/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

A U-process is a collection of U-statistics. Concretely, a U-process over a family H of kernels h of m variables, based on a probability measure P on (S,S), is the collection $\{ U_{n}^{{(m)}}(h,P):h \in \mathcal{H}\} $ of U-statistics. This chapter is devoted to the asymptotic theory of U-processes: we are interested in finding conditions on H and P ensuring that the law of large numbers, or the central limit theorem, or the law of the iterated logarithm, hold for $ U_n^{(m)} (h,P), $uniformly in h ∈ H. The theory of empirical processes deals with the same questions for the case m = 1, and U-process theory is patterned after it. This is a relatively new subject, at least in the generality presented here (H being an arbitrary collection of kernels which are defined on a general measurable space). There is therefore a need to indicate its usefulness. To this effect, a section on applications is added at the end of the chapter (Section 5.5); there we illustrate the use of each of the main theorems in this chapter by deriving properties of certain multidimensional generalizations of the median, and in general, of M-estimators, and by studying estimators of the cumulative hazard and distribution functions of a random variable based on truncated data.

/pdf/limit-theorems-for-u-processes-4shx4sj76c.pdf

Limit Theorems for $U$-Processes

Limit theorems for functions of stationary mean-zero Gaussian sequences of vectors satisfying long range dependence conditions are considered. Depending on the rate of decay of the coefficients, the limit law can be either Gaussian or the law of a multiple Ito-Wiener integral. We prove the bootstrap of these limit theorems in the case when the limit is normal. A sufficient bracketing condition for these limit theorems to happen uniformly over a class of functions is presented.

/pdf/limit-theorems-for-nonlinear-functionals-of-a-stationary-4fmmo3nwvn.pdf

Limit Theorems for Nonlinear Functionals of a Stationary Gaussian Sequence of Vectors

Bootstrap distributional limit theorems for $U$ and $V$ statistics are proved. They hold a.s., under weak moment conditions and without restrictions on the bootstrap sample size (as long as it tends to $\infty$), regardless of the degree of degeneracy of $U$ and $V$. A testing procedure based on these results is outlined.

/pdf/on-the-bootstrap-of-u-and-v-statistics-33k4pnikei.pdf

On the Bootstrap of $U$ and $V$ Statistics

This paper gives sufficient conditions for the weak convergence to Gaussian processes of empirical processes andU-processes from stationary β mixing sequences indexed byV-C subgraph classes of functions. If the envelope function of theV-C subgraph class is inL

 p
 for some 2<p<∞, we obtain a uniform central limit theorem for the empirical process under the β mixing condition
 
$$k^{p/(p - 2)} (\log k)^{2(p - 1)/(p - 2)} \beta _k \to 0 as k \to \infty $$

 In the case that the functions in theV-C subgraph class are uniformly bounded, we obtain uniform central limit theorems for the empirical process and theU-process, provided that the decay rate of the β mixing coefficient satisfies β
 k
 =O(k
−r
) for somer>1. These conditions are almost minimal.

Central limit theorems for empirical and U -processes of stationary mixing sequences

Nous etudions le theoreme central limite «bootstrap» presque sur et en probabilite, pour des variables aleatoires X de second moment infini, dans le domaine d'attraction de la loi normale ou des autres lois stables, ou dans le domaine partiel d'attraction d'une loi infiniment divisible. La taille m n de l'echantillon «bootstrap» n'est pas necessairement egale (et meme dans certains cas elle est necessairement differente) a la taille n de l'echantillon original

Miguel A. Arcones

Papers

Limit Theorems for $U$-Processes

Limit Theorems for Nonlinear Functionals of a Stationary Gaussian Sequence of Vectors

On the Bootstrap of $U$ and $V$ Statistics

Central limit theorems for empirical and U -processes of stationary mixing sequences

The bootstrap of the mean with arbitrary bootstrap sample size