires producers to have a better sense of space and artistic sense, there is must be a good use of all kinds of 3D animation production software. The image design of animated characters from the perspective of content, design styles and styles vary greatly from reality to non-reality. However, whether taken from nature or elsewhere, they are inseparable from their character as a vehicle for expressing human spirit and emotion. Since the birth of animation, its prominent entertainment function has become the value of the existence of this art style, and it has the possibility of development. And animated characters are more iconic than any other element of animation. The purpose of animation character design is to give appeal and vitality to each animation character art.

The derivation of the sample covariance is difficult compared to that of the distribution of the sample correlation coefficient. This paper deals with the distributions of covariance structures appearing in real scalar/vector/matrix variables. Covariance structure is a bilinear structure. Consider a bilinear form u = X ′ A Y where X and Y are p × 1 and q × 1 real vectors and A is a constant p × q matrix. The basic aim in this paper is to derive the distribution of such a structure when the components are scalar/vector/matrix Gaussian variables. The procedure used is to examine the Laplace transform or the moment generating function (mgf) coming from such a bilinear form in real scalar/vector/matrix variables. Covariance structures in several situations are shown to produce a mgf of the type (1 − λ 2 t 2) − α , λ > 0 , α > 0 , − 1 λ < R (t) < 1 λ where t is the mgf parameter, R (·) means the real part of (·) , and λ and α are real scalar parameters. Explicit evaluation of the density of u is considered when α is a positive integer as well as for a general α. It is shown that the exact densities can be written as linear functions of double gamma densities and double exponential or Laplace densities when α is a positive integer. For the general value of α, it is shown that the exact density can be written in terms of double Mittag-Leffler or a double confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

The classical autoregressive type models are widely used in time series modelling. Recently, a class of models known as generalized autoregressive, recognized by an additional parameter, has been proposed in order to reveal some hidden features which cannot be characterized by the standard autoregressive models. In this paper, the generalized autoregressive models are extended to the vector-valued autoregressive models which provide a flexible framework for modelling the dependent data. The properties of the new model such as stationary conditions, some explicit form of the auto-covariance function and the spectral density matrices are investigated. Unknown parameters are then estimated and compared with other kinds of traditional methods. The numerical results obtained by means of simulation studies are then reported. Finally, the traditional autoregressive model and generalized autoregressive model are fitted to a well-known bivariate time series, respectively, and the performance of the proposed models and the estimation methods are discussed. [ABSTRACT FROM AUTHOR]

In this paper a new family of Harris skewed normal distributions (HSND) is proposed. The probability density function of the HSND can be both positively and negatively skewed, unimodal and multimodal and can be symmetric with heavy tails. Monte Carlo simulations are conducted to evaluate the applicability of the proposed family of distributions. The HSND family is used to fit three different real data sets. [ABSTRACT FROM AUTHOR]

The precise modeling of bivariate continuous characteristics remains an actual challenge in probability and statistics. In this paper, we explore a new strategy based on the combination of a simple polynomial-sine copula and the Burr XII distribution. The idea is to use the oscillating functionalities of the polynomial-sine copula and the flexibility of the Burr XII distribution to propose a serious bivariate solution for the modelling of bivariate lifetime phenomena. Both theory and practice are developed. In particular, we determine the main functions related to the distribution, like the cumulative distribution function, probability density function, conditional density function, and hazard rate function, and perform a moment analysis, including various useful measures for bivariate modeling. On the practical plan, we derive the maximum likelihood and Bayes estimates for the unknown parameters. Also, the bootstrap confidence interval and the highest posterior density interval are obtained. The performance of the proposed bivariate distributions is examined using a simulation study. Finally, one data set is considered to illustrate its flexibility for real-life applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider a flexible semiparametric approach for estimating multivariate probability mass functions. The corresponding estimator is governed by a parametric starter, for instance a multivariate Poisson distribution with nonnegative cross correlations which is basically estimated through an expectation–maximization algorithm, and a nonparametric part which is an unknown weight discrete function to be smoothed through multiple binomial kernels. Our central focus is upon the selection matrix of bandwidths by the local Bayesian method. We additionally discuss the diagnostic model to enact an appropriate choice between the parametric, semiparametric and nonparametric approaches. Retaining a pure nonparametric method implies losing parametric benefices in this modelling framework. Practical applications, including a tail probability estimation, on multivariate count datasets are analyzed under several scenarios of correlations and dispersions. This semiparametic approach demonstrates superior performances and better interpretations compared to parametric and nonparametric ones. [ABSTRACT FROM AUTHOR]

Let X and Y be independent m × m symmetric positive definite random matrices. Assume that X follows a matrix variate beta distribution with parameters a and b and that Y has a matrix variate beta distribution with parameters a + b and c. Define R = I m − Y + Y 1 / 2 X Y 1 / 2 − 1 / 2 Y 1 / 2 X Y 1 / 2 I m − Y + Y 1 / 2 X Y 1 / 2 − 1 / 2 and S = I m − Y + Y 1 / 2 X Y 1 / 2 , where Im is an identity matrix and A1/2 is the unique symmetric positive definite square root of A. In this paper, we have shown that random matrices R and S are independent and follow matrix variate beta distributions generalizing an independence property established by Jones and Balakrishnan (Statistics and Probability Letters, 170 (2021), article id 109011) in the univariate case. [ABSTRACT FROM AUTHOR]

Let X n be n × N random complex matrices, and let R n and T n be non-random complex matrices with dimensions n × N and n × n , respectively. We assume that the entries of X n are normalized independent random variables satisfying the Lindeberg condition, T n are nonnegative definite Hermitian matrices and commutative with R n R n ∗ , i.e., T n R n R n ∗ = R n R n ∗ T n . The general information-plus-noise-type matrices are defined by C n = 1 N T n 1 2 R n + X n R n + X n ∗ T n 1 2 . In this paper, we establish the limiting spectral distribution of the large-dimensional general information-plus-noise-type matrices C n . Specifically, we show that as n and N tend to infinity proportionally, the empirical distribution of the eigenvalues of C n converges weakly to a non-random probability distribution, which is characterized in terms of a system of equations of its Stieltjes transform. [ABSTRACT FROM AUTHOR]

This paper constructs a mental health probability space based on the law of large numbers. First, this paper defines and predicts the transition of personal emotion and the transmission of emotion. This study treats emotion as two random stages. At the same time, according to the initial parameters of each different stage model, a psychological model with individual characteristics based on the Markov chain is established. It can predict the outcome of emotional activities emotionally. Finally, numerical simulation is carried out with Matlab. The results show that the prediction method can well reflect the emotional state transmission. It can be used in psychological prediction simulations of individuals. The mental health statistical model based on the law of large numbers established in this paper opens up a new way for individual psychological counseling and mental health training.

This article aims to provide a method of regression for multivariate multiple inflated count responses assuming the responses follow a negative binomial distribution. Negative binomial regression models are common in the literature for modeling univariate and multivariate count data. However, two problems commonly arise in modeling such data: choice of the multivariate form of the underlying distribution and modeling the zero-inflated structure of the data. Copula functions have become a popular solution to the former problem because they can model the response variables’ dependence structure. The latter problem is often solved by modeling an assumed latent variable ZZ generating excess zero-valued counts in addition to the underlying distribution. However, despite their flexibility, zero-inflation models do not account for the case of mm additional inflated values at k1,k2,…,km{{\bf{k}}}_{1},{{\bf{k}}}_{2},\ldots ,{{\bf{k}}}_{m}. We propose a multivariate multiple-inflated negative binomial regression model for modeling such cases. Furthermore, we present an iterative procedure for estimating model parameters using maximum likelihood estimation. The multivariate distribution functions considering the dependence structure of the response vectors are found using copula methods. The proposed method is illustrated using simulated data and applied to real data.

The difficulty to efficiently compute the null distribution of the largest eigenvalue of a MANOVA matrix has hindered the wider applicability of Roy's Largest Root Test (RLRT) though it was proposed over six decades ago. Recent progress made by Johnstone, Butler and Paige and Chiani has greatly simplified the approximate and exact computation of the critical values of RLRT. When datasets are high dimensional (HD), Chiani's numerical algorithm of exact computation may not give reliable results due to truncation error, and Johnstone's approximation method via Tracy-Widom distribution is likely to give a good approximation. In this paper, we conduct comparative studies to study in which region the exact method gives reliable numerical values, and in which region Johnstone's method gives a good quality approximation. We formulate recommendations to inform practitioners of RLRT. We also conduct simulation studies in the high dimensional setting to examine the robustness of RLRT against normality assumption in populations. Our study provides support of RLRT robustness against non-normality in HD. [ABSTRACT FROM AUTHOR]

A bivariate extreme-value copula is characterized by its Pickands dependence function, i.e., a convex function defined on the unit interval satisfying boundary conditions. This paper investigates the large-sample behavior of a nonparametric estimator of this function due to Cormier et al. (Extremes 17:633–659, 2014). These authors showed how to construct this estimator through constrained quadratic median B-spline smoothing of pairs of pseudo-observations derived from a random sample. Their estimator is shown here to exist whatever the order m ≥ 3 of the B-spline basis, and its consistency is established under minimal conditions. The large-sample distribution of this estimator is also determined under the additional assumption that the underlying Pickands dependence function is a B-spline of given order with a known set of knots. [ABSTRACT FROM AUTHOR]

The structure testing of a high-dimensional covariance matrix plays an important role in financial stock analyses, genetic series analyses, and many other fields. Testing that the covariance matrix is block-diagonal under the high-dimensional setting is the main focus of this paper. Several test procedures that rely on normality assumptions, two-diagonal block assumptions, or sub-block dimensionality assumptions have been proposed to tackle this problem. To relax these assumptions, we develop a test framework based on U-statistics, and the asymptotic distributions of the U-statistics are established under the null and local alternative hypotheses. Moreover, a test approach is developed for alternatives with different sparsity levels. Finally, both a simulation study and real data analysis demonstrate the performance of our proposed methods. [ABSTRACT FROM AUTHOR]

Gaussian Markov random fields (GMRF) and their multivariate extensions (MGMRFs) are powerful tools for modelling probabilistic interactions of directly related variables. As an important category of graphical models, the potentials of (M)GMRF application are far-reaching. In this article, we review a class of coregionalized MGMRFs that coordinates, integrates, and generalizes the key MGMRFs in the literature. Theoretical and analytic results, including simulation and case studies, yield important insights into the model class, its options of parameterization, and the nature of asymmetry modelled by an asymmetric matrix of spatial parameters and its adaptive extensions. We show that while the Markovian interpretation of latent conditionals may be the main appeal of MGMRFs for some applications, another attraction of this model class is its coregionalization models that harness multidimensional interactions, dependencies, correlations, and variabilities for analysis of covariance structure. The latter is further illustrated by presenting models for shared component analysis, principal component analysis, and dimension reduction. The model class and its wide-ranging options for generalization are discussed for their richness and broad applicability to spatial and image data analytics and beyond. [ABSTRACT FROM AUTHOR]

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X=(X1,…,Xd){\boldsymbol{X}}=\left({X}_{1},\ldots ,{X}_{d}) of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on X{\boldsymbol{X}} is shown to be strictly weaker than the assumption that the density of the random vector (∣X1∣,…,∣Xd∣)\left(| {X}_{1}| ,\ldots ,| {X}_{d}| ) is multivariate totally positive of order 2, abbreviated MTP2{\text{MTP}}_{2}, for which the GPI is already known to hold. Under this condition, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.

This article explores the parameter space of multivariate skew normal families having identical distributed marginal distributions under a few autoregressive-moving average correlation structures ( Ω ¯ ): namely, MA(1), MA(2), and AR(2) correlation structures. Such an undertaking escapes triviality since the efficacious { Ω ¯ , δ } parametrization, in terms of analysis of marginal distributions, restricts the parameter space of the correlation and shape parameters in an intertwined fashion, which upon unraveling illuminates properties of the multivariate skew normal's correlation matrix, where under identical marginals, δ = δ 1. Using the { Ω ¯ , δ } parametrization of the multivariate skew normal distribution, the support of δ is found using a limiting argument for the cases considered herein, which can be approximated well via nonlinear regression using a ratio of rational functions of the sample size, k, for an MA(q) correlation structure. Moreover, a range of values for δ is found, when possible, such that the parameter space of the correlation parameters under skew-normality agrees with that under normality. For fixed δ's, the parameter space of the correlation parameters is found by numerical inversion, where plots have been included to visualize the regions. From the parameter space, inequalities on the component-wise correlations of the skew normal families are developed and displayed pictorially. Finally, transformations are offered to convert the multivariate skew normal families to the more prevalent { Ω ¯ , α } parametrization, with α representing a shape parameter. [ABSTRACT FROM AUTHOR]

This paper studies a prediction problem using time series data and machine learning algorithms. The case study is related to the quality control of bumper beams in the automotive industry. These parts are milled during the production process, and the locations of the milled holes are subject to strict tolerance limits. Machine learning models are used to predict the location of milled holes in the next beam. By doing so, tolerance violations are detected at an early stage, and the production flow can be improved. A standard neural network, a long short term memory network (LSTM), and random forest algorithms are implemented and trained with historical data, including a time series of previous product measurements. Experiments indicate that all models have similar predictive capabilities with a slight dominance for the LSTM and random forest. The results show that some holes can be predicted with good quality, and the predictions can be used to improve the quality control process. However, other holes show poor results and support the claim that real data problems are challenged by inappropriate information or a lack of relevant information. [ABSTRACT FROM AUTHOR]

We investigate multiple testing and variable selection using the Least Angle Regression (LARS) algorithm in high dimensions under the assumption of Gaussian noise. LARS is known to produce a piecewise affine solution path with change points referred to as the knots of the LARS path. The key to our results is an expression in closed form of the exact joint law of a |$K$| -tuple of knots conditional on the variables selected by LARS, the so-called post-selection joint law of the LARS knots. Numerical experiments demonstrate the perfect fit of our findings. This paper makes three main contributions. First, we build testing procedures on variables entering the model along the LARS path in the general design case when the noise level can be unknown. These testing procedures are referred to as the Generalized |$t$| -Spacing tests and we prove that they have an exact non-asymptotic level (i.e. the Type I error is exactly controlled). This extends work of [ 31 ] where the spacing test works for consecutive knots and known variance. Second, we introduce a new exact multiple testing procedure after model selection in the general design case when the noise level may be unknown. We prove that this testing procedure has exact non-asymptotic level for general design and unknown noise level. Third, we prove exact control of the false discovery rate under orthogonal design assumption. Monte-Carlo simulations and a real data experiment are provided to illustrate our results in this case. Of independent interest, we introduce an equivalent formulation of the LARS algorithm based on a recursive function. [ABSTRACT FROM AUTHOR]

In a previous paper (Huang et al., Advances in Computational Mathematics 47(5):1–34, 2021), we presented the fundamentals of a new hierarchical algorithm for computing the expectation of a N-dimensional function H (X) where X satisfies the truncated multi-variate normal (TMVN) distribution. The algorithm assumes that H (X) is low-rank and the covariance matrix Σ and precision matrix A = Σ - 1 have low-rank blocks with low-dimensional features. Analysis and numerical results were presented when A is tridiagonal or given by the exponential model. In this paper, we first demonstrate how the hierarchical algorithm structure allows the simultaneous calculations of all the order M and less moments E (H (X) = X 1 m 1 ⋯ X N m N | a i < X i < b i , i = 1 , … , N) , ∑ i m i ≤ M using asymptotically optimal O (N M) operations when M ≥ 2 and O (N log (N)) operations when M = 1 . These O (N M) moments are often required in the Expectation Maximization (EM) algorithms. We illustrate the algorithm ideas using the case when A is tridiagonal or the exponential model where the off-diagonal matrix block has rank K = 1 and number of effective variables P ≤ 2 for each function associated with a hierarchical tree node. The smaller K and P values allow the use of existing FFT and Non-uniform FFT (NuFFT) solvers to accelerate the computation of the compressed features in the system. To handle cases with higher K and P values, we introduce the sparse grid technique aimed at problems with K + P ≈ 5 ∼ 20 . We present numerical results for computing both the moments and higher K and P values to demonstrate the accuracy and efficiency of the algorithms. Finally, we summarize our results and discuss the limitations and generalizations, in particular, our algorithm capability is limited by the availability of mathematical tools in higher dimensions. When K + P is greater than 20, as far as we know, there are no practical tools available for problems with 20 truly independent variables. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the problem of the estimation of a parameter θ , in Banach spaces, maximizing some criterion function which depends on an unknown nuisance parameter h, possibly infinite-dimensional. The classical estimation methods are mainly based on maximizing the corresponding empirical criterion by substituting the nuisance parameter by a nonparametric estimator. We show that the M-estimators converge weakly to maximizers of Gaussian processes under rather general conditions. The conventional bootstrap method fails in general to consistently estimate the limit law. We show that the m out of n bootstrap, in this extended setting, is weakly consistent under conditions similar to those required for weak convergence of the M-estimators. The aim of this paper is therefore to extend the existing theory on the bootstrap of the M-estimators. Examples of applications from the literature are given to illustrate the generality and the usefulness of our results. Finally, we investigate the performance of the methodology for small samples through a short simulation study. [ABSTRACT FROM AUTHOR]