Asymptotic normality of generalized L-statistics with unbounded scores

Helmers, Roelof; Ruymgaart, F.H.

doi:10.1016/0378-3758(88)90051-1

Cited by 15 publications

(5 citation statements)

References 18 publications

(17 reference statements)

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“…A partial list consists of Chernoff et al [6], Bickel [2], Shorack [19,20], Stigler [23,24], Csörgo et al [8], Griffin and Pruitt [11], Cheng [5], Mason and Shorack [16]. For m ≥ 2, under various sets of regularity conditions, asymptotic normality of various types of generalized L-statistics has been investigated by Silverman [21], Serfling [18], Akritas [1], Janssen et al [15], Helmers and Ruymgaart [12], Gijbels et al [10] and Hössjer [14].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Asymptotic distributions for a class of generalized $L$-statistics

Borovskikh¹,

Weber²

2010

Bernoulli

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We adapt the techniques in Stigler [Ann. Statist. 1 (1973) 472-477] to obtain a new, general asymptotic result for trimmed U -statistics via the generalized L-statistic representation introduced by Serfling [Ann. Statist. 12 (1984) 76-86]. Unlike existing results, we do not require continuity of an associated distribution at the truncation points. Our results are quite general and are expressed in terms of the quantile function associated with the distribution of the Ustatistic summands. This approach leads to improved conditions for the asymptotic normality of these trimmed U -statistics.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Asymptotic distributions for a class of generalized $L$-statistics

Borovskikh¹,

Weber²

2010

Bernoulli

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“…Results on asymptotic normality of GL-statistics T(H ) are developed in n Sering (1984) and Helmers and Ruymgaart (1988) for for T(1) a classical Lfunctional with bounded scores and unbounded scores, respectively, and in Janssen, Sering, and Veraverbeke (1984) for T(1) having the more general form (1.2.2). Under moderate regularity conditions, these statistics satisfy For T(1) g i ven by t h e more general functional (1.2.2), the treatment i s somewhat more complicated.…”

Section: Asymptotic Normality and The Lil Bmentioning

confidence: 99%

Robust Estimation via Generalized L-Statistics: Theory, Applications, and Perspectives

Serfling¹

2019

Advances on Methodological and Applied Aspects of Probability and Statistics

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Generalized L-statistics, i ntroduced in Sering (1984) and including classical U-statistics and L-statistics, are linear functions based on the ordered evaluations of a kernel over subsets of the sample observations. In particular, generalized median s t a tistics fall within this class and are found to fulll an interesting and potent principle, that \smoothing" followed by \medianing" yields a very favorable combination of eciency and robustness. Extensive asymptotic theory now available for generalized L-statistics is reviewed, including a s ymptotic normality, strong convergence, large deviation, sequential xed-width condence interval, j a c kknife, and bootstrap results, as well as Glivenko-Cantelli theory for associated empirical processes of U-statistic structure. Illustrative a pplications are treated, including nonparametric and robust location and spread estimation, nonparametric analysis of linear models, nonparametric regression, and robust parametric scale estimation for exponential distributions, equivalently tail index estimation for Pareto distributions.

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“…The asymptotic properties of the generalized L-statistics have been extensively studied. We refer to Serfling [12,13] Borovshikh and Weber [1] for the asymptotic normality and for the general theory and applications, Cai [2] for the moderate and large deviations, and Helmers and Ruymgaart [6], Helmers, Jansseen and Serfling [5] for the uniform Berry-Esseen bounds. Let H (y) be the distribution function of the random variable function h(X 1 , .…”

Section: Weighted U-statisticsmentioning

confidence: 99%