Cramér type large deviations for trimmed L-statistics

Abstract

In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with ones in Callaert et al. (1982) the first and, as far as we know, the single article, where some results on probabilities of large deviations for the trimmed L-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed L-statistic by a non-trimmed L-statistic (with smooth weight function) based on Winsorized random variables. Using this method, we establish the Cramér type large deviation results for the trimmed L-statistics under quite mild and natural conditions.

Cramér type moderate deviations for trimmed L-statistics

“…In this article we supplement our previous work on Cramér type large deviations for trimmed L-statistics (cf. Gribkova (2016)) by some results on moderate deviations. Our approach here is the same as in Gribkova (2016): we approximate the trimmed L-statistic by a non-trimmed L-statistic with coefficients generated by a smooth on (0, 1) weight function, where the approximating (non-trimmed) L-statistic is based on order statistics corresponding to a sample of auxiliary i.i.d.…”

“…The following lemma provides us a useful representation which is crucial in our proofs. This lemma is proved in (Gribkova, 2016, Lemma 2.1), therefore here we present only its statement.…”

“…For the case of heavy truncated L-statistics, i.e., the case when the weight function is zero outside some interval [α, β] ⊂ (0, 1), a result on Cramér type large deviations was first obtained by Callaert et al (1982); more recently, the latter result was extended and strengthened in Gribkova (2016), where a different approach than in Callaert et al (1982) was proposed and implemented.…”

“…Remark 2.1 It should be noted that the method based on the L-statistic approximation was first applied in Gribkova (2016); it can be viewed as a development of the approach proposed in Gribkova and Helmers (2006, 2007, 2014 (2012)).…”

Weighted allocations, their concomitant-based estimators, and asymptotics

“…While π w is a linear combination of order statistics (e.g., Gribkova, 2017; and references therein), which is a less technically demanding object, the estimators Π w , Π w , and ∆ w are linear combinations of concomitants, which require much more sophisticated methods of analysis. In what follows, we establish conditions under which these estimators are consistent and asymptotically normal.…”