We discuss briefly the very interesting concept of Brownian distance covariance developed by Szekely and Rizzo (2009) and describe two possible extensions. The first extension is for high dimensional data that can be coerced into a Hilbert space, including certain high throughput screening and functional data settings. The second extension involves very simple modifications that may yield increased power in some settings. We commend Szekely and Rizzo for their very interesting work and recognize that this general idea has potential to have a large impact on the way in which statisticians evaluate dependency in data.

/pdf/brownian-distance-covariance-1xjy2lylae.pdf

Brownian distance covariance

This paper proposes a general approach and a computationally convenient estimation procedure for the structural analysis of auctioni data. Considering first-price sealed-bid auction models within the independent private value paradigm, we show that the underlying distribution of bidders' private values is identified from observed bids and the number of actual bidders without any parametric assumptions. Using the theoiy of minimax, we establish the best rate of uniform convergence at which the latent density of private values can be estimated nonparametrically from available data. We then propose a two-step kernel-based estimator that converges at the optimal rate.

Optimal Nonparametric Estimation of First-price Auctions

In 1959 A. Renyi proposed a set of axioms for a measure of dependence for pairs of random variables. In the same year A. Sklar introduced the general notion of a copula. This is a function which links an $n$-dimensional distribution function to its one-dimensional margins and is itself a continuous distribution function on the unit $n$-cube, with uniform margins. We show that the copula of a pair of random variables $X, Y$ is invariant under a.s. strictly increasing transformations of $X$ and $Y$, and that any property of the joint distribution function of $X$ and $Y$ which is invariant under such transformations is solely a function of their copula. Exploiting these facts, we use copulas to define several natural nonparametric measures of dependence for pairs of random variables. We show that these measures satisfy reasonable modifications of Renyi's conditions and compare them to various known measures of dependence, e.g., the correlation coefficient and Spearman's $\rho$.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176345528

On Nonparametric Measures of Dependence for Random Variables

The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. How and why the nuisance parameter approach works and how it can be extended bear closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banach-valued random elements and applying the Banach central limit theorem and law of iterated logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted.

Consistent specification testing with nuisance parameters present only under the alternative

The aim of this paper is to present a few techniques which may be useful in the analysis of time series when a failure is suspected. We present two categories of tests and investigate their asymptotic properties: one, of nonparametric type, is intended to detect a general failure in spectrum; the other investigates the properties of likelihood ratio tests in parametric models which have a non-standard behaviour in this situation. Finally, we obtain the asymptotic distribution of the likelihood estimators of the change parameters.

Testing and Estimating Change-Points in Time Series

Certain tests of independence based on the sample distribution function (d.f.) possess power properties superior to those of other tests of independence previously discussed in the literature. The characteristic functions of the limiting d.f.'s of a class of such test criteria are obtained, and the corresponding d.f. is tabled in the bivariate case, where the test is equivalent to one originally proposed by Hoeffding [4]. A discussion is included of the computational problems which arise in the inversion of characteristic functions of this type. Techniques for computing the statistics and for approximating the tail probabilities are considered.

/pdf/distribution-free-tests-of-independence-based-on-the-sample-3gxsspzb1y.pdf

Distribution Free Tests of Independence Based on the Sample Distribution Function

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Distribution Free Tests of Independence Based on the Sample Distribution Function