/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

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

Convergence of Probability Measures

This paper develops the method of matching as an econometric evaluation estimator. A rigorous distribution theory for kernel-based matching is presented. The method of matching is extended to more general conditions than the ones assumed in the statistical literature on the topic. We focus on the method of propensity score matching and show that it is not necessarily better, in the sense of reducing the variance of the resulting estimator, to use the propensity score method even if propensity score is known. We extend the statistical literature on the propensity score by considering the case when it is estimated both parametrically and nonparametrically. We examine the benefits of separability and exclusion restrictions in improving the efficiency of the estimator. Our methods also apply to the econometric selection bias estimator. Matching is a widely-used method of evaluation. It is based on the intuitively attractive idea of contrasting the outcomes of programme participants (denoted Y1) with the outcomes of "comparable" nonparticipants (denoted Y0). Differences in the outcomes between the two groups are attributed to the programme. Let 1 and 11 denote the set of indices for nonparticipants and participants, respectively. The following framework describes conventional matching methods as well as the smoothed versions of these methods analysed in this paper. To estimate a treatment effect for each treated person iecI, outcome Yli is compared to an average of the outcomes Yoj for matched persons je10 in the untreated sample. Matches are constructed on the basis of observed characteristics X in Rd. Typically, when the observed characteristics of an untreated person are closer to those of the treated person ieI1, using a specific distance measure, the untreated person gets a higher weight in constructing the match. The estimated gain for each person i in the treated sample is

/pdf/matching-as-an-econometric-evaluation-estimator-2g3vl15lyt.pdf

Matching As An Econometric Evaluation Estimator

We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD).We present two distribution free tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

/pdf/a-kernel-two-sample-test-2mr8hwo4ml.pdf

A kernel two-sample test

On robust estimation of the location parameter

We consider the following linear regression model:where are independent and identically distributed random variables, Yi, is real, Zi has values in Rm, Ui, is independent of Zi, and θ0 is an m-dimensional parameter to be estimated. The Lp estimator of θ0 is the value 6n such thatHere, we will give the exact Bahadur-Kiefer representation of θn, for each p ≥ 1. Explicitly, we will see that, under regularity conditions,where and c is a positive constant, which depends on p and on the random variable X.

The Bahadur-Kiefer Representation of Lp Regression Estimators

We study the convergence in distribution of M -estimators over a convex kernel. Under convexity, the limit distribution of M -estimators can be obtained under minimal assumptions. We consider the case when the limit is arbitrary, not necessarily normal. If some Taylor expansions hold, the limit distribution is stable. As an application, we examine the limit distribution of M -estimators for the multivariate linear regression model. We obtain the distributional convergence of M -estimators for the multivariate linear regression model for a wide range of sequences of regressors and different types of conditions on the sequence of errors.

Asymptotic theory for m -estimators over a convex kernel

We consider the distributional and the almost sure pointwise Bahadur-Kiefer representation for U-quantiles We show that the order of this representation depends on the order of the local variance of the empirical process of U-statistic structure at the U-quantile Our results indicate that U-quantiles can be smoother than quantiles U-quantiles can either be as unsmooth as quantiles or can behave as differentiable statistical functionals

https://projecteuclid.org/download/pdf_1/euclid.aos/1032526976

The Bahadur-Kiefer representation for U-quantiles

Some inaccuracies in [2] are corrected and some additional results are presented. The bootstrap central limit theorem in the domain of attraction case is improved to include convergence of bootstrap moments. Self-normalized limit theorems for variables in the domain of attraction of a p-stable law are bootstrapped, thus freeing the bootstrap from the index p and the norming constants {bn}. Simultations on the bootstrap of the self-normalized sums for a few values of p and n are also included. RESUME. Nous corrigeons quelques inexactitudes de l’article [2] et nous présentons certains resultats complementaires. Nous ameliorons le théorème central limite « bootstrap » pour obtenir la convergence des moments « bootstrap ». Des theoremes limites auto-normalises pour des variables dans le domaines d’attraction d’une loi p-stable sont donnes sous forme bootstrap, ce qui libere le bootstrap de l’indice p et des constantes de normalisation (bn). On presente aussi des simulations du bootstrap des sommes auto-normalisee pour quelques valeurs de p et n. (*) Research partially supported by N.S.F. grant No. DMS-9000132 and PSC-CUNNY grant No. 661376. Annales de l’lnstitut Henri Poincaré Probabilités et Statistiques 0246-0203 Vol. 27/91/04/583i 13’~ 330/0 Gauthier-Villars 584 M. A. ARCONES AND E. GINÉ

Additions and correction to “The bootstrap of the mean with arbitrary bootstrap sample”

We identify the asymptotic behavior of the estimators proposed by Rojo and Samaniego and Mukerjee of a distribution $F$ assumed to be uniformly stochastically smaller than a known baseline distribution $G$.We show that these estimators are $\sqrt{n}$-convergent to a limit distribution with mean squared error smaller than or equal to the mean squared error of the empirical survival function. By examining the mean squared error of the limit distribution, we are able to identify the optimal estimator within Mukerjee’s class under a variety of different assumptions on $F$ and $G$. Similar theoretical results are developed for the two-sample problem where$F$ and $G$ are both unknown. The asymptotic distribution theory is applied to obtain confidence bands for the survival function $\bar{F}$ based on published data from an accelerated life testing experiment.

/pdf/on-the-asymptotic-distribution-theory-of-a-class-of-5buewo52pt.pdf

Miguel A. Arcones

Papers

The Bahadur-Kiefer Representation of Lp Regression Estimators

Asymptotic theory for m -estimators over a convex kernel

The Bahadur-Kiefer representation for U-quantiles

Additions and correction to “The bootstrap of the mean with arbitrary bootstrap sample”

On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint