Moments of the generalized hyperbolic distribution

In this paper we demonstrate a recursive method for obtaining the moments of the generalized hyperbolic distribution. The method is readily programmable for numerical evaluation of moments. For low order moments we also give an alternative derivation of the moments of the generalized hyperbolic distribution. The expressions given for these moments may be used to obtain moments for special cases such as the hyperbolic and normal inverse Gaussian distributions. Moments for limiting cases such as the skew hyperbolic t and variance gamma distributions can be found using the same approach.     

Generalized canonical correlation analysis with missing values

Generalized canonical correlation analysis is a versatile technique that allows the joint analysis of several sets of data matrices. The generalized canonical correlation analysis solution can be obtained through an eigenequation and distributional assumptions are not required. When dealing with multiple set data, the situation frequently occurs that some values are missing. In this paper, two new methods for dealing with missing values in generalized canonical correlation analysis are introduced. The first approach, which does not require iterations, is a generalization of the Test Equating method available for principal component analysis. In the second approach, missing values are imputed in such a way that the generalized canonical correlation analysis objective function does not increase in subsequent steps. Convergence is achieved when the value of the objective function remains constant. By means of a simulation study, we assess the performance of the new methods. We compare the results with those of two available methods; the missing-data passive method, introduced in Gifi’s homogeneity analysis framework, and the GENCOM algorithm developed by Green and Carroll. An application using world bank data is used to illustrate the proposed methods.     

A generalized separation for the variance contributions of input variables and their distribution parameters

For the structural system with both the uncertainties of input variables and their distribution parameters, this work investigates the generalized separation approach by transforming the original variable into the auxiliary variable with arbitrary distribution. Based on the variance based sensitivity analysis, the generalized sensitivity measures can be given, which are used to identify the influences of the auxiliary variables and distribution parameters simultaneously. For the different auxiliary variables, the variance contributions are proved to be identical, which illustrates the correctness of the generalized separation approach. Then the relationship of the variance contributions of original variables with those of the auxiliary variables and distribution parameters is investigated. Several examples are employed to demonstrate the rationality of the generalized separation approach.     

Classification accuracy and correlation: LDA in failure prediction

The relationship between canonical correlation and classification accuracy in linear discriminant analysis is explored mathematically. The discriminant score is assumed to conform to a uniform distribution on the interval (0, 1]. This distribution is used as a reference distribution to extract a minimum correlation for certain classification accuracy. Four different cases are analyzed. First, a case for equal group size is considered for an overall accuracy of 100%. Second, the results are generalized for unequal group size. Third, existence of discordant observations is allowed. Fourth, the effect of concentration is analyzed for the first case. The results are demonstrated by numerical examples. In addition, a sample of 2092 default and 63,072 non-default Finnish firms are used to empirically illustrate the results in the context of failure prediction. The results show that group size of default firms, number of discordant observations, and bipolar concentration of observations strongly affect both canonical correlation and classification accuracy.     

The lifetime of a financial bubble

We combine both a mathematical analysis of financial bubbles and a statistical procedure for determining when a given stock is in a bubble, with an analysis of a large data set, in order to compute the empirical distribution of the lifetime of financial bubbles. We find that it follows a generalized gamma distribution, and we provide estimates for its parameters. We also perform goodness of fit tests, and we provide a derivation, within the context of bubbles, that explains why the generalized gamma distribution might be the natural one to expect for the lifetimes of financial bubbles.     

Inference in canonical correlation analysis

The asymptotic behavior, for large sample size, is given for the distribution of the canonical correlation coefficients. The result is used to examine the Bartlett-Lawley test that the residual population canonical correlation coefficients are zero. A marginal likelihood function for the population coefficients is obtained and the maximum marginal likelihood estimates are shown to provide a bias correction.     

Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions

Dynamics of a system of hard spheres with inelastic collisions is investigated. This system is a model for granular flow. The map induced by a shift along the trajectory does not preserve the volume of the phase space, and the corresponding Jacobian is different from one. A special distribution function is defined as the product of the usual distribution function and the squared Jacobian. For this distribution function, the Liouville equation with boundary condition is derived. A sequence of correlation functions is defined for canonical and grand canonical ensemble. The generalized BBGKY hierarchy and boundary condition are deduced for correlation functions. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 818–839, June, 2005.     

A test for additional information in canonical correlation analysis

Summary In canonical correlation analysis a hypothesis concerning the relevance of a subset of variables from each of the two given variable sets is formulated. The likelihood ratio statistic for the hypothesis and an asymptotic expansion for its null distribution are obtained. In discriminant analysis various alternative forms of a hypothesis concerning the relevance of a specified variable subset are also discussed.     

Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

It is shown that any generalized Jordan (triple-)derivation on a 2–torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2–torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.     

A condition for null robustness

Sufficient conditions are given that certain statistics have a common distribution under a wide class of underlying distributions. Invariance methods are the primary technical tool in establishing the theoretical results. These results are applied to MANOVA problems, problems involving canonical correlations, and certain statistics associated with the complex normal distribution.     

On Generalized Linearity of Quadratic Fractional Functions

Quadratic fractional functions are proved to be quasilinear if and only if they are pseudo-linear. For these classes of functions, some characterizations are provided by means of the inertia of the quadratic form and the behavior of the gradient of the function itself. The study is then developed showing that generalized linear quadratic fractional functions share a particular structure. Therefore it is possible to suggest a sort of “canonical form” for those functions. A wider class of functions given by the sum of a quadratic fractional function and a linear one is also studied. In this case generalized linearity is characterized by means of simple conditions. Finally, it is deepened on the role played by generalized linear quadratic fractional functions in optimization problems.     

The spectral decomposition of a covariance matrix for the balanced mixed analysis of variance model

We derive the spectral decomposition of a covariance matrix for the balanced mixed analysis of variance model. The derivation is based on determining the distinct eigenvalues of a covariance matrix and then obtaining a principal idempotent matrix for each distinct eigenvalue. Examples are given to illustrate the results.     

CANONICAL CORRELATION EQUATIONS AND THEIR EXTREMAL PROPERTIES

The canonical variables and canonical correlation coefficients satisfy a matrix equation which is called the canonical correlation equation. There are some different forms of the canonical correlation e-quations given in the literature. In this paper, we discuss four different forms of the canonical correlation equations. The purpose of this paper is to give extremal properties of the solutions of the canonical correlation equations. The results show that canonical variables maximize the determinant of the dispersion matrix of the transformed variables.     

Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem

We propose a new algorithm for sparse estimation of eigenvectors in generalized eigenvalue problems (GEPs). The GEP arises in a number of modern data-analytic situations and statistical methods, including principal component analysis (PCA), multiclass linear discriminant analysis (LDA), canonical correlation analysis (CCA), sufficient dimension reduction (SDR), and invariant co-ordinate selection. We propose to modify the standard generalized orthogonal iteration with a sparsity-inducing penalty for the eigenvectors. To achieve this goal, we generalize the equation-solving step of orthogonal iteration to a penalized convex optimization problem. The resulting algorithm, called penalized orthogonal iteration, provides accurate estimation of the true eigenspace, when it is sparse. Also proposed is a computationally more efficient alternative, which works well for PCA and LDA problems. Numerical studies reveal that the proposed algorithms are competitive, and that our tuning procedure works well. We demonstrate applications of the proposed algorithm to obtain sparse estimates for PCA, multiclass LDA, CCA, and SDR. Supplementary materials for this article are available online.     

Use of the multinomial jack-knife and bootstrap in generalized non-linear canonical correlation analysis

In this paper we discuss the estimation of mean and standard errors of the eigenvalues and category quantifications in generalized non-linear canonical correlation analysis (OVERALS). Starting points are the delta method equations, but the jack-knife and bootstrap are used to provide finite difference approximations to the derivatives.     

Variance minimization for continuous-time Markov decision processes: two approaches

This paper studies the limit average variance criterion for continuous-time Markov decision processes in Polish spaces. Based on two approaches, this paper proves not only the existence of solutions to the variance minimization optimality equation and the existence of a variance minimal policy that is canonical, but also the existence of solutions to the two variance minimization optimality inequalities and the existence of a variance minimal policy which may not be canonical. An example is given to illustrate all of our conditions.     

Optimal reparametrization and large sample likelihood inference for the location-scale skew-normal model

Motivated by results in Rotnitzky et al. (2000), a family of parametrizations of the location-scale skew-normal model is introduced, and it is shown that, under each member of this class, the hypothesis H ₀: ?? = 0 is invariant, where ?? is the asymmetry parameter. Using the trace of the inverse variance matrix associated to a generalized gradient as a selection index, a subclass of optimal parametrizations is identified, and it is proved that a slight variant of Azzalini??s centred parametrization is optimal. Next, via an arbitrary optimal parametrization, a simple derivation of the limit behavior of maximum likelihood estimators is given under H ₀, and the asymptotic distribution of the corresponding likelihood ratio statistic for this composite hypothesis is determined.     

The Equality Constrained Indefinite Least Squares Problem: Theory and Algorithms

We present theory and algorithms for the equality constrained indefinite least squares problem, which requires minimization of an indefinite quadratic form subject to a linear equality constraint. A generalized hyperbolic QR factorization is introduced and used in the derivation of perturbation bounds and to construct a numerical method. An alternative method is obtained by employing a generalized QR factorization in combination with a Cholesky factorization. Rounding error analysis is given to show that both methods have satisfactory numerical stability properties and numerical experiments are given for illustration. This work builds on recent work on the unconstrained indefinite least squares problem by Chandrasekaran, Gu, and Sayed and by the present authors.     

On a generalization of the Keldysh theorem

The Keldysh theorem for an elliptic equation with characteristic parabolic degeneration is generalized for the case of an elliptic equation of the second-order canonical form with order and type degeneration. The criteria under which the Dirichlet or Keldysh problems are correct are given in a one-sided neighborhood of the degeneration segment, enabling one to write the criteria in a single form. Moreover, some cases are pointed out in which it is even nessesary to give a criterion in the neighborhood because it is impossible to establish it on the segment of degeneracy of the equation.     

两组变量间相关关系的统计分析方法

研究典型相关分析的原理、典型成分的计算方法及计算步骤.把两组变量X与Y转化为具有最大相关性的若干对典型成分,直到两组变量的相关性被分解.通过典型相关系数及其显著性检验.选择典型成分分析两组变量的相关性.实例表明只有第一个典型相关系数能通过显著性检验,而其它两个典型相关系数显著为零,故应选取第一对典型成分F1和G1做分析...