A Modern Course on Statistical Distributions in Scientific Work

These three volumes constitute the edited Proceedings of the NATO Advanced Study Institute on Statistical Distributions in Scientific Work held at the University of Calgary from July 29 to August 10, ~. 974. The general title of the volumes is "Statistical Distributions in Scientific Work". The individual volumes are: Volume 1 - Models and Structures; Volume 2 - Model Building and Model Selection; and Volume 3 - Characterizations and Applications. These correspond to the three advanced seminars of the Institute devoted to the respective subject areas. The planned activities of the Institute consisted of main lectures and expositions, seminar lectures and study group dis­ cussions, tutorials and individual study. The activities included meetings of editorial committees to discuss editorial matters for these proceedings which consist of contributions that have gone through the usual refereeing process. A special session was organized to consider the potential of introducing a course on statistical distributions in scientific modeling in the curriculum of statistics and quantitative studies. This session is reported in Volume 2. The overall perspective for the Institute is provided by the Institute Director, Professor G. P. Pati1, in his inaugural address which appears in Volume 1. The Linnik Memorial Inaugural Lecture given by Professor C. R. Rao for the Characterizations Seminar is included in Volume 3. As discussed in the Institute inaugural address, not mL.

1. Linnik Memorial Inaugural Lecture.- 1.1 Some Problems in the Characterization of the Multivariate Normal Distribution.- 2. Mathematical Tools for Characterization Problems.- 2.1 Properties of Non-Negative Definite Solutions of Certain Ordinary Differential Equations.- 2.2 General Solution of a Functional Equation Connected With a Characterization of Statistical Distributions.- 2.3 An Extension of the Phragmén-Lindelöf Theory Which is Relevant for Characterization Theory.- 3. Characterizations Using Order Statistics.- 3.1 Characterizations of Probability Distributions by Properties of Order Statistics I.- 3.2 Characterizations of Probability Distributions by Properties of Order Statistics II.- 3.3 Characterization of Distributions Based on the “Scores” Arising in the Theory of Rank Tests.- 3.4 Characterization of Distributions by the Form of Predictors of Order Statistics.- 3.5 Characterization of the Exponential Distribution Using Lower Moments of Order Statistics.- 3.6 A Characterization of the Exponential Distribution.- 3. 7 L1-Completeness of a Class of Beta Densities.- 3.8 Expected Value of the Spacings Between Order Statistics.- 4. Characterizations by other Statistical Properties.- 4.1 Characterizations of Distributions by Statistical Properties on Groups.- 4.2 Transcendental Functions of a Vector Variable and a Characterization of a Multivariate Poisson Distribution.- 4.3 On the Conditions of Asymptotic ?-Admissibility of the Polynomial Pitman Estimators of a Location Parameter and Certain Properties of Information Measures of Closeness.- 4.4 Characterizations of the Gamma Distribution and Related Law.- 4.5 Characterization of Normal Law by Constancy of Regression.- 4.6 A Characterization of the Normal Distribution.- 5. Characterizations on Spaces and Processes.- 5.1 On Characterization of Probability Distributions by Conditional Expectations.- 5.2 On a Characterization of Probability Distributions on Locally Compact Abelian Groups-II.- 5.3 Some Characterizations of the Exchangeable Processes and Distribution-Free Tests.- 5.4 Characterization by Functional Equations of Branching Process Limit Laws.- 6. Characterization Problems for Damaged Observations.- 6.1 Problems of Damaged Random Variables and Related Characterizations.- 6.2 On Some Characterizations of the Binomial and Poisson Distributions Based on a Damage Model.- 6.3 Some New Characterizations of Discrete Lagrangian Distributions.- 6.4 A Characterization of a Dependent Bivariate Poisson Distribution.- 7. Characterizations Using Entropy Measures and Related Problems.- 7.1 Maximum Entropy Characterizations of some Distributions.- 7.2 On Fisher’s Amount of Information for Location Family.- 7.3 Characterizations Based on Second-Order Entropy.- 7.4 Some Recent Applications of Functional Equations and Inequalities to Characterizations of Probability Distributions, Combinatorics, Information Theory and Mathematical Economics.- 8. Characterizations for Discrete Distributions and Families.- 8.1 Some Characterizations Involving Additivity and Infinite Divisibility and Their Applications to Poisson Mixtures and Poisson Sums.- 8.2 On Characterizing the Hypergeometric and Multivariate Hypergeometric Distributions.- 8.3 Characterizations of Certain Discrete Distributions.- 9. Characterizations for Continuous Distributions and Families.- 9.1 Characterizations of Directional Distributions.- 9.2 Maximum Likelihood Characterization of the von Mises Distribution.- 9.3 On Characterizing Univariate and Multivariate Exponential Distributions With Applications.- 9.4 Characterizations of Linear Exponential Families.