scholarly journals Cramér type large deviations for trimmed L-statistics

2018 ◽  
Vol 37 (1) ◽  
pp. 101-118 ◽  
Author(s):  
Nadezhda Gribkova
Keyword(s):  
Large Deviations ◽  
Weight Function ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Natural Conditions ◽  
New Approach ◽  
Probabilities Of Large Deviations ◽  
Large Deviation Problem ◽  
Deviation Problem

CRAMÉR TYPE LARGE DEVIATIONS FOR TRIMMED L-STATISTICSIn this paper, we propose a new approach to the investigationof asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with those 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 onWinsorized random variables. Using this method, we establish the Cramér type large deviation results for the trimmed L-statistics under quite mild and natural conditions.

A large deviations heuristic made precise

2000 ◽  
Vol 128 (3) ◽  
pp. 561-569 ◽  
Author(s):  
NEIL O'CONNELL
Keyword(s):  
Large Deviations ◽  
Bin Packing ◽  
Symmetric Functions ◽  
Large Deviation ◽  
Random Variables ◽  
Packing Problem ◽  
Rate Function ◽  
Empirical Measures ◽  
Underlying Distribution ◽  
Uniformly Lipschitz

Sanov's Theorem states that the sequence of empirical measures associated with a sequence of i.d.d. random variables satisfies the large deviation principle (LDP) in the weak topology with rate function given by a relative entropy. We present a derivative which allows one to establish LDPs for symmetric functions of many i.d.d. random variables under the condition that (i) a law of large numbers holds whatever the underlying distribution and (ii) the functions are uniformly Lipschitz. The heuristic (of the title) is that the LDP follows from (i) provided the functions are ‘sufficiently smooth’. As an application, we obtain large deviations results for the stochastic bin-packing problem.

scholarly journals On the Lower Tail Variational Problem for Random Graphs

2016 ◽  
Vol 26 (2) ◽  
pp. 301-320 ◽  
Author(s):  
YUFEI ZHAO
Keyword(s):  
Variational Problem ◽  
Random Graph ◽  
Random Graphs ◽  
Large Deviation ◽  
Tail Probability ◽  
Edge Density ◽  
Lower Tail ◽  
Lower Tail Probability ◽  
Large Deviation Problem ◽  
Deviation Problem

We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Rényi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$ for fixed 0 < δ < 1.Thanks to the results of Chatterjee, Dembo and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p ≥ n−αH (and conjecturally for a larger range of p). We study this variational problem and provide a partial characterization of the so-called ‘replica symmetric’ phase. Informally, our main result says that for every H, and 0 < δ < δH for some δH > 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1.

scholarly journals On the probabilities of large deviations for sums of random number of independent random variables

1967 ◽  
Vol 7 (3) ◽  
pp. 513-516
Author(s):  
V. Statulevičius
Keyword(s):  
Large Deviations ◽  
Random Number ◽  
Random Variables ◽  
Independent Random Variables ◽  
Probabilities Of Large Deviations

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: В. А. Статулявичюс. О вероятностях больших уклонений для суммы случайного числа независимых случайных величин V. Statulevičius. Didelių atsilenkimų tikimybių nepriklausomų atsitiktinių dydžių sumai su atsitiktiniu dėmenų skaičiumi klausimu

The probabilities of large deviations for a certain class of statistics associated with multinomial distribution

2020 ◽  
Vol 24 ◽  
pp. 581-606
Author(s):  
Sherzod M. Mirakhmedov
Keyword(s):  
Large Deviations ◽  
Random Vector ◽  
Large Deviation ◽  
Multinomial Distribution ◽  
Negative Integer ◽  
Probabilities Of Large Deviations ◽  
Divergence Statistics ◽  
Power Divergence

Let η = (η1, …, ηN) be a multinomial random vector with parameters n = η1 + ⋯ + ηN and pm > 0, m = 1, …, N, p1 + ⋯ + pN = 1. We assume that N →∞ and maxpm → 0 as n →∞. The probabilities of large deviations for statistics of the form h1(η1) + ⋯ + hN(ηN) are studied, where hm(x) is a real-valued function of a non-negative integer-valued argument. The new large deviation results for the power-divergence statistics and its most popular special variants, as well as for several count statistics are derived as consequences of the general theorems.

scholarly journals Large deviations for exchangeable observations with applications

2004 ◽  
Vol 2004 (55) ◽  
pp. 2947-2958
Author(s):  
Jinwen Chen
Keyword(s):  
Large Deviations ◽  
Large Deviation ◽  
Random Variables ◽  
Moment Conditions

We first prove some large deviation results for a mixture of i.i.d. random variables. Compared with most of the known results in the literature, our results are built on relaxing some restrictive conditions that may not be easy to be checked in certain typical cases. The main feature in our main results is that we require little knowledge of (continuity of) the component measures and/or of the compactness of the support of the mixing measure. Instead, we pose certain moment conditions, which may be more practical in applications. We then use the large deviation approach to study the problem of estimating the component and the mixing measures.

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