A new moment matching algorithm for sampling from partially specified symmetric distributions

A new algorithm is proposed for generating scenarios from a partially specified symmetric multivariate distribution. The algorithm generates samples which match the first two moments exactly, and match the marginal fourth moments approximately, using a semidefinite programming procedure. The performance of the algorithm is illustrated by a numerical example.     

多元$t$分布数据的局部影响分析

对于多元$t$分布数据, 直接应用其概率密度进行影响分析是困难的\bd 本文通过引入服从Gamma分布的权重, 将其表示为特定多元正态分布的混合\bd 在此基础上, 进而将权重视为缺失数据, 引入EM算法; 从而利用基于完全数据似然函数的条件期望进行局部影响分析\bd 本文进一步系统研究了加权扰动模型下的局部影响分析, 得到了相应的诊断统计量; 并通过两个实例说明了这种方法的有效性.     

On numerical calculation of probabilities according to Dirichlet distribution

The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation of the probability values according to the underlying multivariate probability distribution. In addition, when we are using a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives may also be necessary.     

An accept-reject algorithm for the positive multivariate normal distribution

The need to simulate from a positive multivariate normal distribution arises in several settings, specifically in Bayesian analysis. A variety of algorithms can be used to sample from this distribution, but most of these algorithms involve Gibbs sampling. Since the sample is generated from a Markov chain, the user has to account for the fact that sequential draws in the sample depend on one another and that the sample generated only follows a positive multivariate normal distribution asymptotically. The user would not have to account for such issues if the sample generated was i.i.d. In this paper, an accept-reject algorithm is introduced in which variates from a positive multivariate normal distribution are proposed from a multivariate skew-normal distribution. This new algorithm generates an i.i.d. sample and is shown, under certain conditions, to be very efficient.     

高维正态概率积分的降维算法与L_1逼近

高维正态概率积分计算一直是统计学家关注的课题．早期工作已由Gupta（1963）[1]评价，并给出大量的参考文献．近期工作则可参考Tong（1990）[2]的专著．虽然有关的文献很多，但是除了二、三维问题已有较好的算法外（例如见Zhana－Yana，1993[3]），更高维问题尚无公认的有效算法．在维数m＞3的高维情形，多数文章常假设积分域或相关阵有特殊形式，否则只有使用MonteCarlo方法［4］或拟MonteCarlo方法（亦称数论网格方法，例如见Fang－Wang，1994[5]）．但即使是被认为较好的拟MonteCarlo方法，其收敛阶仅为O（n－2／m），因此对于真…     

A multivariate stochastic model with non-stationary trend component

The purposes of this paper are to introduce a multivariate non-stationary stochastic time series model without individual detrending and to extract the multiple relationships between variables. To infer the statistical relation between variables, we attempt to estimate the co-movement of multivariate non-stationary time series components. The model is expressed in state-space form, and time series components are estimated by the maximum likelihood method using numerical optimization algorithm. The Kalman filter algorithm is used to compute the likelihood of the model. The AIC procedure gives a criterion for selecting the best model fit for the data. The multiple relationship becomes clear by analysing estimated AR coefficients. Real economic data are used for a numerical example.     

Evaluating Nearly Singular Multinormal Expectations with Application to Wave Distributions

The numerical computation of expectations for (nearly) singular multivariate normal distribution is a difficult problem, which frequently occurs in widely varying statistical contexts. In this article we discuss several strategies to improve the algorithm proposed by Genz and Kwong (2000) when either a moderate accuracy is requested, the correlation structure is strong, and, most importantly, the dimension of the integral is large. Test results for typical problems show an average speedup of 10 using the modified algorithm, but even more is gained as the dimension of the problem increases. We apply the modified algorithm to compute long-run distributions of Gaussian wave characteristics, a difficult problem where previous algorithms fail to compute accurate values in reasonable time. AMS 2000 Subject Classification 65C60, 65D15, 68W25     

Multivariate Gaussian Simulation Outside Arbitrary Ellipsoids

Methods for simulation from multivariate Gaussian distributions restricted to be from outside an arbitrary ellipsoidal region are often needed in applications. A standard rejection algorithm that draws a sample from a multivariate Gaussian distribution and accepts it if it is outside the ellipsoid is often employed; however, this is computationally inefficient if the probability of that ellipsoid under the multivariate normal distribution is substantial. We provide a two-stage rejection sampling scheme for drawing samples from such a truncated distribution. Experiments show that the added complexity of the two-stage approach results in the standard algorithm being more efficient for small ellipsoids (i.e., with small rejection probability). However, as the size of the ellipsoid increases, the efficiency of the two-stage approach relative to the standard algorithm increases indefinitely. The relative efficiency also increases as the number of dimensions increases, as the centers of the ellipsoid and the multivariate Gaussian distribution come closer, and as the shape of the ellipsoid becomes more spherical. We provide results of simulation experiments conducted to quantify the relative efficiency over a range of parameter settings.     

用广义交互确认方法选择良好参数进行多元多项式散乱数据自然样条光顺

1．简介多元样条函数在多元逼近中发挥很大作用，已有数量相当多的综合报告和研究论文正式发表，就在1996年6月在法国召开的第三届国际曲线与曲面会议上便有不少多元样条方面的报告，不过总的感觉是仍然缺乏对噪声数据特别是散乱数据的有效光顺方法．李岳生、崔锦泰、关履泰、胡日章等讨论广义调配样条与张量积函数，并用希氏空间样条方法处理多元散乱数据样条插值与光顺，提出多元多项式自然样条，推广了相应一元的结果．我们知道，在样条光顺中有一个如何选择参数的问题，用广义交互确认方法（generalizedcross－validation，以下简称GC…     

Polynomial bases on the numerical solution of the multivariate discrete moment problem

The multivariate discrete moment problem (MDMP) has been introduced by Prékopa. The objective of the MDMP is to find the minimum and/or maximum of the expected value of a function of a random vector with a discrete finite support where the probability distribution is unknown, but some of the moments are given. The MDMP can be formulated as a linear programming problem, however, the coefficient matrix is very ill-conditioned. Hence, the LP problem usually cannot be solved in a regular way. In the univariate case Prékopa developed a numerically stable dual method for the solution. It is based on the knowledge of all dual feasible bases under some conditions on the objective function. In the multidimensional case the recent results are also about the dual feasible basis structures. Unfortunately, at higher dimensions, the whole structure has not been found under any circumstances. This means that a dual method, similar to Prékopa??s, cannot be developed. Only bounds on the objective function value are given, which can be far from the optimum. This paper introduces a different approach to treat the numerical difficulties. The method is based on multivariate polynomial bases. Our algorithm, in most cases, yields the optimum of the MDMP without any assumption on the objective function. The efficiency of the method is tested on several numerical examples.     

Estimates of MM type for the multivariate linear model

We propose a class of robust estimates for multivariate linear models. Based on the approach of MM-estimation (Yohai 1987, [24]), we estimate the regression coefficients and the covariance matrix of the errors simultaneously. These estimates have both a high breakdown point and high asymptotic efficiency under Gaussian errors. We prove consistency and asymptotic normality assuming errors with an elliptical distribution. We describe an iterative algorithm for the numerical calculation of these estimates. The advantages of the proposed estimates over their competitors are demonstrated through both simulated and real data.     

Wavelet Shrinkage on Paths for Denoising of Scattered Data

We propose a new algorithm for denoising of multivariate function values given at scattered points in ${\mathbb{R}^{d}}$ . The method is based on the one-dimensional wavelet transform that is applied along suitably chosen path vectors at each transform level. The idea can be seen as a generalization of the relaxed easy path wavelet transform by Plonka (Multiscale Model Simul 7:1474–1496, 2009) to the case of multivariate scattered data. The choice of the path vectors is crucial for the success of the algorithm. We propose two adaptive path constructions that take the distribution of the scattered points as well as the corresponding function values into account. Further, we present some theoretical results on the wavelet transform along path vectors in order to indicate that the wavelet shrinkage along path vectors can really remove noise. The numerical results show the efficiency of the proposed denoising method.     

A one-step worst-case optimal algorithm for bi-objective univariate optimization

A bi-objective optimization problem with univariate non-convex objectives is considered. The objective functions are assumed Lipschitz continuous. An algorithm is developed implementing the concept of one-step worst case optimality. The possibility of extension of the proposed algorithm to the multivariate case is discussed. Some illustrative numerical examples are included where the proposed algorithm is compared with an average case optimal algorithm.     

Numerical stability of nonequispaced fast Fourier transforms

This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case study, we show that both approximation error and roundoff error have a strong influence on the numerical stability of NFFT. Numerical tests confirm the theoretical estimates of numerical stability.     

Modeling fat tails in stock returns: a multivariate stable-GARCH approach

In this paper a new multivariate volatility model is proposed. It combines the appealing properties of the stable Paretian distribution to model the heavy tails with the GARCH model to capture the volatility clustering. Returns on assets are assumed to follow a sub-Gaussian distribution, which is a particular multivariate stable distribution. In this way the characteristic function of the fitted returns has a tractable expression and the density function can be recovered by numerical methods. A multivariate GARCH structure is then adopted to model the covariance matrix of the Gaussian vectors underlying the sub-Gaussian system. The model is applied to a bivariate series of daily U.S. stock returns. Value-at-risk for long and short positions is computed and compared with the one obtained using the multivariate normal and the multivariate Student’s t distribution. Finally, exploiting the recent developments in the vast dimensional time-varying covariances modeling, possible feasible extensions of our model to higher dimensions are suggested and an illustrative example using the Dow Jones index components is presented.     

Efficient ML Estimation of the Multivariate Normal Distribution from Incomplete Data

It is well known that the maximum likelihood estimates (MLEs) of a multivariate normal distribution from incomplete data with a monotone pattern have closed-form expressions and that the MLEs from incomplete data with a general missing-data pattern can be obtained using the Expectation-Maximization (EM) algorithm. This article gives closed-form expressions, analogous to the extension of the Bartlett decomposition, for both the MLEs of the parameters and the associated Fisher information matrix from incomplete data with a monotone missing-data pattern. For MLEs of the parameters from incomplete data with a general missing-data pattern, we implement EM and Expectation-Constrained-Maximization-Either (ECME), by augmenting the observed data into a complete monotone sample. We also provide a numerical example, which shows that the monotone EM (MEM) and monotone ECME (MECME) algorithms converge much faster than the EM algorithm.     

Estimation of parameters in latent class models using fuzzy clustering algorithms

A mixture approach to clustering is an important technique in cluster analysis. A mixture of multivariate multinomial distributions is usually used to analyze categorical data with latent class model. The parameter estimation is an important step for a mixture distribution. Described here are four approaches to estimating the parameters of a mixture of multivariate multinomial distributions. The first approach is an extended maximum likelihood (ML) method. The second approach is based on the well-known expectation maximization (EM) algorithm. The third approach is the classification maximum likelihood (CML) algorithm. In this paper, we propose a new approach using the so-called fuzzy class model and then create the fuzzy classification maximum likelihood (FCML) approach for categorical data. The accuracy, robustness and effectiveness of these four types of algorithms for estimating the parameters of multivariate binomial mixtures are compared using real empirical data and samples drawn from the multivariate binomial mixtures of two classes. The results show that the proposed FCML algorithm presents better accuracy, robustness and effectiveness. Overall, the FCML algorithm has the superiority over the ML, EM and CML algorithms. Thus, we recommend FCML as another good tool for estimating the parameters of mixture multivariate multinomial models.     

条件正定矩阵及其在多元插值计算中的应用

这里A一般不是正定的,按后面定义只是“条件正定”的,特别,A的对角线元素往往是零.这给方程组的求解带来了困难.我们的目的是如何利用“条件正定”的特点建立有效的算法,减少计算量和机器时间.为此,先讨论“条件正定”矩阵及与之相关的“条件正定”函数的某些性质,以便于判定A的条件正定性.然后利用这个性质构造有效算法.最后的平板样条数值结果表明,应用“条件正定”作工具建立的算法,比通常算法求解(1)的效率提高四倍以上.     

On the convergence of row-modification algorithm for matrix projections

This paper proposes an algorithm for matrix minimum-distance projection, with respect to a metric induced from an inner product that is the sum of inner products of column vectors, onto the collection of all matrices with their rows restricted in closed convex sets. This algorithm produces a sequence of matrices by modifying a matrix row by row, over and over again. It is shown that the sequence is convergent, and it converges to the desired projection. The implementation of the algorithm for multivariate isotonic regressions and numerical examples are also presented in the paper.     

Comparison of Methods for the Computation of Multivariate t Probabilities

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