Characterizations of the exponential distribution by some properties of the record values

A sequence {X_n, n≥1} of independent and identically distributed random variables with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max {X₁, X₂, ..., X_j?1}. We define L(n)=min {j|j>L(n?1), X_j>X_L(n?1)}, with L(0)=1. Let Z_n,m=X_L(n)?X_L(m), n>m≥0. Some characterizations of the exponential distribution are considered in terms of Z_n,m and X_L(m).     

Criteria for the Unique Determination of Probability Distributions by Moments

A positive probability law has a density function of the general form Q ( x )exp(− x ^1/λ L ( x )), where Q is subject to growth restrictions, and L is slowly varying at infinity. This law is determined by its moment sequence when λ< 2, and not determined when λ> 2. It is still determined when λ= 2 and L ( x ) does not tend to zero too quickly. This paper explores the consequences for the induced power and doubled laws, and for mixtures. The proofs couple the Carleman and Krein criteria with elementary comparison arguments.     

BOOK REVIEWS

Book reviewed in this article:
Pollard, A. H. A Service Course in Statistics.
Penǵ, K. C. The Design and Analysis of Scientijic Experiments.
Neyman, J. A Selection of Early Statistical Papers of J. Neyman.
Chiang, C. L. Introduction to Stochastic Processes in Biostatistics.
Jenkins, G. M., and Watts, D. G. Spectral Analysis and its Applications.
Harris, Bernard (Editor). Spectral Analysis of Time Serie. Proceedings of an Advanced Seminar conducted by the Mathematics Research Center, U.S. Army, and the Statistics Department at the University of Wisconsin, Madison, October 3–5, 1966.
Sarkadi, K., and Vincze, I. (Eds.). Studies in Mathematical Statistics: Theory and Applications.
Pearson, Karl (Ed.). Tables of the Incomplete Beta-Function (with a new Introduction by E. S. Pearson and N. L. Johnson).
Kendall, M. G., and Doiǵ, A. G. Bibliography of Statistical Literature: pre-1940 with supplements to the volumes for 1940-49 and 1950-58.
Albert, A. E., and Gardner, L. A., Jr. Stochastic Approximation and Nonlinear Regression.     

Record values of exponentially distributed random variables

A sequence {X_n, n≥1} of independent and identically distributed random variables with absolutely continuous (with respect to Lebesque measure) cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max(X₁,…,X_j?1), j>1. Let {X_L(n), n≥0} with L(o)=1 be the sequence of such record values and Z_n,n?1=X_L(n)–X_L(n?1). Some properties of Z_n,n?1 are studied and characterizations of the exponential distribution are discussed in terms of the expectation and the hazard rate of z_n,n?1.     

BOOK REVIEWS

Book reviewed in this article:
Winkler, Robert L. An Introduction to Bayesian Inference and Decision.
Bliss, C. I. Statistics in Biology.
Sard, A., and Weintraub, S. A Book of Splines.
Kshirsagar, Anant M. Multivariate Analysis.
Ghosh, B. K. Sequential Tests of Statistical Hypotheses.
Rao, C. Radhakrishna, and Mitra, Sujit Kumar. Generalised Inverse of Matrices and Its Applications.
Balaam, L. N. Fundamentals of Biometry.
Simmons, Donald M. Linear Programming for Operations Research.
Strum, Jay E. Introduction to Linear Programming.
Newman, T. G., and Odell, P. L. The Generation of Random Variates.
Rényi, Alfred. Foundations of Probability.
Romanovsky, V. I. Discrete Markov Chains.     

Discussion of paper “nonparametric Bayesian inference in applications” by Peter Müller,Fernando A. Quintana and Garritt L. Page

This is an invited discussion of review paper “Nonparametric Bayesian Inference in Applications” by Peter Müller, Fernando A. Quintana and Garritt L. Page.     

New asymptotic inferences about the difference,ratio, and linear combination of two independent proportions

In statistics it is customary to realize asymptotic inferences about the difference d, the ratio R or a linear combination L of two independent proportions. In this article the authors evaluate ten inference methods, and conclude that for α = 1% some of the new procedures behave better than the classical. In cases d, R, or L, the optimal method consists in adding 0.5, 0.5 or 1 to all the data, respectively, and then applying a modification of the arc-sine transformation (d or R) or the likelihood ratio test (L). A free program may be obtained at ULR http://www.ugr.es/local/bioest/Z_LINEAR_K.EXE.     

On principles and arguments to likelihood

Birnbaum (1962a) argued that the conditionality principle (C) and the sufficiency principle (S) implied the likelihood principle (L); he then argued (Birnbaum 1972) that C and a mathematical equivalence principle M implied L. Evans, Fraser, and Monette (1985a) gave reference details, and this paper gives proof that C alone implies L. The level of support by the profession for L is sharply less than that for S or even for C; thus the paradoxical nature of these results. In this regard, we elaborate on the Monette example (Fraser, Monette, and Ng 1984), which provides a strong case against L. We also examine closely the various proofs linking the principles and find that S and C can each be used operationally to suppress information otherwise deemed relevant. From another viewpoint this says that S and C can each be used in contexts that directly conflict with the original examples and motivations supporting them; the principles can thus be viewed as inappropriately used, or more strongly, as invalid. In either case, the result that C and S imply L or that C implies L can be regarded as noneffective in the context of discriminating applications. A resolution of the apparent anomalies can be obtained by allowing the statistical model to include ingredients additional to those usually present (particularly for subsequent use with conditionality), or alternatively by restricting the application of the principles to contexts where the conflicts would seem not to arise.     

BOOK REVIEWS

Book reviewed in this article:
Markov Chain Monte Carlo in Practice. Eds W.R. Gilks, S. Richardson & D.J. Spiegelhalter. London
Workshop Statistics - Discovery with Data. By Allan J. Rossman. New York: Springer-Verlag
Activity-based Statistics, Instructor Resources. By Richard L. Scheaffer, Mrudulla Gnanadesikan, Ann Watkins, Jeffrey A. Witmer. New York
Activity-based Statistics, Student Guide. Richard L. Scheaffer, Mrudulla Gnanadesikan, Ann Watkins, Jeffrey A. Witmer. New York
Biplots. By J.C. Gower and D.J. Hand. London: Chapman and Hall
The Economic Laws of Scientific Research. By Terence Kealey. London     

Reviews

Abstract

Occupation Statistics of the Twelfth Census Reviewed by W. C. Mitchell

Economic Conditions of the Jews in Russia Reviewed by E. A. Goldesweiser

Misleading Statistics of Cost of Production Reviewed by David L. Wing

Records of Charity Reviewed by Lilian Brandt

Consumers' Societies in Germany Reviewed by Frank Wetzlar

Wages and Family Budgets in Berlin Reviewed by C.-E. A. Winslow     

SEMI-PARAMETRIC ANALYSIS OF COVARIANCE UNDER DEPENDENCE CONDITIONS WITHIN EACH GROUP

Consider the problem of covariance analysis based on regression models whose regression function is the sum of a linear and a non-parametric component. We propose a parametric and a non-parametric statistical test to compare the effects of the linear and non-parametric components, respectively, on the response variable in   L ≥ 2  groups. Serially correlated errors within each group are allowed. The first (second) test compares the differences between the estimates of the parametric (non-parametric) components of each group by means of a Mahalanobis  ( L ²)  distance. The asymptotic distribution of each statistic under the null hypothesis is obtained. A modest simulation study and an application to a real data set illustrate our methodology.     

A note on score statistic for grouped data

In this paper, we consider modifying the score statistic proposed by Prentice and Gloeckler [Prentice, R. L., & Gloeckler, L. A. (1978). Regression analysis of grouped data with applications to breast cancer data. Biometrics, 34, 57–67] for the grouped data under the proportional hazards model. For this matter, we apply the likelihood method and derive the scores without re-parameterization as a discrete model. Then we illustrate the test with an example and compare the efficiency with the test of Prentice and Gloeckler’s statistic by obtaining empirical powers through simulation study. Also we discuss some possible extension and estimated variances of the score statistic as concluding remarks.     

ON INCOMPLETE BLOCK DESIGNS FOR SYMMETRIC PARALLEL LINE ASSAYS

Incomplete block designs for symmetric parallel line assays, which estimate L_p, L₁and L1/1 with full efficiency, are constructed in a unified manner. Necessary and sufficient conditions are given for existence. A complete solution is presented when the number of doses is even. A general systematic method of construction is given when the number of doses is odd, and all possible designs for up to 15 doses of each preparation are tabulated.     

Book Reviews

Books reviewed:
J.K. Lindsey, Introductory Statistics: A Modelling Approach
D.R. Brillinger, L.T. Fernholz & S. Morgenthaler (eds), The Practice of Data Analysis: Essays in Honor of John W. Tukey
Thomas P. Ryan, Modern Regression Methods
Samuel Kotz & Cynthia R. Lovelace, Process Capability Indices in Theory and Practice
David A. Harville, Matrix Algebra from a Statistician's Perspective
Steven G. Krantz, A Primer of Mathematical Writing
Nicholas J. Higham, Handbook of Writing for the Mathematical Sciences     

BOOK REVIEWS

Book reviewed in this article:
Anderson, T. W. The Statistical Analysis of lime Series.
Freedman, D. Brownian Motion and Diffusion.
Hobson, A. Concepts in Statistical Mechanics.
Howard, Ronald A. Dynamic Probabilistic Systems. Volume I: Markov Models.
Kempthorne, O., and Folks, L. Probability, Statistics and Data Analysis.
Kotz, S. Russian–English/English–Russian Glossary of Statistical Terms.
Kendall, M. G., and Buckland, W. R. A Dictionary of Statistical Terms.
Kendall, M. G., and Buckland, W. R. Russian–English Dictionary of Statistical Terms and Expressions.
Kendall, M. G., and Buckland, W. R. Slovmik Polsko-Rosyjsko-Angielski Statystyki Matematycznej i Statystycznej Kontroli Jakosci Produkcji.
Ladell, J. L., and Dillon, M. J. Curves for the Rapid Estimation of Partial Correlation Coefficients.
Pringle, R. M., and Rayner, A. A. Generalised Inverse Matrices with Application to Statistics.
Puri, Madan Lal, and Sen, Pranab. Nonparametric Methods in Multivariate Analysis.
Steutel, F. W. Preservation of Infinite Divisibility under Mixing and Related Topics.
Whittle, Peter. Optimisation under Constraints.     

A local moment type estimator for an extreme quantile in regression with random covariates

A conditional extreme quantile estimator is proposed in the presence of random covariates. It is based on an adaptation of the moment estimator introduced by Dekkers et al. (1989 Dekkers, A.L.M., Einmahl, J.H.J., de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17:1833–1855.[Crossref], [Web of Science ®] , [Google Scholar]) in the classical univariate setting, and thus it is valid in the domain of attraction of the extreme value distribution, i.e., whatever the sign of the extreme value index is. Asymptotic normality of the estimator is established under suitable assumptions, and its finite sample behavior is evaluated with a small simulation study, where a comparison with an alternative estimator already proposed in the literature is provided. An illustration to a real dataset concerning the world catalogue of earthquake magnitudes is also proposed.     

BOOK REVIEWS

Book reviewed in this article:
Cox, D. R., and Lewis, P. A. W. The Statistical Analysis of Series of Events.
Rao, C. Radhakrishna. Linear Statistical Inference and Its Applications.
Bennett, B. M., and Horst, C. Supplement to Tables for Testing Significance in a 2 × 2 Contingency Table.
Huitson, A. The Analysis of Variance.
Moran, P. A. P., and Smith, C. A. B. The Correlation between Relatives on the Supposition of Mendelian Inheritance.
Searle, S. R. Matrix Algebra for the Biological Sciences.
Feller, W. An Introduction to Probability Theory and Its Applications.
Ashton, Winifred D. The Theory of Road Traffic Flow.
Kemeny, J. G., Snell, J. L., and Thompson, G. L. Introduction to Finite Mathematics.
Coale, A. J., and Demeny, P. Regional Model Life Tables and Stable Populations.     

Rank-based group variable selection

A robust rank-based estimator for variable selection in linear models, with grouped predictors, is studied. The proposed estimation procedure extends the existing rank-based variable selection [Johnson, B.A., and Peng, L. (2008), ‘Rank-based Variable Selection’, Journal of Nonparametric Statistics, 20(3):241–252] and the ww-scad [Wang, L., and Li, R. (2009), ‘Weighted Wilcoxon-type Smoothly Clipped Absolute Deviation Method’, Biometrics, 65(2):564–571] to linear regression models with grouped variables. The resulting estimator is robust to contamination or deviations in both the response and the design space.The Oracle property and asymptotic normality of the estimator are established under some regularity conditions. Simulation studies reveal that the proposed method performs better than the existing rank-based methods [Johnson, B.A., and Peng, L. (2008), ‘Rank-based Variable Selection’, Journal of Nonparametric Statistics, 20(3):241–252; Wang, L., and Li, R. (2009), ‘Weighted Wilcoxon-type Smoothly Clipped Absolute Deviation Method’, Biometrics, 65(2):564–571] for grouped variables models. This estimation procedure also outperforms the adaptive hlasso [Zhou, N., and Zhu, J. (2010), ‘Group Variable Selection Via a Hierarchical Lasso and its Oracle Property’, Interface, 3(4):557–574] in the presence of local contamination in the design space or for heavy-tailed error distribution.     

Book Reviews

Books reviewed:
Bosq, D. Nonparametric Statistics for Stochastic Processes: Estimation and Prediction
Khuri, A.I., Mathew, T. & Sinha, B.K. Statistical Tests for Mixed Linear Models
Latouche, G. & Ramaswami, V. Introduction to Matrix Analytic Methods in Stochastic Modeling
Maitra, A.P. & Sudderth, W.D. Discrete Gambling and Stochastic Games
Rolski, T., Schmidli, H., Schmidt, V. & Teugels, J. Stochastic Processes for Insurance and Finance
Chung, K.L. & Williams, R.J. Introduction to Stochastic Integration