2010/09/27 by Martin Wendler, Wendler, Martin
Computer Science · Mathematics · #60F17 #60G10 #62G30 #Advanced Statistical Methods and Models #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Statistical Methods and Inference #Statistics Theory (math.ST) #math.PR #math.ST #msc:60F17 #msc:60G10 #msc:62G30 #stat.TH
paper · pdf · doi:10.48550/arxiv.1009.5337
24 pages
openalex publication_date 2010/09/27 · arxiv created 2011/08/18 · arxiv updated 2011/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Generalized linear statistics are an unifying class that contains U-statistics, U-quantiles, L-statistics as well as trimmed and winsorized U-statistics. For example, many commonly used estimators of scale fall into this class. GL-statistics only have been studied under independence; in this paper, we develop an asymptotic theory for GL-statistics of sequences which are strongly mixing or L1 near epoch dependent on an absolutely regular process. For this purpose, we prove an almost sure approximation of the empirical U-process by a Gaussian process. With the help of a generalized Bahadur representation, it follows that such a strong invariance principle also holds for the empirical U-quantile process and consequently for GL-statistics. We obtain central limit theorems and laws of the iterated logarithm for U-processes, U-quantile processes and GL-statistics as straightforward corollaries.