We present a general non-parametric statistical inference theory for integrals of quantiles without assuming any specific sampling design or dependence structure. Technical considerations are accompanied by examples and discussions, including those pertaining to the bias empirical estimators. To illustrate how results can be adapted situations, we derive - at stroke under minimal conditions consistency asymptotic normality tail-value-at-risk, Lorenz Gini curves probability level in case...

We show that the $M\ll N$ (N is original data sample size, M denotes size of bootstrap resample; $M/N\to~0$, as $M\to~\infty$) approximation distribution trimmed mean consistent without any conditions upon population F, whereas Efron's naive (i.e., $M=N$) bootstrap, well normal approximation, fails to be if F has gaps at those two quantiles where trimming occurs.

We investigate the second order asymptotic behavior of distributions statistics $T_n=\frac 1n \sum_{i=k_{n}+1}^{n-m_{n}}{X_{i:n}}$, where $k_n$, $m_n$ are sequences integers, $0\le k_n < n-m_n \le n$, and $r_n:=\min(k_n, m_n) \to \infty$ as ${n \infty}$, ${X_{i:n}}$'s denote corresponding to a sample $X_1,\dots,X_n$ $n$ independent identically distributed random variables. In particular, we focus on case slightly trimmed means with vanishing trimming percentages; i.e., assume that...

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

Background, or systematic, risks are integral parts of many systems and models in insurance finance. These can, for example, be economic nature, they can carry more technical connotations, such as errors intrusions, which could intentional unintentional. A most natural question arises from the practical point view: is given system really affected by these risks? In this paper we offer an algorithm answering question, input-output data appropriately constructed statistics, rely on order...

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

Abstract The prominence of the Euler allocation rule (EAR) is rooted in fact that it only return on risk-adjusted capital (RORAC) compatible rule. When total regulatory set using value-at-risk (VaR), EAR becomes – a statistical term quantile-regression (QR) function. Although cumulative QR function (i.e., an integral function) has received considerable attention literature, fully developed inference theory for itself been elusive. In present paper, we develop such based empirical estimator,...

In a number of research areas, such as non-convex optimization and machine learning, determining assessing regions monotonicity functions is pivotal. Numerically, it can be done using the proportion positive (or negative) increments transformed ordered inputs. When inputs grows, tends to an index increase decrease) underlying function. this paper, we introduce most general provide its interpretation in all practically relevant scenarios, including those that arise when distribution has jumps...

In this article, we establish Cramér-type moderate deviation results for (intermediate) trimmed means Tn = n− 1∑n − mni kn + 1Xi: n, where Xi: n's are the order statistics corresponding to first n observations in a sequence X1, X2, … of independent identically distributed random variables with F. We consider two cases intermediate and heavy trimming. former case, when max (αn, βn) → 0 (αn kn/n, βn mn/n) min (kn, mn) ∞ as ∞, obtain our under natural moment assumption mild condition on rate at...