A limit theorem on moment convergence of centered spectral statistics of random matrices

ABSTRACT

The eigenvalues of a random matrix are a sequence of specific dependent random variables, the limiting properties of which are one of interesting topics in probability theory. The aim of the article is to extend some probability limiting properties of i.i.d. random variables in the context of the complete moment convergence to the centered spectral statistics of random matrices. Some precise asymptotic results related to the complete convergence of p-order conditional moment of Wigner matrices and sample covariance matrices are obtained. The proofs mainly depend on the central limit theorem and large deviation inequalities of spectral statistics.     

Estimating the Intercept in an Orthogonally Blocked Experiment when the Block Effects are Random

Abstract

For an orthogonally blocked experiment, Khuri [Khuri, A. I. (1992). Response surface models with random block effects. Technometrics 34:26–37] has shown that the ordinary least squares estimator, the generalized least squares estimator and the intra-block estimator of the factor effects in a response surface model with random block effects coincide. The ordinary least squares estimator ignores the blocks, whereas the generalized least squares and the intra-block estimators treat the block effects as random and fixed, respectively. As shown in this paper, the equivalence does not hold for the estimation of the intercept when the block sizes are heterogeneous. Practical examples are given to illustrate the theoretical results.     

Some optimal block designs with nested rows and columns for research on alternative methods of limiting slug damage

Some criteria of optimality of a block design with nested rows and columns are considered. The criteria are based on the eigenvalues of the information matrix C or on the eigenvalues of the matrix C with respect to a diagonal matrix R of treatment replications. New constructions of some optimal block designs with nested rows and columns are presented for application to special plant protection experiments.     

Some applications,properties and conjectures for higher order cumulants of a markovian stepping-stone model

This paper investigates certain tridiagonal coefficient matrices from linear differential equations for the joint cumulants of a Markovian stepping-stone model. The model is used to describe the dynamics of insect population growth, and is illustrated with an application to the spread of Africanized honey bees. The coefficient matrices are shown to have certain structure. A conjecture concerning their eigenvalues is proven for a special case of interest, and is also investigated numerically and by symbolic mathematical analysis.     

Generating Correlation Matrices With Specified Eigenvalues Using the Method of Alternating Projections

ABSTRACT

This article describes a new algorithm for generating correlation matrices with specified eigenvalues. The algorithm uses the method of alternating projections (MAP) that was first described by Neumann. The MAP algorithm for generating correlation matrices is both easy to understand and to program in higher-level computer languages, making this method accessible to applied researchers with no formal training in advanced mathematics. Simulations indicate that the new algorithm has excellent convergence properties. Correlation matrices with specified eigenvalues can be profitably used in Monte Carlo research in statistics, psychometrics, computer science, and related disciplines. To encourage such use, R code (R Core Team) for implementing the algorithm is provided in the supplementary material.     

CONVERGENCE AND PREDICTION OF PRINCIPAL COMPONENT SCORES IN HIGH-DIMENSIONAL SETTINGS

A number of settings arise in which it is of interest to predict Principal Component (PC) scores for new observations using data from an initial sample. In this paper, we demonstrate that naive approaches to PC score prediction can be substantially biased towards 0 in the analysis of large matrices. This phenomenon is largely related to known inconsistency results for sample eigenvalues and eigenvectors as both dimensions of the matrix increase. For the spiked eigenvalue model for random matrices, we expand the generality of these results, and propose bias-adjusted PC score prediction. In addition, we compute the asymptotic correlation coefficient between PC scores from sample and population eigenvectors. Simulation and real data examples from the genetics literature show the improved bias and numerical properties of our estimators.     

Approximated Transient Queue Length and Waiting Time Distributions via Steady State Analysis

Abstract

We propose a method to approximate the transient performance measures of a discrete time queueing system via a steady state analysis. The main idea is to approximate the system state at time slot t or on the n-th arrival–-depending on whether we are studying the transient queue length or waiting time distribution–-by the system state after a negative binomially distributed number of slots or arrivals. By increasing the number of phases k of the negative binomial distribution, an accurate approximation of the transient distribution of interest can be obtained.

In order to efficiently obtain the system state after a negative binomially distributed number of slots or arrivals, we introduce so-called reset Markov chains, by inserting reset events into the evolution of the queueing system under consideration. When computing the steady state vector of such a reset Markov chain, we exploit the block triangular block Toeplitz structure of the transition matrices involved and we directly obtain the approximation from its steady state vector. The concept of the reset Markov chains can be applied to a broad class of queueing systems and is demonstrated in full detail on a discrete-time queue with Markovian arrivals and phase-type services (i.e., the D-MAP/PH/1 queue). We focus on the queue length distribution at time t and the waiting time distribution of the n-th customer. Other distributions, e.g., the amount of work left behind by the n-th customer, that can be acquired in a similar way, are briefly touched upon.

Using various numerical examples, it is shown that the method provides good to excellent approximations at low computational costs–-as opposed to a recursive algorithm or a numerical inversion of the Laplace transform or generating function involved–-offering new perspectives to the transient analysis of practical queueing systems.     

A Comparison of Two Random Number Generators for the Birnbaum–Saunders Distribution

Abstract

The Birnbaum–Saunders distribution was developed to describe fatigue failure lifetimes, however, the distribution has been shown to be applicable for a variety of situations that frequently occur in the engineering sciences. In general, the distribution can be used for situations that involve stochastic wear–out failure. The distribution does not have an exponential family structure, and it is often necessary to use simulation methods to study the properties of statistical inference procedures for this distribution. Two random number generators for the Birnbaum–Saunders distribution have appeared in the literature. The purpose of this article is to present and compare these two random number generators to determine which is more efficient. It is shown that one of these generators is a special case of the other and is simpler and more efficient to use.     

Discrimination of AR,MA and ARMA time series models

The problem of discrimination between two stationary ARMA time series models is considered, and in particular AR(p), MA(p), ARMA(1,1) models. The discriminant based on the likelihood ration leads to a quadratic form that is generally too complicated to evaluated explicitly. The discriminant can be expressed approximately as a linear combination of independent chi–squared random varianles each with one degree of freedom, the coefficients, of which are eigenvalues of cumbersome matrices. An analytical solution which gives the coefficients approximately is suggested.     

Efficient three-level screening designs using weighing matrices

Screening is the first stage of many industrial experiments and is used to determine efficiently and effectively a small number of potential factors among a large number of factors which may affect a particular response. In a recent paper, Jones and Nachtsheim [A class of three-level designs for definitive screening in the presence of second-order effects. J. Qual. Technol. 2011;43:1–15] have given a class of three-level designs for screening in the presence of second-order effects using a variant of the coordinate exchange algorithm as it was given by Meyer and Nachtsheim [The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 1995;37:60–69]. Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8] have used conference matrices to construct definitive screening designs with good properties. In this paper, we propose a method for the construction of efficient three-level screening designs based on weighing matrices and their complete foldover. This method can be considered as a generalization of the method proposed by Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8]. Many new orthogonal three-level screening designs are constructed and their properties are explored. These designs are highly D-efficient and provide uncorrelated estimates of main effects that are unbiased by any second-order effect. Our approach is relatively straightforward and no computer search is needed since our designs are constructed using known weighing matrices.     

Spatial circulants, with applications

The cumulants of quadratic forms associated to the so-called spatial design matrices are often needed for inference in the context of isotropic processes on uniform grids. Because the eigenvalues of the matrices involved are generally unknown, the computation of the cumulants can be very demanding if the grids are large. This paper first replaces the spatial design matrices with circular counterparts having known eigenvalues. It then studies some of the properties of the approximating matrices, and analyzes their performance in a number of applications to well-known inferential procedures.     

Optimal sample size for record data and associated cost analysis for exponential distribution

Estimation of the mean of an exponential distribution based on record data has been treated by Samaniego and Whitaker [F.J. Samaniego, and L.R. Whitaker, On estimating popular characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Quart. 33 (1986), pp. 531–543] and Doostparast [M. Doostparast, A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. When a random sample Y ₁, …, Y _n is examined sequentially and successive minimum values are recorded, Samaniego and Whitaker [F.J. Samaniego, and L.R. Whitaker, On estimating popular characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Quart. 33 (1986), pp. 531–543] obtained a maximum likelihood estimator of the mean of the population and showed its convergence in probability. We establish here its convergence in mean square error, which is stronger than the convergence in probability. Next, we discuss the optimal sample size for estimating the mean based on a criterion involving a cost function as well as the Fisher information based on records arising from a random sample. Finally, a comparison between complete data and record is carried out and some special cases are discussed in detail.     

Complete convergence for weighted sums of END random variables and its application to nonparametric regression models

In this article, the complete convergence for weighted sums of extended negatively dependent (END, for short) random variables is investigated. Some sufficient conditions for the complete convergence are provided. In addition, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of END random variables is obtained. The results obtained in the article generalise and improve the corresponding one of Wang et al. [(2014b), ‘On Complete Convergence for an Extended Negatively Dependent Sequence’, Communications in Statistics-Theory and Methods, 43, 2923–2937]. As an application, the complete consistency for the estimator of nonparametric regression model is established.     

Generalized Sobol sensitivity indices for dependent variables: numerical methods

The hierarchically orthogonal functional decomposition of any measurable function η of a random vector X=(X₁,?…?, X_p) consists in decomposing η(X) into a sum of increasing dimension functions depending only on a subvector of X. Even when X₁,?…?, X_p are assumed to be dependent, this decomposition is unique if the components are hierarchically orthogonal. That is, two of the components are orthogonal whenever all the variables involved in one of the summands are a subset of the variables involved in the other. Setting Y=η(X), this decomposition leads to the definition of generalized sensitivity indices able to quantify the uncertainty of Y due to each dependent input in X [Chastaing G, Gamboa F, Prieur C. Generalized Hoeffding–Sobol decomposition for dependent variables – application to sensitivity analysis. Electron J Statist. 2012;6:2420–2448]. In this paper, a numerical method is developed to identify the component functions of the decomposition using the hierarchical orthogonality property. Furthermore, the asymptotic properties of the components estimation is studied, as well as the numerical estimation of the generalized sensitivity indices of a toy model. Lastly, the method is applied to a model arising from a real-world problem.     

Some reliability characteristics and stochastic ordering of series and parallel systems of bivariate generalized exponential distribution

In this paper, the reliability properties of two-component parallel and series systems are considered for bivariate generalized exponential (BVGE) distribution introduced by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivar Anal. 2009;100:581–593]. For this model, the moments and mean residual life functions of these systems and the regression mean residual life function are derived. Stochastic comparisons of series and parallel systems of BVGE distribution are investigated. Moreover, various ordering results for the comparisons of series and parallel systems arising from BVGE random vectors are obtained with respect to the usual stochastic, reversed hazard rate and likelihood ratio orderings.     

Isotropic Random Fields with the Von Kármán–Whittle Type Correlation Structure

This article proposes a subclass of stationary Gaussian or elliptically contoured random fields whose covariance functions are isotropic and are the linear combination of the von Kármán–Whittle structure. A particular example of this type of random fields is the so-called spartan random field recently appeared in the applied literature.     

New results on the convergence of random matrices

This paper extends the previous convergence results in Cerqueti and Costantini (2008) to a more general case using larger normed set of functions. In this regard, the weight-based convergence of the random matrices and their generalized eigenvalues is obtained under less restrictive requirements for the weights.     

Is categorization of random data necessary for parallel analysis on Likert-type data?

Random eigenvalues are the key elements in parallel analysis. When analyzing Likert-type data, is it necessary to convert the continuous random data to discrete type before estimating eigenvalues? The study compared the random eigenvalues obtained from continuous and categorized random data from two popular computer programs to be used as the basis for comparison in conducting parallel analysis on Likert-type data. Results indicated that categorized random data gave eigenvalues and number of factors similar to those obtained from continuous random data. It is suggested that when conducting parallel analysis on Likert-type data by the two programs, the conversion is unnecessary.     

On the moment generating functions for integer valued random variables

Abstract

In this note, we give explicit expressions of moment generating functions for integer valued random variables in both univariate and multivariate cases, which extend the results obtained by Nadarajah and Mitov [Communications in Statistics–Theory and Methods, 32, 2003, 47–60] and more recently by Chakraborti, Jardim and Epprecht [The American Statistician, 2017], Kwong and Nadarajah [Communications in Statistics–Theory and Methods, 2017]. Some examples are also discussed.     

Continuous Time Wishart Process for Stochastic Risk

Risks are usually represented and measured by volatility–covolatility matrices. Wishart processes are models for a dynamic analysis of multivariate risk and describe the evolution of stochastic volatility–covolatility matrices, constrained to be symmetric positive definite. The autoregressive Wishart process (WAR) is the multivariate extension of the Cox, Ingersoll, Ross (CIR) process introduced for scalar stochastic volatility. As a CIR process it allows for closed-form solutions for a number of financial problems, such as term structure of T-bonds and corporate bonds, derivative pricing in a multivariate stochastic volatility model, and the structural model for credit risk. Moreover, the Wishart dynamics are very flexible and are serious competitors for less structural multivariate ARCH models.