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

Convergence of Probability Measures

We discuss the analysis of the extremes of data by modelling the sizes and occurrence of exceedances over high thresholds. The natural distribution for such exceedances, the generalized Pareto distribution, is described and its properties elucidated. Estimation and model-checking procedures for univariate and regression data are developed, and the influence of and information contained in the most extreme observations in a sample are studied. Models for seasonality and serial dependence in the point process of exceedances are described. Sets of data on river flows and wave heights are discussed, and an application to the siting of nuclear installations is described

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1990.tb01796.x

Models for exceedances over high thresholds

Le choix de metrique de probabilite est une decision tres importante lorsqu'on etudie la convergence des mesures. Nous vous fournissons avec un sommaire de plusieurs metriques/distances de probabilite couramment utilisees par des statisticiens(nes) at par des probabilistes, ainsi que certains nouveaux resultats qui se rapportent a leurs bornes. Avoir connaissance d'autres metriques peut vous fournir avec un moyen de deriver des bornes pour une autre metrique dans un probleme applique. Le fait de prendre en consideration plusieurs metriques vous permettra d'approcher des problemes d'une maniere differente. Ainsi, nous vous demontrons que les taux de convergence peuvent dependre de facon importante sur votre choix de metrique. Il est donc important de tout considerer lorsqu'on doit choisir une metrique.

/pdf/on-choosing-and-bounding-probability-metrics-1291rrx16o.pdf

On Choosing and Bounding Probability Metrics

This introductory book on the most useful parts of large-sample theory is designed to be accessible to scientists outside statistics and certainly to master’s-level statistics students who ignore most of measure theory. According to the author, “the subject of this book, first-order large- sample theory, constitutes a coherent body of concepts and results that are central to both theoretical and applied statistics.” All of the other existing books published on the subject over the last 20 years, from Ibragimov and Has’minskii in 1979 to the most recent by Van der Waart in 1998 have a common prerequisite in mathematical sophistication (measure theory in particular) that do not make the concepts available to a wide audience.

/pdf/elements-of-large-sample-theory-2ij6yyaggv.pdf

Elements of large-sample theory

Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation. Specifically, we study the usefulness of continuous bimodal priors to model hypervariance parameters, and the effect scaling has on the posterior mean through its relationship to penalization. Several model selection strategies, some frequentist and some Bayesian in nature, are developed and studied theoretically. We demonstrate the importance of selective shrinkage for effective variable selection in terms of risk misclassification, and show this is achieved using the posterior from a rescaled spike and slab model. We also show how to verify a procedure’s ability to reduce model uncertainty in finite samples using a specialized forward selection strategy. Using this tool, we illustrate the effectiveness of rescaled spike and slab models in reducing model uncertainty.

/pdf/spike-and-slab-variable-selection-frequentist-and-bayesian-3zlof3dzue.pdf

Spike and slab variable selection: Frequentist and Bayesian strategies

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

Asymptotics in statistics

In this chapter we describe some of the asymptotic properties of Bayes procedures. These are obtained by using on the parameter set Θ a finite positive measure μ and minimizing the average risk $\int R(\theta,\rho)\mu(d\theta)$. (See Chapter 2 for notation). The procedure ρ that achieves this minimum will of course depend on the choice of μ. However the literature contains numerous statements to the effect that, for large samples, the choice of μ matters little. This cannot be generally true, but we start with a proposition to this effect. If instead of μ one uses A dominated by μ and if the density $\frac{d\lambda}{d\mu}$ can be closely estimated, then a procedure that is nearly Bayes for μ is also nearly Bayes for λ.

https://link.springer.com/content/pdf/10.1007%2F978-1-4684-0377-0_7.pdf

On Bayes Procedures

A double censoring mechanism is such that each variable $X$ in the sample is observable if and only if $X$ is within the observation interval $\lbrack Z, Y \rbrack$. Otherwise, we can only determine whether $X$ is less than $Z$ or greater than $Y$ and observe $Z$ or $Y$ correspondingly. This kind of censoring occurs often in collecting lifetime data. Our problem is to estimate the survival function of $X, S_X(t) = P \lbrack X > t \rbrack$, from a doubly censored sample, where $X$ is assumed to be independent of the random interval $\lbrack Z, Y \rbrack$. We establish sufficient conditions for which $S_X(t)$ is identifiable and then prove the strong consistency of the self-consistent estimator $\hat{S}_X(t)$ for $S_X(t)$. This investigation generalizes the results available for the right censored data.

/pdf/strong-consistency-of-a-nonparametric-estimator-of-the-2a85mkgwuh.pdf

Strong Consistency of a Nonparametric Estimator of the Survival Function with Doubly Censored Data

In the analysis of life tables one biometric function of interest is the life expectancy at age $x, e_x = E\lbrack X - x\mid X > x\rbrack$. Estimation of $e_x$ is considered, the standard estimator used in life tables is shown to be asymptotically unbiased, uniformly strong consistent, and converges in distribution to a Gaussian process. The connections of the estimator studied in this article and that used in reliability theory are illustrated.

/pdf/estimation-of-a-biometric-function-1suyilxe2b.pdf

Grace L. Yang

Papers

Asymptotics in statistics : some basic concepts

Asymptotics in statistics

On Bayes Procedures

Strong Consistency of a Nonparametric Estimator of the Survival Function with Doubly Censored Data

Estimation of a biometric function