Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized L-statistics

Helmers, Roelof; Janssen, Paul; Serfling, Robert

doi:10.1007/bf00319105

Cited by 50 publications

(35 citation statements)

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“…random variables. Further properties of U -processes were investigated by Dehling, Denker and Philipp [17], Helmers, Janssen and Serfling [28], Nolan and Pollard [36], and others. For weakly dependent observations, weak convergence of U -processes has been established by Arcones and Yu [3] and Borovkova [10].…”

Section: Motivation and Examplesmentioning

confidence: 99%

Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation

Borovkova

Burton

Dehling

2001

Trans. Amer. Math. Soc.

123

137

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Abstract. In this paper we develop a general approach for investigating the asymptotic distribution of functionals Xn = f ((Z n+k ) k∈Z ) of absolutely regular stochastic processes (Zn) n∈Z . Such functionals occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U -statisticswith symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U -processes (Un(h)) h , indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.

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Section: Motivation and Examplesmentioning

confidence: 99%

Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation

Borovkova

Burton

Dehling

2001

Trans. Amer. Math. Soc.

123

137

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“…Also note that it is easily inferred from Corollary 3.1 of Helmers et al (1985) that Tn is not only weakly consistent, but strongly consistent as well in estimating r.…”

Section: An Examplementioning

confidence: 81%

“…Let Osa<bs 1 be arbitrary but fixed and let as.s<ts.b. Then we have Serfling (1980, Section 5.6), see also Helmers et al (1985), it will be shown that an analogous inequality holds true for the processes Un in (2.2). Throughout the remainder of this section the symbols A and C will be used as generic constants in (0, oo) that are independent of all the relevant parameters (n, )., a and b).…”

Section: Properties Of the Empirical Df Of U-statistic Structurementioning

confidence: 82%

“…A strong law for statistics of the general form (1.4), implying almost sure convergence of Tn to its natural limit µ(H) = J~ J(t) lff H(t) dt for some limiting score function J, is contained in Corollary 3.1 of Helmers et al (1985). We also refer to Janssen et al (1984) for the asymptotic normality of a general class of statistical functions.…”

Section: = I Ln(hn(m)(t)) Ijih(t) Dhn(m)(t) Asmentioning

confidence: 99%

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Asymptotic normality of generalized L-statistics with unbounded scores

Helmers¹,

Ruymgaart

1988

Journal of Statistical Planning and Inference

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A central limit theorem for linear combinations of a function of generalized order statistics with unbounded scores is established. The result supplements previous work of Silverman (1983), Serfling (1984) and Akritas (1986) concerning the asymptotic normality of generalized L-statistics. Our proof is patterned after the well-known Chernoff-Savage approach. A linear bound for the empirical distribution function of U-statistic structure is also derived and subsequently applied in the treatment of certain remainder terms.

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“…The theory of U -empirical df's was developed in 1980s (see, e.g., [31], [33], [39]) and is similar to the theory of usual empirical df's. By the GlivenkoCantelli theorem for U -empirical df's we have (wp 1) as n → ∞ :…”

Section: Tests Based On Characterizations and Their Efficienciesmentioning

confidence: 99%

Tests based on characterizations, and their efficiencies: a survey

Nikitin

2017

ACUTM

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Abstract. A survey of goodness-of-fit and symmetry tests based on the characterization properties of distributions is presented. This approach became popular in recent years. In most cases the test statistics are functionals of U -empirical processes. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various parametric alternatives is calculated and compared with each other as well as with diverse previously known tests. We also describe new directions of possible research in this domain.

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Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized L-statistics

Cited by 50 publications

References 12 publications

Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation

Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation

Asymptotic normality of generalized L-statistics with unbounded scores

Tests based on characterizations, and their efficiencies: a survey

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