Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Subjectivism has become the dominant philosophical foundation for Bayesian inference. Yet in practice, most Bayesian analyses are performed with so-called “noninformative” priors, that is, priors constructed by some formal rule. We review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used. We give special emphasis to Jeffreys's rules and discuss the evolution of his viewpoint about the interpretation of priors, away from unique representation of ignorance toward the notion that they should be chosen by convention. We conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly: When sample sizes are small (relative to the number of parameters being estimated), it is dangerous to put faith in any “default” solution; but when asymptotics take over, Jeffreys's rules and their variants remain reasonable choices. We also provide an annotated b...

The Selection of Prior Distributions by Formal Rules

A comprehensive account of economic size distributions around the world and throughout the years In the course of the past 100 years, economists and applied statisticians have developed a remarkably diverse variety of income distribution models, yet no single resource convincingly accounts for all of these models, analyzing their strengths and weaknesses, similarities and differences. Statistical Size Distributions in Economics and Actuarial Sciences is the first collection to systematically investigate a wide variety of parametric models that deal with income, wealth, and related notions. Christian Kleiber and Samuel Kotz survey, compliment, compare, and unify all of the disparate models of income distribution, highlighting at times a lack of coordination between them that can result in unnecessary duplication. Considering models from eight languages and all continents, the authors discuss the social and economic implications of each as well as distributions of size of loss in actuarial applications. Specific models covered include: Pareto distributions Lognormal distributions Gamma-type size distributions Beta-type size distributions Miscellaneous size distributions Three appendices provide brief biographies of some of the leading players along with the basic properties of each of the distributions. Actuaries, economists, market researchers, social scientists, and physicists interested in econophysics will find Statistical Size Distributions in Economics and Actuarial Sciences to be a truly one-of-a-kind addition to the professional literature.

Statistical Size Distributions in Economics and Actuarial Sciences

Weak Convergence and Empirical Processes - A. W. van der Vaart; J. A. Wellner.

1 Introduction.- 2 Experiments, Deficiencies, Distances v.- 2.1 Comparing Risk Functions.- 2.2 Deficiency and Distance between Experiments.- 2.3 Likelihood Ratios and Blackwell's Representation.- 2.4 Further Remarks on the Convergence of Distri butions of Likelihood Ratios.- 2.5 Historical Remarks.- 3 Contiguity - Hellinger Transforms.- 3.1 Contiguity.- 3.2 Hellinger Distances, Hellinger Transforms.- 3.3 Historical Remarks.- 4 Gaussian Shift and Poisson Experiments.- 4.1 Introduction.- 4.2 Gaussian Experiments.- 4.3 Poisson Experiments.- 4.4 Historical Remarks.- 5 Limit Laws for Likelihood Ratios.- 5.1 Introduction.- 5.2 Auxiliary Results.- 5.2.1 Lindeberg's Procedure.- 5.2.2 Levy Splittings.- 5.2.3 Paul Levy's Symmetrization Inequalities.- 5.2.4 Conditions for Shift-Compactness.- 5.2.5 A Central Limit Theorem for Infinitesimal Arrays.- 5.2.6 The Special Case of Gaussian Limits.- 5.2.7 Peano Differentiable Functions.- 5.3 Limits for Binary Experiments.- 5.4 Gaussian Limits.- 5.5 Historical Remarks.- 6 Local Asymptotic Normality.- 6.1 Introduction.- 6.2 Locally Asymptotically Quadratic Families.- 6.3 A Method of Construction of Estimates.- 6.4 Some Local Bayes Properties.- 6.5 Invariance and Regularity.- 6.6 The LAMN and LAN Conditions.- 6.7 Additional Remarks on the LAN Conditions.- 6.8 Wald's Tests and Confidence Ellipsoids.- 6.9 Possible Extensions.- 6.10 Historical Remarks.- 7 Independent, Identically Distributed Observations.- 7.1 Introduction.- 7.2 The Standard i.i.d. Case: Differentiability in Quadratic Mean.- 7.3 Some Examples.- 7.4 Some Nonparametric Considerations.- 7.5 Bounds on the Risk of Estimates.- 7.6 Some Cases Where the Number of Observations Is Random.- 7.7 Historical Remarks.- 8 On Bayes Procedures.- 8.1 Introduction.- 8.2 Bayes Procedures Behave Nicely.- 8.3 The Bernstein-von Mises Phenomenon.- 8.4 A Bernstein-von Mises Result for the i.i.d. Case.- 8.5 Bayes Procedures Behave Miserably.- 8.6 Historical Remarks.- Author Index.

Asymptotics in Statistics–Some Basic Concepts

Let \((X_n)\) be a sequence of random variables, adapted to a filtration \((\mathcal {G}_n)\), and let \(\mu _n=(1/n)\,\sum _{i=1}^n\delta _{X_i}\) and \(a_n(\cdot )=P(X_{n+1}\in \cdot \mid \mathcal {G}_n)\) be the empirical and the predictive measures. We focus on \(||\mu _n-a_n||=\sup _{B\in \mathcal {D}}\,|\mu _n(B)-a_n(B)|\), where \(\mathcal {D}\) is a class of measurable sets. Conditions for \(||\mu _n-a_n||\rightarrow 0\), almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of \(r_n\,||\mu _n-a_n||\) is investigated for suitable constants \(r_n\). Special attention is paid to \(r_n=\sqrt{n}\). The sequence \((X_n)\) is exchangeable or, more generally, conditionally identically distributed.

Asymptotics of Predictive Distributions

Finitely additive mixtures of probability measures

Let X be the set of real functions on [0, ∞) which are cadlag or caglad, B the Borel σ-algebra on X (with respect to the topology of uniform convergence on compacts), and BV = {x ∈ X: x has bounded variation on compacts}. Fix a probability P on B such that P(C[0,∞)) = 1 and the coordinate process X(t,x) = x(t) is a local martingale under P, and define the Ito integral I(t) = ∫ t 0 YdX for a suitable process Y: [0,∞) × X → R. For x ∈ BV, define also the Lebesgue-Stieltjes integral J(x)(t) = ∫ [ 0 , t ] Y(s, x)dx(s). A finitely additive probability Q on B is constructed such that Q(BV) = 1, and, for a large class of integrands Y, P{I E A} = Q{J E A} whenever A E B and P{I E ∂A} = 0. In particular, P{I(t 1 ) < a 1 ,...,I(t k ) < a k } = Q{J(t 1 ) < a 1 ,...,J(t k ) ≤ a k } provided P{I(t i ) = a i } = 0 for all i. Such Q is useful for connecting Ito integral I with pathwise (Lebesgue-Stieltjes) integral J.

Integration with respect to Brownian-like processes

A counterexample to a transfer result in [5] (Theorem 2.4, Chap. 4) is given. A new result, which provides a reasonable substitute for the disproved one, is proved as well. This result yields, in particular, a transfer theorem for processes whose paths are right-continuous but not necessarily cadlag.

Transfer theorems and right-continuous processes

take values in an arbitrarymeasurable space, is dealt with in Subsection 4.2.(ii) In Bayesian statistics, π is the prior distribution. And priors are typicallyassumed absolutely continuous with respect to Lebesgue measure (possibly, withsmooth densities). The results mentioned above, thus, apply to most Bayesianproblems.(iii) Let p > 1 and c > 0. Those π which are absolutely continuous with respectto Lebesgue measure, with a density f such that R

Pietro Rigo

Papers

Asymptotics of Predictive Distributions

Finitely additive mixtures of probability measures

Integration with respect to Brownian-like processes

Transfer theorems and right-continuous processes

Rate of convergence for predictive distributions of exchangeable indicators