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

Convergence of Probability Measures

SUMMARY This paper investigates the properties of a semiparametric method for estimating the dependence parameters in a family of multivariate distributions. The proposed estimator, obtained as a solution of a pseudo-likelihood equation, is shown to be consistent, asymptotically normal and fully efficient at independence. A natural estimator of its asymptotic variance is proved to be consistent. Comparisons are made with alternative semiparametric estimators in the special case of Clayton's model for association in bivariate data.

https://www.jstor.org/stable/pdfplus/2337532.pdf

A semiparametric estimation procedure of dependence parameters in multivariate families of distributions

Weak convergence of the empirical copula process has been established by Deheuvels in the case of independent marginal distributions. Van der Vaart and Wellner utilize the functional delta method to show convergence in l ∞ ([a,b] 2) for some 0<a<b<1, under restrictions on the distribution functions. We extend their results by proving the weak convergence of this process in l ∞ ([0,1] 2) under minimal conditions on the copula function, which coincides with the result obtained by Gaenssler and Stute. It is argued that the condition on the copula function is necessary. The proof uses the functional delta method and, as a consequence, the convergence of the bootstrap counterpart of the empirical copula process follows immediately. In addition, weak convergence of the smoothed empirical copula process is established.

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Weak convergence of empirical copula processes

. Dependencies between extreme events (extremal dependencies) are attracting an increasing attention in modern risk management. In practice, the concept of tail dependence represents the current standard to describe the amount of extremal dependence. In theory, multi-variate extreme-value theory turns out to be the natural choice to model the latter dependencies. The present paper embeds tail dependence into the concept of tail copulae which describes the dependence structure in the tail of multivariate distributions but works more generally. Various non-parametric estimators for tail copulae and tail dependence are discussed, and weak convergence, asymptotic normality, and strong consistency of these estimators are shown by means of a functional delta method. Further, weak convergence of a general upper-order rank-statistics for extreme events is investigated and the relationship to tail dependence is provided. A simulation study compares the introduced estimators and two financial data sets were analysed by our methods.

Non‐parametric Estimation of Tail Dependence

By means of a general weak convergence theorem some invariance principles are proven for the multivariate sequential empirical process and for the multivariate rank order process w.r.t. stronger metrics than the generalized Skorohod metric. The underlying random variables are neither assumed to be independent nor to be stationary. These results are then applied to derive convergence of the weighted empirical cumulatives and for the weighted rank order process. Finally by a new representation asymptotic normality is proven for a general class of linear multivariate rank order statistics.

/pdf/asymptotic-distributions-of-multivariate-rank-order-4y6mv9gw8b.pdf

Asymptotic Distributions of Multivariate Rank Order Statistics

Asymptotic normality of nonparametric tests for independence

Asymptotic normality of linear rank statistics for testing the hypothesis of independence is established under fixed alternatives. A generalization of a result of Bhuchongkul [1] is obtained both with respect to the conditions concerning the orders of magnitude of the score functions and with respect to the smoothness conditions on these functions.

/pdf/asymptotic-normality-of-nonparametric-tests-for-independence-3o12lrkzjg.pdf

G. R. Shorack

Papers

Asymptotic normality of nonparametric tests for independence

Asymptotic Normality of Nonparametric Tests for Independence