A study of generalized skew-normal distribution

Following the paper by Genton and Loperfido [Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math. 57 (2005), pp. 389–401], we say that Z has a generalized skew-normal distribution, if its probability density function (p.d.f.) is given by f(z)=2φ _p (z; ξ, Ω)π (z?ξ), z∈? ^p , where φ _p (·; ξ, Ω) is the p-dimensional normal p.d.f. with location vector ξ and scale matrix Ω, ξ∈? ^p , Ω>0, and π is a skewing function from ? ^p to ?, that is 0≤π (z)≤1 and π (?z)=1?π (z), ? z∈? ^p . First the distribution of linear transformations of Z are studied, and some moments of Z and its quadratic forms are derived. Next we obtain the joint moment-generating functions (m.g.f.’s) of linear and quadratic forms of Z and then investigate conditions for their independence. Finally explicit forms for the above distributions, m.g.f.’s and moments are derived when π (z)=κ (α′z), where α∈? ^p and κ is the normal, Laplace, logistic or uniform distribution function.     

Three-term asymptotic expansion: A semi-Markovian random walk with a generalized beta distributed interference of chance

A semi-Markovian random walk process (X(t)) with a generalized beta distribution of chance is considered. The asymptotic expansions for the first four moments of the ergodic distribution of the process are obtained as E(ζ_n) → ∞ when the random variable ζ_n has a generalized beta distribution with parameters (s, S, α, β); ?α, β > 1,?0? ? s < S < ∞. Finally, the accuracy of the asymptotic expansions is examined by using the Monte Carlo simulation method.     

Characterizing Relationships Between Estimations Under a General Linear Model with Explicit and Implicit Restrictions by Rank of Matrix

In the investigation of the restricted linear model ? _r  = {y, X β | A β = b, σ² Σ}, the parameter constraints A β = b are often handled by transforming the model into certain implicitly restricted model. Any estimation derived from the explicitly and implicitly restricted models on the vector β and its functions should be equivalent, although the expressions of the estimation under the two models may be different. However, people more likely want to directly compare different expressions of estimations and yield a conclusion on their equivalence by using some algebraic operations on expressions of estimations. In this article, we give some results on equivalence of the well-known OLSEs and BLUEs under the explicitly and implicitly restricted linear models by using some expansion formulas for ranks of matrices.     

Reviews and Notices

Abstract

Longstaff's Studies in Statistics: Studies in Statistics. Social, Political, and Medical. By George Blundell Longstaff, M.A., M.B., etc. London: Edward Stanford. 1891. 8vo. Pp. 455. Maps and diagrams. Reviewed by S. W. Abbott.

Keynes on Statistics: The Scope and Method of Political Economy. By John Neville Keynes, M.A., London. Macmillan and Company. 1891. Pp. xiv, 359. Reviewed by Davis R. Dewey.

United States Census Bulletins: 1No. 39. March 16, 1891. Wealth and Resources of Alaska. By Ivan Petroff. Pp. 15.

United States Census Bulletins: No. 40. March 17. Population by Counties, North Central Division. Pp. 9.

United States Census Bulletins: No. 41. March 19. Agriculture, Truck Farming. By J. H. Hale. Pp. 12.

United States Census Bulletins: No. 42. March 20. Population by Counties, South Central and Western Divisions. Pp. 9.

United States Census Bulletins: No. 43. March 21. Coal Product West of the Mississippi River. By John H. Jones. Pp. 8.

United States Census Bulletins: No. 44. March 25. Distribution of Population in Accordance with Mean Relative Humidity of the Atmosphere. By Henry Gannett. Pp. 3.

United States Census Bulletins: No. 45. March 26. Granite. By William C. Day. Pp. 41.

United States Census Bulletins: No. 46. March 27. Railway Statistics of the New England States. By Henry C. Adams. Pp. 18.

United States Census Bulletins: No. 47. March 28. Distribution of Population by Drainage Basins. By Henry Gannett. Pp. 5.

United States Census Bulletins: No. 48. April 7. The White and Colored Population of the South. 1890. Pp. 27.

United States Census Bulletins: No. 49. April 14. Precious and Ornamental Stones and Diamond Cutting. By George Frederick Kunz. Pp. 8.

United States Census Bulletins: No. 50. April 15. Population of Rhode Island by Minor Civil Divisions. Pp. 3.

United States Census Bulletins: No. 51. April 16. Population of Vermont by Minor Civil Divisions. Pp. 4.

United States Census Bulletins: No. 52. April 17. Urban Population in 1890. Cities Containing 8000 Inhabitants or more. Pp. 9.

United States Census Bulletins: No. 53. April 20. Statistics of Education. Alaska, Arkansas, Delaware, Missouri, Iowa, Michigan, Minnesota, Mississippi, New Mexico, New York, North Dakota, Oregon, Texas, Utah, Washington, West Virginia, and forty-two Cities. By James H. Blodgett. Pp. 34.

United States Census Bulletins: No. 54. April 23. Public School Finances. By J. K. Upton. Pp. 12.

United States Census Bulletins: No. 55. April 24. The Relative Economy of Cable, Electric, and Animal Motive Power for Street Railways. By Charles H. Cooley. Pp. 17.

United States Census Bulletins: No. 56. April 25. Population of Maine by Minor Civil Divisions. Pp. 7.

United States Census Bulletins: No. 57. April 27. Population of Delaware by Minor Civil Divisions. Pp. 3.

United States Census Bulletins: No. 58. April 28. Population of Connecticut by Minor Civil Divisions. Pp. 3.

United States Census Bulletins: No. 59. April 29. Commercial Floriculture. By J. H. Hale. Pp. 11.

United States Census Bulletins: No. 60. April 30. Irrigation in New Mexico. By F. H. Newell. Pp. 14.

United States Census Bulletins: No. 61. May 8. The Production of Mica. By L. J. Childs. Pp. 6.

United States Census Bulletins: No. 62. May 9. Asylums for the Insane in the United States. By Dr. John S. Billings and W. H. Olcott. Pp. 32.

United States Census Bulletins: No. 63. May 11. Distribution of Population in Accordance with Latitude and Longitude. With diagrams. By Henry Gannett. Pp. 7.

United States Census Bulletins: No. 64. May 12. Foreign, National, State, and County Indebtedness. By J. K. Upton. Pp. 52.

United States Census Bulletins: No. 65. May 13. Distribution of Population in Accordance with Topographic Features. By Henry Gannett. Pp. 7.

United States Census Bulletins: No. 66. May 14. Floating Equipment on the Great Lakes. By Henry C. Adams. Pp. 11.

United States Census Bulletins: No. 68. May 16. Production of Manganese Ores. By Joseph D. Weeks. Pp. 5.

United States Census Bulletins: Mo. 69. May 18. Population of New Jersey by Minor Civil Divisions. Pp. 6.

United States Census Bulletins: No. 70. May 22. Statistics of Churches. By Henry K. Carroll. Pp. 27.

United States Census Bulletins: No. 71. May 23. Production of Bluestone. By William C. Day. Pp. 6.

Congress of Demography.

Old Age and Pauperism in England.

Mortality of English Clergymen.

Fire Statistics.     

Best Quadratic Unbiased Prediction in a General Linear Model with Stochastic Regression Coefficients

In this article, we discuss on how to predict a combined quadratic parametric function of the form β′ H β + hσ² in a general linear model with stochastic regression coefficients denoted by y  =  X β +  e . Firstly, the quadratic predictability of β′ H β + hσ² is investigated to obtain a quadratic unbiased predictor (QUP) via a general method of structuring an unbiased estimator. This QUP is also optimal in some situations and therefore we hope it will be a fine predictor. To show this idea, we apply the Lagrange multipliers method to this problem and finally reach the expected conclusion through permutation matrix techniques.     

Combined synthetic and |S| chart for monitoring process dispersion

This article proposes a multivariate control chart, the syn-|S| chart, which comprises a standard |S| subchart and a multivariate synthetic sample generalized variance |S| (synthetic |S|) subchart, for detecting shifts in the covariance matrix of a multivariate normally distributed process. A procedure for the optimal design of the syn-|S| chart by minimizing the average extra quadratic loss is provided. The syn-|S| chart has better overall performance compared to the synthetic |S| chart and the standard |S| chart, based on the zero-state and steady-state modes. An example is given to illustrate the operation of the synthetic |S| chart.     

A Note on Fitting a Regression without an Intercept Term

By entering the data (y _i ,x _i ) followed by (–y _i ,–x _i ), one can obtain an intercept-free regression Y = Xβ + ε from a program package that normally uses an intercept term. There is no bias in the resultant regression coefficients, but a minor postanalysis adjustment is needed to the residual variance and standard errors.     

A note on efficient simulation of multidimensional spatial autoregressive processes

In applications of spatial statistics, it is necessary to compute the product of some matrix W of spatial weights and a vector y of observations. The weighting matrix often needs to be adapted to the specific problems, such that the computation of Wy cannot necessarily be done with available R-packages. Hence, this article suggests one possibility treating such issues. The proposed technique avoids the computation of the matrix product by calculating each entry of Wy separately. Initially, a specific spatial autoregressive process is introduced. The performance of the proposed program is briefly compared to a basic program using the matrix multiplication.     

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

Improved ridge estimators in a linear regression model

In this paper, the notion of the improved ridge estimator (IRE) is put forward in the linear regression model y=X β+e. The problem arises if augmenting the equation 0=c′α+ε instead of 0=C α+? to the model. Three special IREs are considered and studied under the mean-squared error criterion and the prediction error sum of squares criterion. The simulations demonstrate that the proposed estimators are effective and recommendable, especially when multicollinearity is severe.     

Comparing the BLUEs Under Two Linear Models

In this article, we consider two linear models, ?₁ = {y, X β, V ₁} and ?₂ = {y, X β, V ₂}, which differ only in their covariance matrices. Our main focus lies on the difference of the best linear unbiased estimators, BLUEs, of X β under these models. The corresponding problems between the models {y, X β, I _n } and {y, X β, V}, i.e., between the OLSE (ordinary least squares estimator) and BLUE, are pretty well studied. Our purpose is to review the corresponding considerations between the BLUEs of X β under ?₁ and ?₂. This article is an expository one presenting also new results.     

On the BLUEs in Two Linear Models via C. R. Rao's Pandora's Box

We consider the estimation of the parameters in two partitioned linear models, denoted by 𝒜 = {y, X ₁ β ₁ + X ₂ β ₂, V _𝒜} and ? = {y, X ₁ β ₁ + X ₂ β ₂, V _?}, which we call full models. Correspondingly, we define submodels 𝒜₁ = {y, X ₁ β ₁, V _𝒜} and ?₁ = {y, X ₁ β ₁, V _?}. Using the so-called Pandora's Box approach introduced by Rao (1971 Rao , C. R. ( 1971 ). Unified theory of linear estimation . Sankhy?, Ser. A 33 : 371 – 394 . [Corrigendum (1972), 34, p. 194, 477.]  [Google Scholar], we give new necessary and sufficient conditions for the equality between the best linear unbiased estimators (BLUEs) of X ₁ β ₁ under 𝒜₁ and ?₁ as well as under 𝒜 and ?. In our considerations we will utilise the Frisch–Waugh–Lovell theorem which provides a connection between the full model 𝒜 and the reduced model 𝒜 _r  = {M ₂ y, M ₂ X ₁ β ₁, M ₂ V _𝒜 M ₂} with M ₂ being an appropriate orthogonal projector. Moreover, we consider the equality of the BLUEs under the full models assuming that they are equal under the submodels.     

Optimal designs of multivariate synthetic |S| control chart based on median run length

The multivariate synthetic generalized sample variance |S| (synthetic |S|) chart is a combination of the |S| sub-chart and the conforming run length sub-chart. A procedure for optimal designs of the synthetic |S| chart, based on the median run length (MRL), for both zero and steady-state modes are provided by minimizing the out-of-control MRL. The comparative results show that the synthetic |S| chart performs better than the standard |S| chart for detecting shifts in the covariance matrix of a multivariate normally distributed process, in terms of the MRL. An example is given to illustrate the operation of the synthetic |S| chart.     

Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

A characterization of the dilation order and its applications

LetX andY be two random variables with finite expectationsE X andE Y, respectively. ThenX is said to be smaller thanY in the dilation order ifE[ϕ(X-E X)]≤E[ϕ(Y-E Y)] for any convex functionϕ for which the expectations exist. In this paper we obtain a new characterization of the dilation order. This characterization enables us to give new interpretations to the dilation order, and using them we identify conditions which imply the dilation order. A sample of applications of the new characterization is given. Partially supported by MURST 40% Program on Non-Linear Systems and Applications. Partially supported by “Gruppo Nazionale per l'Analisi Funzionale e sue Applicazioni”—CNR.     

On the Distribution of Matrix Quadratic Forms

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.     

Reviews and Notices

Abstract

Studies in Jewish Statistics: Studies in Jewish Statistics, social, vital, and anthropometric. By Joseph Jacobs. London, 1891. Pp. 59, lxix 13. Reviewed by Charles F. A. Currier.

The Wool Book: A Statistical Manual Containing the Latest Official Data of the Production, Movement, and Consumption of Wool in all Countries. Compiled for the National Association of Wool Manufacturers, by S. N. D. North, Secretary, Boston, 1892.

Statistical Articles in German Periodicals and Journals: Die Entwickelung der Briteschen Landwirthschaft unter dem Druck ausländischer Konkurrenz. By Prof. Dr. Paasche. Reviewed by Victor Rosewater.

Statistical Articles in German Periodicals and Journals: JahrbüUcher füUr NationalsöUkonomie und Statistik. February, 1892.

Statistical Articles in German Periodicals and Journals: Schmoller's Jahrbnch. XVI. No. 1. Der Wiener Weltpost Kongress. By J. Jung. Reviewed by Victor Rosewater.

Statistical Articles in German Periodicals and Journals: Allgemeines Statistisches Archiv. II. No. 1. Die Jahresschwankungen in der Häufigkeit verschiedener bevölkerungs und moralstatistischer Erscheinungen. By Dr. K. Becker. Reviewed by Victor Rosewater.

Statistical Articles in German Periodicals and Journals: Zeitschrift für Schweizerische Statistik. XXVII. No. 3. Zehn Basler Arbeiterhaushaltungen. By Carl Landolt. Reviewed by Victor Rosewater.

Statistical Articles in German Periodicals and Journals: Vierteljahrshefte zur Statistik des Deutschen Reichs. Herausgegeben vom Kaiserlichen Statistischen Amt. Erster Jahrgang, 1892. Erster Heft. Berlin, 1892. Reviewed by C. F. A. Currier.

Statistische Monatschrift. Vienna, January, 1892. Die überseeische österreichische Auswanderung, insbesondere in den Jahren 1889 und 1890. By Dr. Friedrich ProbSt.

Die gewerbliche Genossenschaften in Oesterreich zu Ende des Jahres 1891. By Dr. Schmid.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.