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

/pdf/on-the-existence-of-continuous-processes-with-given-one-4mcgrhbw37.pdf

On the existence of continuous processes with given one-dimensional distributions

Let L be a linear space of real-bounded random variables on the probability space . A finitely additive probability P on such thatis called equivalent martingale finitely additive probability (EMFA). In this paper, EMFAs are investigated in case P0 is atomic. Existence of EMFAs is characterized and a question raised in Berti et al. (2012, J. Theoret. Probab. Available at http://economia.unipv.it/pagp/pagine_personali/prigo/arb.pdf) is answered. Some results of the following type are obtained as well. Let and Y be a bounded random variable. Then , for some sequence , provided EMFAs exist and for each EMFA P.

Price uniqueness and fundamental theorem of asset pricing with finitely additive probabilities

The duality theory of the Monge–Kantorovich transport problem is investigated in an abstract measure theoretic framework. Let 
$$(\mathcal {X},\mathcal {F},\mu )$$

 and 
$$(\mathcal {Y},\mathcal {G},\nu )$$

 be any probability spaces and 
$$c:\mathcal {X}\times \mathcal {Y}\rightarrow \mathbb {R}$$

 a measurable cost function such that 
$$f_1+g_1\le c\le f_2+g_2$$

 for some 
$$f_1,\,f_2\in L_1(\mu )$$

 and 
$$g_1,\,g_2\in L_1(\nu )$$

. Define 
$$\alpha (c)=\inf _P\int c\,dP$$

 and 
$$\alpha ^*(c)=\sup _P\int c\,dP$$

, where 
$$\inf $$

 and 
$$\sup $$

 are over the probabilities P on 
$$\mathcal {F}\otimes \mathcal {G}$$

 with marginals 
$$\mu $$

 and 
$$\nu $$

. Some duality theorems for 
$$\alpha (c)$$

 and 
$$\alpha ^*(c)$$

, not requiring 
$$\mu $$

 or 
$$\nu $$

 to be perfect, are proved. As an example, suppose 
$$\mathcal {X}$$

 and 
$$\mathcal {Y}$$

 are metric spaces and 
$$\mu $$

 is separable. Then, duality holds for 
$$\alpha (c)$$

 (for 
$$\alpha ^*(c)$$

) provided c is upper-semicontinuous (lower-semicontinuous). Moreover, duality holds for both 
$$\alpha (c)$$

 and 
$$\alpha ^*(c)$$

 if the maps 
$$x\mapsto c(x,y)$$

 and 
$$y\mapsto c(x,y)$$

 are continuous, or if c is bounded and 
$$x\mapsto c(x,y)$$

 is continuous. This improves the existing results in Ramachandran and Ruschendorf (Probab Theory Relat Fields 101:311–319, 1995) if c satisfies the quoted conditions and the cardinalities of 
$$\mathcal {X}$$

 and 
$$\mathcal {Y}$$

 do not exceed the continuum.

A Note on Duality Theorems in Mass Transportation

We give a central limit theorem, which has applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced generalized Polya urns.

A Central Limit Theorem and its Applications to Multicolor Randomly Reinforced Urns

Given a statistical modelP = {Pθ : θ ∈ x} and a surjective functiong: ϑ→Λ the problem of transformingP into a new modelQ= {λ : λ ∈ Λ} indexed by Λ is investigated. Two characterizations are given for those modelsQ of the form Qλ = ∫ Pθ πλ(dθ), where πλ is some probability such that πλ(g=λ)=1. The first is related to a geometric property ofQ, while the second rests on the inferential implications of adoptingQ. Also, in the first πλ is allowed to be finitely additive, while in the second πλ is σ-additive. Finally, integrated likelihoods are revisited in light of the second characterization.

Pietro Rigo

Papers

On the existence of continuous processes with given one-dimensional distributions

Price uniqueness and fundamental theorem of asset pricing with finitely additive probabilities

A Note on Duality Theorems in Mass Transportation

A Central Limit Theorem and its Applications to Multicolor Randomly Reinforced Urns

Eliminating nuisance parameters: two characterizations