Abstract-We investigate the second order accuracy of the M out of N bootstrap for a Studentized trimmed mean using the Edgeworth expansion derived in a previous paper. Some simulations, which support our theoretical results, are also given. The effect of extrapolation in conjunction with the M out of N bootstrap for Studentized trimmed means is briefly discussed. As an auxiliary result we obtain a Bahadur's type representation for an M out of N bootstrap quantile. 1. INTRODUCTION This article is closely connected with our previous paper [11], where the validity of the one-term Edgeworth expansion (EE) and the empirical Edgeworth expansion (EEE) for a Studentized trimmed mean was established and explicit formulas for the leading terms of the EE were obtained. We study two second-order approximations to the distribution function (df ) of a Studentized trimmed mean: EE and bootstrap.During the past twenty five years the attention of many authors was focussed on Efron's bootstrap ([8]), and nowadays there exists a voluminous literature on this topic. The consistency of the standard nonparametric, or naive, bootstrap was proved for many interesting statistics, at least for the asymptotically normal ones (see [1,4,6,7,12,13,15,25] and references therein). One of the main reasons of interest in the bootstrap and its application in statistics is the second order accuracy property: under proper conditions the bootstrap approximation to the distribution function (df ) of a pivotal statistic is more accurate than the normal one. This beneficial property of the bootstrap was proved for the sample mean [25], for the class of Hoeffding's U -statistics (cf. [15]) and for some other statistics. The usual way to prove this fact is based on the congruence of the one-term EE for the pivotal statistic (in the 'real world') and the EE for its bootstrap counterpart (in the 'bootstrap world'). Typically the structure of the one-term EE for the bootstrapped statistic is the same as the one for the pivotal statistic, when the parameters of the formula of the first leading term of EE are replaced by their empirical counterparts (plug-in estimators). So, the application of a relevant version of the Law of Large Numbers implies the second order accuracy of the bootstrap (cf. [12,15,25]). However, the case of the trimmed mean is a special one. The problem is connected with the difficulty in obtaining the explicit formula for the oneterm EE (cf. [13]).In this paper we establish the validity of a one-term EE for the bootstrapped Studentized trimmed mean (in the bootstrap world). We also obtain an explicit formula for the M −1/2 -term (correcting for

No abstract

When dealing with control systems, it is useful and even necessary to assess the performance of underlying transfer functions. The functions may or may not be linear, may or may not be even monotonic. In addition, they may have structural breaks and other abberations that require monitoring and quantification to aid decision making. The present paper develops such a methodology, which is based on an index of increase that naturally arises as the solution to an optimization problem. We show theoretically and illustrate numerically that the empirical counterpart of the index needs to be used with great care and indepth knowledge of the problem at hand in order to achieve desired large-sample properties, such as consistency.

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.

Control systems are exposed to unintentional errors, deliberate intrusions, false data injection attacks, and various other disruptions. In this paper we propose, justify, and illustrate a rule of thumb for detecting, or confirming the absence of, such disruptions. To facilitate the use of the rule, we rigorously discuss background results that delineate the boundaries of the rule's applicability. We also discuss ways to further widen the applicability of the proposed intrusion-detection methodology.

No abstract

We investigate the second order asymptotic behavior of distributions of statistics Tn = 1 n n−mn i=kn+1 X i:n , where kn, mn are sequences of integers, 0 kn < n − mn n, and rn := min(kn, mn) → ∞ as n → ∞, and the X i:n 's denote the order statistics corresponding to a sample X 1 , . . . , Xn of n independent identically distributed random variables. In particular, we focus on the case of slightly trimmed means with vanishing trimming percentages; i.e., we assume that max(kn, mn)/n → 0 as n → ∞, and heavy tailed distribution F ; i.e., the common distribution of the observations F is assumed to have an infinite variance. We derive optimal bounds of Berry-Esseen type of the order O(r −1/2 n ) for the normal approximation to Tn and, in addition, establish one-term expansions of the Edgeworth type for slightly trimmed means and their Studentized versions. Our results supplement previous work on first order approximations for slightly trimmed sums by Csörgő, Haeusler, and Mason [Ann. Probab., 16 (1988), pp. 672-699] and on second order approximations for (Studentized) trimmed means with fixed trimming percentages by Gribkova and Helmers [Math.

Systems are exposed to a variety of risks, including those known as background or systematic risks. Therefore, advanced economic, financial, and engineering models incorporate such risks, thus inevitably making the models more challenging to explore. A number of natural questions arise. First and foremost, is the given system affected by any of such risks? If so, then is the system affected by the risks at the input or output stage, or at both stages? In the present paper we construct an algorithm that answers such questions. Even though the algorithm is based on intricate probabilistic considerations, its practical implementation is easy. Keywords