Matrix geometric approach for random walks: Stability condition and equilibrium distribution

ABSTRACT

In this paper, we analyze a sub-class of two-dimensional homogeneous nearest neighbor (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions^[30 Neuts, M.F., Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach; The Johns Hopkins University Press: Baltimore, 1981. [Google Scholar]^] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions.^[13 Fayolle, G.; Iasnogorodski, R.; Malyshev, V., Random Walks in the Quarter Plane; Springer-Verlag: New York, 1999. [Google Scholar]^] Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix R appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix R.     

The Homogeneous Markov System (HMS) as an Elastic Medium. The Three-Dimensional Case

Every attainable structure of the so-called continuous-time Homogeneous Markov System (HMS) with fixed size and state space S = {1, 2,…, n} is considered as a particle of R ⁿ and, consequently, the motion of the structure corresponds to the motion of the particle. Under the assumption that “the motion of every particle-structure at every time point is due to its interaction with its surroundings,” R ⁿ becomes a continuum (Tsaklidis, 1998 Tsaklidis , G. ( 1998 ). The continuous time homogeneous Markov system with fixed size as a Newtonian fluid? Appl. Stoch. Mod. Data Anal. 13 : 177 – 182 .[Crossref] , [Google Scholar]). Then the evolution of the set of the attainable structures corresponds to the motion of the continuum. For the case of a three-state HMS it is stated that the concept of the two-dimensional isotropic elasticity can further be used to interpret the three-state HMS's evolution.     

On asymptotics of multivariate integrals with applications to records

Let { X _n ,n≥1} be a sequence of iid. Gaussian random vectors in R ^d , d≥2, with nonsingular distribution function F. In this paper the asymptotics for the sequence of integrals I _F,n (G _n )?n∫ _{R ^d} G _n ^n?1( X ) dF( X ) is considered with G _n some distribution function on R ^d . In the case G _n =F the integral I _F,n (F)/n is the probability that a record occurs in X ₁,…, X _n at index n. [1] Gnedin, A.V. 1998. Records from a Multivariate Normal Sample. Statist. Probab. Lett., 39: 11–15. [Crossref], [Web of Science ®] , [Google Scholar] obtained lower and upper asymptotic bounds for this case, whereas [2] Ledford, W.A. and Twan, A.J. 1998. On the Tail Concomitant Behaviour for Extremes. Adv. Appl. Probab., 30: 197–215. [Crossref], [Web of Science ®] , [Google Scholar] showed the rate of convergence if d=2. In this paper we derive the exact rate of convergence of I _F,n (G _n ) for d≥2 under some restrictions on the distribution function G _n . Some related results for multivariate Gaussian tails are discussed also.     

Admissible Estimation for Linear Combination of Fixed and Random Effects in General Mixed Linear Models

The general mixed linear model can be denoted by y  =  X β +  Z u  +  e , where β is a vector of fixed effects, u is a vector of random effects, and e is a vector of random errors. In this article, the problem of admissibility of Q y and Q y  +  q for estimating linear functions, ? =  L ′β +  M ′ u , of the fixed and random effects is considered, and the necessary and sufficient conditions for Q y (resp. Q y  +  q ) to be admissible in the set of homogeneous (resp. potentially inhomogeneous) linear estimators with respect to the MSE and MSEM criteria are investigated. We provide a straightforward alternative proof to the method that was utilized by Wu (1988 Wu , Q. G. ( 1988 ). Several results on admissibility of a linear estimate of stochastic regression coefficients and parameters . Acta Mathemaica Applicatae Sinica 11 ( 1 ): 95 – 106 . (in Chinese)  [Google Scholar]), Baksalary and Markiewicz (1990 Baksalary , J. K. , Markiewicz , A. ( 1990 ). Admissible linear estimators of an arbitrary vector of parametric functions in the general Gauss–Markov model . J. Stat. Plann. Infer. 26 : 161 – 171 . [Google Scholar]), and Groß and Markiewicz (1999 Groß , J. , Markiewicz , A. ( 1999 ). On admissibility of linear estimators with respect to the mean square error matrix criterion under the general mixed linear model . Statistics 33 : 57 – 71 .[Taylor & Francis Online] , [Google Scholar]). In addition, we derive the corresponding results on the admissibility problem under the generalized MSE criterion.     

BARTLETT-TYPE CORRECTIONS FOR TWO-PARAMETER EXPONENTIAL FAMILY MODELS

ABSTRACT

Cordeiro and Ferrari[1] Cordeiro, G.M. and Ferrari, S.L.P. 1991. A Modified Score Test Statistic Having Chi-Squared Distribution to Order n^?1. Biometrika, 78: 573–582. [Web of Science ®] , [Google Scholar] obtained a Bartlett-type correction to the score statistic that is given by a polynomial of second degree in the statistic itself with coefficients that depend on cumulants of log–likelihood derivatives. Although the corrected statistic has good size properties, it is not always a monotone transformation of the original statistic. Recently, some monotone transformations of the score statistic have been proposed as alternatives to the polynomial transformation. In this paper we derive simple formulae for various modified score statistics for testing a scalar parameter in two-parameter exponential models which do not require knowledge of cumulants. The formulae are readily applicable to cover many important and commonly used distributions and involve only trivial operations on certain functions and their derivatives.     

Progressively Type-II censored conditionally N-ordered statistics from a unified elliptically contoured copula

Progressively Type-II censored conditionally N-ordered statistics (PCCOS-N) arising from iid random vectors X_i = (X¹_i, X²_i, …, X_i^p), i = 1, 2…, n, were investigated by Bairamov (2006 Bairamov, I. (2006). Progressive Type II censored order statistics for multivariate observations. J. Mult. Anal. 97:797–809.[Crossref], [Web of Science ®] , [Google Scholar]), with respect to the magnitudes of N(X_i), i = 1, 2, …, n, where N( · ) is a p-variate measurable function defined on the support set of X₁ satisfying certain regularity conditions and N(X_i) denotes the lifetime of the random vector X_i, i = 1, …, n. Under the PCCOS-N sampling scheme, n independent units are placed on a life-test and after the ith failure, R_i (i = 1, …, m) of the surviving units are removed at random from the remaining observations. In this article, we consider PCCOS-N arising from a vector with identical as well as non identical dependent components, jointly distributed according to a unified elliptically contoured copula (PCCOSDUECC-N). Results established here contain the previous results as particular cases. Illustrative examples and simulation studies show that PCCOSDUECC-N enables us to analyze the lifetime of several systems, including repairable systems and systems with standby components, more efficiently than PCCOS-N.     

EXPLICIT NONPARAMETRIC CONFIDENCE INTERVALS FOR THE VARIANCE WITH GUARANTEED COVERAGE

ABSTRACT

In this paper, we provide a method for constructing confidence intervals for the variance which exhibits guaranteed coverage probability for any sample size, uniformly over a wide class of probability distributions. In contrast, standard methods achieve guaranteed coverage only in the limit for a fixed distribution or for any sample size over a very restrictive (parametric) class of probability distributions. Of course, it is impossible to construct effective confidence intervals for the variance without some restriction, due to a result of Bahadur and Savage.[1] Bahadur, R. and Savage, L. 1956. The Nonexistence of Certain Statistical Procedures in Nonparametric Problems. Annals of Mathematical Statistics, 25: 1115–1122.  [Google Scholar] However, it is possible if the observations lie in a fixed compact set. We also consider the case of lower confidence bounds without any support restriction. Our method is based on the behavior of the variance over distributions that lie within a Kolmogorov–Smirnov confidence band for the underlying distribution. The method is a generalization of an idea of Anderson,[2] Anderson, T. 1967. Confidence Limits for the Expected Value of an Arbitrary Bounded Random Variable with a Continuous Distribution Function. Bull. ISI, 43: 249–251.  [Google Scholar] who considered only the case of the mean; it applies to very general parameters, and particularly the variance. While typically it is not clear how to compute these intervals explicitly, for the special case of the variance we provide an algorithm to do so. Asymptotically, the length of the intervals is of order n ^?/2 (in probability), so that, while providing guaranteed coverage, they are not overly conservative. A small simulation study examines the finite sample behavior of the proposed intervals.     

Some tests for the allometric extension model in regression

The allometric extension model is a multivariate regression model recently proposed by Tarpey and Ivey (2006 Tarpey, T., Ivey, C.T. (2006). Allometric extension for multivariate regression. J. Data Sci. 4:479–495. [Google Scholar]). This model holds when the matrix of covariances between the variables in the response vector y and the variables in the vector of regressors x has a particular structure. In this paper, we consider tests of hypotheses for this structure when (y′, x′)′ has a multivariate normal distribution. In particular, we investigate the likelihood ratio test and a Wald test.     

A study of the Gamma-Pareto (IV) distribution and its applications

ABSTRACT

Pareto distributions and their close relatives and generalizations provide very flexible families of heavy-tailed distributions that may be used to model income distributions as well as a wide variety of other social and economic distributions. On the other hand, gamma distribution has a wide application in various social and economic spheres such as survival analysis, to model aggregate insurance claims, and the amount of rainfall accumulated in a reservoir etc. Combining the above two heavy-tailed distributions, using the technique by Alzaatreh et al. (2012 Alzaatreh, A., Famoye, F., Lee, C. (2012). Gamma-Pareto distribution and its applications. J. Modern Appl. Stat. Methods. 11:78–94.[Crossref] , [Google Scholar]), we define a new distribution, namely Gamma-Pareto (IV) distribution, hereafter called as GPD^(IV) distribution. Various properties of the GPD^(IV) are investigated such as limiting behavior, moments, mode, and Shannon entropy. Also some characterizations of the GPD^(IV) distribution are mentioned in this paper. Maximum likelihood method is proposed for estimating the model parameters. For illustrative purposes, real data sets are considered as applications of the GPD^(IV) distribution.     

SIZE AND SHAPE AS LINEAR COMBINATIONS OF UNTRANSFORMED MEASUREMENTS

A numerical specification of ‘size’ and ‘shape’ is of interest for making interpretations in morphometrics. Starting from a, possibly large, set m ₁,…, m_r of size measurements, e.g. m ₁= height, m ₂= sitting height, etc., a preliminary analysis provides the set x ₁,…,x_p of size measurements to be used, e.g. x ₁= m ₁? m ₂= subischial leg length, x ₂= m ₂= sitting height, and x ₃= head circumference. In general these x_j are constructed as appropriately scaled linear combinations of the original measurements. A constant term should not be included because size measurements have to be 0 if all x_j are 0. Our theory requires a (compromise) vector μof means and a matrix Σof (co)variances. Size being specified as an optimalsize characteristic of the form c ′ x , the remaining morphological information is expressed by, at most, p? 1 components of shapeof the form d ′ x. Relations with Darroch-Mosimann [9] Darroch, J. N. and Mosimann, J. E. 1985. Canonical and Principal Components of Shape. Biometrika, 72: 241–252. [Crossref], [Web of Science ®] , [Google Scholar]are indicated. An application to human growth is made and other applications are suggested.

Don't read my book, think for yourself.

C. R. Rao, personal communications, 1981     

ROBUST TESTS FOR THE SIGNIFICANCE OF ORTHANT RESTRICTED MEAN VECTOR

Mudholkar and Srivastava [1] Mudholkar, G. S. and Srivastava, D. K. A class of robust stepwise alternatives to Hotelling's T²tests. Submitted to the Journal of Applied Statistics 1999 [Google Scholar]adapted Mudholkar and Subbaiah's [2] Mudholkar, G. S. and Subbaiah, P. 1980. Testing significance of a mean vector–a possible alternative to Hotelling's T². Ann. Inst. Statist. Math., 32(A): 43–52.  [Google Scholar]modified stepwise procedure, using the trimmed means in place of the means and appropriate studentization, to construct robust tests for the significance of a mean vector. They concluded that the robust alternatives provide excellent type I error control, and a substantial gain in power over Hotelling's T ²test in case of heavy tailed populations without significant loss of power when the population is normal. In this paper we adapt the modified stepwise approach to construct simple tests for the significance of the orthant constrained mean vector of a p-variate normal population with unknown covariance matrix, and also for constructing robust tests without assuming normality. The simple normal theory tests have exact type I error, whereas the robust tests provide a reasonably type I error control and substantial power advantage over Perlman's [3] Perlman, M. D. 1969. One-sided testing problems in multivariate analysis. Annals of Mathematical Statistics, 40: 549–567. [Crossref] , [Google Scholar]likelihood ratio test.     

ON THE ECONOMIC DESIGN OF MULTIVARIATE CONTROL CHARTS

ABSTRACT

It is an increasingly common practice to monitor several related quality characteristics of a product or process using a multivariate control chart procedure. Several types of multivariate control charts, including Hotelling's χ ² and T ² control charts, have been developed in attempts to improve monitoring by using the correlation structure that exists between quality characteristics. The purpose of this paper is to summarize the assumptions made regarding the out-of-control process shift in the economic design of multivariate control charts and to address their consequences. We study the average run length (ARL) properties of the χ ² control chart using a numerical example and show that this chart can perform ineffectively under the assumed out-of-control conditions when designed using the economic approach. Following Healy,[1] Healy, J.D. 1987. A Note on the Multivariate CUSUM Procedures. Technometrics, 29: 409–412. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] we offer an alternative procedure that has improved ARL properties and overall performance. These results can be important to researchers and practitioners who are interested in using the economic design of multivariate control procedures.     

Culinary Magazines

Abstract

Marvel Comics, along with rival DC Comics, is one of the two powerhouses of the comic book industry and has been for many decades. The company was founded in 1939 by pulp magazine publisher Martin Goodman. Goodman owned various publishing houses responsible for a variety of pulp titles, including Marvel Science Stories. In 1939, a colleague persuaded Goodman that comic books were the upcoming trend in periodical publishing, so Good-man launched a book called Marvel Comics. Issue no. 1 contained a story by Bill Everett about the Sub-Mariner, as well as the first appearance of the Human Torch. Both characters quickly became leading draws of the Golden Age of comics. Soon after, Goodman hired writer Joe Simon and artist Jack Kirby, who were to become giants in the industry. Simon and Kirby created Captain America, a character who remains a comics icon even today. In those early days, Goodman also hired another future legend: writer Stan Lee, who happened to be Good-man's nephew.

Goodman's comic company was known officially as Timely Comics, Inc., then later as Atlas Publishing. The firm did not change its name to Marvel until the early 1960s. Timely benefitted from the comics boom of the early 1940s, but interest in super-hero stories waned later in the decade. Atlas briefly revived its super-hero line-up in 1954, but the company spent most of the next decade publishing romance, western, horror, and humor comics.1 1. Les Daniels, Marvel: Five Decades of the World's Greatest Comics (NY: Abrams, 1991), pp. 17–23, 26–36, 40–61.¹     

On the Closed Form of Optimal Invariant Quadratic Predictors in Finite Populations

For a finite population and its linear model Y  =  X β +  e , the problem of deriving optimal invariant quadratic predictors including optimal invariant quadratic unbiased predictor (OIQUP) and optimal invariant quadratic (potentially) biased predictor (OIQBP) for the population quadratic quantities, Y ′ HY , is of interest and has been previously considered by Liu and Rong (2007 Liu , X. , Rong , J. ( 2007 ). Quadratic prediction problems in finite populations . Statist. Probab. Lett. 77 : 483 – 489 .[Crossref], [Web of Science ®] , [Google Scholar]). In this note, we mainly aim at motivating the problems of OIQUP and OIQBP by showing that the unique closed form of OIQUP and OIQBP is just the one given by Liu and Rong through permutation matrix techniques.     

Almost sure convergence of vertex degree densities in the vertex splitting model

We study the limiting degree distribution of the vertex splitting model introduced in Ref.^[3 David, F.; Dukes, M.; Jonsson, T.; Stefansson, S.Ö. Random tree growth by vertex splitting. J. Statist. Mech. Theory Exp. 2009, 04. doi:10.1088/1742-5468/2009/04/P04009. [Google Scholar]^]. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this, we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.     

Asymptotic Results for the MPL Estimators of the Contact Process of Order p

Let X be a discrete time contact process (CP) of order p on Z ² as defined by Durrett and Levin (1994 Durrett , R. , Levin , S. A. ( 1994 ). Stochastic spatial models: a user's guide to ecological applications . Phil. Trans. Roy. Soc. Lond. B 343 : 329 – 350 .[Crossref], [Web of Science ®] , [Google Scholar]), that is a CP where the function of birth takes p different values. We study the maximum marginal pseudo-likelihood (MPL) estimator of the model based on space-time evolution of X, that is, T + 1 successive observations of X on a finite subset S of sites. When T → ∞, this estimator is consistent and asymptotically normal for a non vanishing supercritical CP. We also propose a statistical test for the parameters of the model and verify the asymptotic results by some simulation studies.     

A Generalization of Hotelling's T 2

ABSTRACT

We derive an asymptotic version of Hotelling's T ² for the multivariate proper dispersion models of Jøtrgensen and Lauritzen (2000 Jørgensen , B. , Lauritzen , S. L. ( 2000 ). Multivariate dispersion models . J. Multivariate Anal. 74 : 267 – 281 . [CSA] [CROSSREF]  [Google Scholar]), the main tool being the saddlepoint approximation. Multivariate dispersion models are suitable for positive, directional, compositional, and other non normal data. We illustrate the results by a multivariate gamma model.