Three inequalities of Tchebycheff type are presented. Two of them give lower bounds for the probability of intervals not necessarily symmetric around the mean. The third one generalizes the extension of Tchebycheff's inequalities given by Miyamoto (1978). They are based on the inequality of Markov. Attainability of lower bounds is also discussed.

The well-known Bennett–Hoeffding bound for sums of independent random variables is refined, by taking into account positive-part third moments, and at that significantly improved by using, instead of the class of all increasing exponential functions, a much larger class of generalized moment functions. The resulting bounds have certain optimality properties. The results can be extended in a standard manner to (the maximal functions of) (super)martingales. The proof of the main result relies on an...

We prove that the sums $S_k$ of independent random vectors satisfy $P(ma{x}_{1\le k\le n}\parallel S_k\parallel >3t)\le 2ma{x}_{1\le k\le n}P(\parallel S_k\parallel >t)$, t ≥ 0.

Let $F_n$ be the empirical distribution function (df) pertaining to independent random variables with continuous df $F$. We investigate the minimizing point ${\widehat{\tau }}_n$ of the empirical process $F_n-F_0$, where $F_0$ is another df which differs from $F$. If $F$ and $F_0$ are locally Hölder-continuous of order $\alpha$ at a point $\tau$ our main result states that $n^{1/\alpha }({\widehat{\tau }}_n-\tau )$ converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation...

Let Fn be the empirical distribution function (df) pertaining to independent random variables with continuous df F. We investigate the minimizing point ${\widehat{\tau }}_n$ of the empirical process Fn - F0, where F0 is another df which differs from F. If F and F0 are locally Hölder-continuous of order α at a point τ our main result states that $n^{1/\alpha }({\widehat{\tau }}_n-\tau )$ converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation...

Let $𝐗=(X,Y)$ be a pair of exchangeable lifetimes whose dependence structure is described by an Archimedean survival copula, and let ${𝐗}_t=\left[(X-t,Y-t)\right|X>t,Y>t]$ denotes the corresponding pair of residual lifetimes after time $t$, with $t\ge 0$. This note deals with stochastic comparisons between $𝐗$ and ${𝐗}_t$: we provide sufficient conditions for their comparison in usual stochastic and lower orthant orders. Some of the results and examples presented here are quite unexpected, since they show that there is not a direct correspondence between univariate...

Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular then weak tail domination implies strong tail domination. In particular, a positive answer to Oleszkiewicz's question would follow from the so-called Bernoulli conjecture. We also prove that any unconditional logarithmically concave distribution is strongly dominated by a product symmetric exponential measure.

We present optimal upper bounds for expectations of order statistics from i.i.d. samples with a common distribution function belonging to the restricted family of probability measures that either precede or follow a given one in the star ordering. The bounds for families with monotone failure density and rate on the average are specified. The results are obtained by projecting functions onto convex cones of Hilbert spaces.

We consider the problem of providing optimal uncertainty quantification (UQ) – and hence rigorous certification – for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter...

In this paper, we consider order statistics and outlier models, and focus primarily on multiple-outlier models and associated robustness issues. We first synthesise recent developments on order statistics arising from independent and non-identically distributed random variables based primarily on the theory of permanents. We then highlight various applications of these results in evaluating the robustness properties of several linear estimators when multiple outliers are possibly present in the...

For stationary Gaussian processes, we obtain the necessary and sufficient conditions for Poincaré inequality and log-Sobolev inequality of process-level and provide the sharp constants. The extension to moving average processes is also presented, as well as several concrete examples.

In this paper, we consider Poincaré inequalities for non-euclidean metrics on ℝd. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give equivalent functional forms of these Poincaré type inequalities in terms of transportation-cost inequalities and inf-convolution inequalities. Workable sufficient conditions are given...