Reliability and expectation bounds for coherent systems with exchangeable components

AbstractSharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components

Projection Method for Moment Bounds on Order Statistics from Restricted Families I. Dependent Case

AbstractWe present a method of projections onto convex cones for establishing the sharp bounds in terms of the first two moments for the expectations ofL-estimates based on samples from restricted families. In this part, we consider the case of possibly dependent identically distributed parent random variables. For the classes of decreasing failure probability, DFR, and symmetric unimodal marginal distributions, we first determine parametric subclasses which contain the distributions attaining the extreme expectations for allL-estimates. Then we derive the bounds for single order statistics. The results provide some new characterizations of uniform and exponential distribution

MAXIMUM VARIANCE OF KTH RECORDS

Abstract Papadatos (1995) provided sharp bounds for the variances of order statistics in population variance units. This paper presents similar results for the variances of kth record values

Conditional Evaluations of Sums of Sample Maxima and Records

We consider sequences of independent and identically absolutely continuously distributed random variables assuming that they have finite expectation and variance. We determine sharp lower and upper bounds on the expectation of the sum of n first sample maxima and n first upper record values under the condition that the value of the jth (1 ≤ j ≤ n) sample maximum and record value, respectively, are known and equal to a given quantile of the parent distribution. The bounds are expressed in terms of the expectation and standard deviation of the parent distribution. Analogous evaluations are presented for the sum of record values in n observations, when the jth sample maximum is known. The theoretical results are numerically compared

Effects of trans-endocardial delivery of bone marrow-derived CD133+ cells on angina and quality of life in patients with refractory angina: A sub-analysis of the REGENT-VSEL trial

Background: The REGENT-VSEL trial demonstrated a neutral effect of transendocardial injection of autologous bone marrow (BM)-derived CD133+ in regard to myocardial ischemia. The current sub-analysis of the REGENT VSEL trial aims to assess the effect stem cell therapy has on quality of life (QoL) in patients with refractory angina.Methods: Thirty-one patients (63.0 ± 6.4 years, 70% male) with recurrent CCS II–IV angina, despite optimal medical therapy, enrolled in the REGENT-VSEL single center, randomized, double-blinded, and placebo-controlled trial. Of the 31 patients, 16 individuals were randomly assigned to the active stem cell group and 15 individuals were randomly assigned to the placebo group on a 1:1 basis. The inducibility of ischemia, (≥ one myocardial segment) was confirmed for each patient using Tc-99m SPECT. QoL was measured using the Seattle Angina Questionnaire. Each patient completed the questionnaire prior to treatment and at the time of their outpatient follow-up visits at 1, 4, 6, and 12 months after cell/placebo treatment.Results: The main finding of the REGENT-VSEL trial sub-analysis was that transendocardial injection of autologous BM-derived CD133+ stem cells in patients with chronic refractory angina did not show significant improvement in QoL in comparison to the control group. Moreover, there was no significant difference between cell therapy and placebo in a number of patients showing improvement of at least 1 Canadian Cardiovascular Society class during the follow-up period.Conclusions: Intra-myocardial delivery of autologous CD133+ stem cells is safe and feasible but does not show a significant improvement in the QoL or angina pectoris symptoms in patients with chronic myocardial ischemia

A class of unbiased kernel estimates of a probability density function

We propose a class of unbiased and strongly consistent nonparametric kernel estimates of a probability density function, based on a random choice of the sample size and the kernel function. The expected sample size can be arbitrarily small and mild conditions on the local behavior of the density function are imposed

The Ryszard Zielinski's works on nonparametric quantile estimators and their use in robust statistics

Celem tej przegladowej pracy jest opis wyników profesora RyszardaZielinskiego dotyczacych nieparametrycznych estymatorów kwantyli w skonczonychpróbach oraz ich zastosowania w odpornej estymacji parametru połozenia. Główneprzesłanie badan Zielinskiego było nastepujace: do estymacji kwantyli nalezy uzywacpojedynczych statystyk pozycyjnych, a juz ich liniowe kombinacje moga byc bardzoniedokładne w duzych modelach nieparametrycznych. Optymalny wybór statystykipozycyjnej zalezy od kryterium oceny błedu estymacji.This is a survey paper describing achievements of professor Ryszard Zieliński in the subject of nonparametric estimation of population quantiles based on samples of fixed size, and applications of the quantile estimators in the robust estimation of location parameter. Zielinski assumed that a finite sequence of independent identically distributed random variables X1, . . . ,Xn is observed, and their common distribution function F belongs to the family F of continuous and strictly increasing distribution functions. He considered the family T of randomized estimators XJ:n which are single order statistics based on X1, . . . ,Xn with a randomly determined number J. The random variable J is independent of the sample and has an arbitrary distribution on the numbers 1, . . . , n. It was proved that T is the maximal class of estimators which are functions of the complete and sufficient statistic (X1:n, . . . ,Xn:n), and are equivariant with respect to the strictly increasing transformations, i.e., satisfy T(φ(X1:n), . . . ,φ(Xn:n)) = φ(T(X1:n, . . . ,Xn:n)) for arbitrary strictly increasing φ. A number of examples showed that the estimators that do not belong to T are very inaccurate for some F€F.   For comparing estimators, there were used various accuracy criteria based on the difference F(T) - q, where 0 < q < 1 is the quantile order. They are invariant with respect to the strictly increasing transformations. Optimal estimators with respect to the mean absolute loss E|F(T)-q|, mean quadratic loss E(F(T)-q)2, expected LINEX loss E[exp(a[F(T)-q])-a[F(T)-q]-1], a≠0, and Pitman closeness measure were explicitly determined. Further, the best estimators in narrower classes of median-unbiased estimators U(q) = {T€T : med(T, F) = F-1(q)},  (where med(T, F) stands for the median of the distribution of estimator T when the parent distribution function is F), and F-unbiased estimators V(q) = {T € T : EF(T) = q} of quantiles F-1 (q), 0 < q < 1, are determined for some accuracy criteria. Also, random confidence intervals for F-1(q), F€F, of the form [XI:n,XJ:n] on a fixed confidence level 0 <  < 1, i.e. satisfying P(XI:n ≤F-1(q) ≤XJ:n)≥γ,  F € F, , and minimizing E(J - I), are described. Median-unbiased estimators of quantiles were applied by Zielinski in the robust estimation of location parameter. For the i.i.d. sample X1, . . . ,Xn from the location model Fμ(x) = F(x - μ), where μ€R and F is a known unimodal distribution function, and the ε-contamination of the model Z(μ) = {G = (1 -ε)Fμ +εH : H - arbitrary distribution function} for some fixed 0 <ε< 1/2 , the most robust translation equivariant estimator with respect to the median oscillation criterion bn(T, μ) = supG1,G2€Z(μ) |med(T,G1) - med(T,G2)| has the form XJ:n - F-1(q*), XJ:n  €U(q*). Number q*  is chosen so to minimize function (ε, 1 - ε)Э q→ F-1(q/(1-ε))-F-1((q-ε)/(1-ε)). If F is unimodal and symmetric, then q* = ½.. However, Zielinski also showed that a slight modification of the ε-contamination for symmetric unimodal F may imply that XJ:n - F-1(q*), XJ:n € U(q*), for some q*≠1/2 is the most robust estimator with respect to the median oscillation criterion. Celem tej przeglądowej pracy jest opis wyników profesora Ryszarda Zielińskiego dotyczącychnieparametrycznych estymatorów kwantyli w skończonych próbach oraz ich zastosowania w odpornej estymacjiparametru położenia. Główne przesłanie badań Zielińskiego było następujące:do estymacji kwantyli należy używać pojedynczych statystyk pozycyjnych, a już ich liniowekombinacje mogą być bardzo niedokładne w dużych modelach nieparametrycznych.Optymalny wybór statystyki pozycyjnej zależy od kryterium oceny błędu estymacji