Javier Rojo

Stochastic comparisons of parallel systems of heterogeneous exponential components

Abstract Let X1, …, Xn be independent exponential random variables with Xi having hazard rate λi, i = 1, …, n. Let λ = (λ1, …, λn). Let Y1, …, Yn be a random sample of size n from an exponential distribution with common hazard rate λ = Σ n i = 1 λ i n . The purpose of this paper is to study stochastic comparisons between the largest order statistics Xn:n and Yn:n from these two samples. It is proved that the hazard rate of Xn:n is smaller than that of Yn:n. This gives a convenient upper bound on the hazard rate of Xn:n in terms of that of Yn:n. It is also proved that Yn:n is smaller than Xn:n according to dispersive ordering. While it is known that the survival function of Xn:n is Schur convex in λ, Boland, El-Neweihi and Proschan [J. Appl. Prohab. 31 (1994) 180–192] have shown that for n > 2, the hazard rate of Xn:n is not Schur concave. It is shown here that, however, the reversed hazard rate of Xn:n is Schur convex in λ.

New properties and characterizations of the dispersive ordering

New characterizations of the dispersive ordering are established. These include a characterization in terms of the stochastic ordering of the sample spacings, preservation of the ordering by monotone convex (concave) transformations, and preservation of the ordering by truncation at the same quantile. The question of when the sample spacings inherit the dispersive ordering is investigated and, for the important special case of F or G being the exponential distribution, it is shown that F and G are ordered in dispersion if and only if the sample spacings also have the same order.

Reliability Assessment of Lifeline Systems with Radial Topology

The increased susceptibility of lifeline systems to failure due to aging and external hazards requires efficient methods to quantify their reliability and related uncertainty. Monte Carlo simulation techniques for network-level reliability and uncertainty assessment usually require large computational experiments. Also, available analytical approaches apply mainly to simple network topologies, and are limited to providing average values, low order moments, or confidence bounds of reliability metrics. This study introduces a closed form technique to obtain the entire probability distribution of a reliability metric of customer service availability (CSA) for generic radial lifeline systems. A special case of this general formulation reduces to a simple sum of products equation, for which a recursive algorithm that exploits its structure is presented. This special-case algorithm computes the probability mass function (PMF) of CSA for systems with M elements in O(M 3 ) operations, relative to conventional O(2 M ) operations, and opens the possibility of finding recursive algorithms for the general radial case. Parametric models that approximate the CSA metric are also explored and their errors quantified. The proposed radial topology reliability assessment tools and resulting probability distributions provide infrastructure owners with critical insights for informed operation and maintenance decision making under uncertainty.

Analysis of recent pediatric orthotopic liver transplantation outcomes indicates that allograft type is no longer a predictor of survivals.

Two strategies to increase the donor allograft pool for pediatric orthotopic liver transplantation (OLT) are deceased donor segmental liver transplantation (DDSLT) and living donor liver transplantation (LDLT). The purpose of this study is to evaluate outcomes after use of these alternative allograft types. Data on all OLT recipients between February 2002 and December 2004 less than 12 years of age were obtained from the United Network for Organ Sharing database. The impact of allograft type on posttransplant survivals was assessed. The number of recipients was 1260. Of these, 52% underwent whole liver transplantation (WLT), 33% underwent DDSLT, and 15% underwent LDLT. There was no difference in retransplantation rates. Immediate posttransplant survivals differed, with WLT patients having improved 30‐day patient survivals compared to DDSLT and LDLT patients (P = 0.004). Although unadjusted 1‐year patient survivals were better for WLT versus DDSLT (P = 0.01), after risk adjustment, 1‐year patient survivals for WLT (94%), DDSLT (91%), and LDLT (93%) were similar (P values > 0.05). Unadjusted allograft survivals were better for WLT and LDLT in comparison with DDSLT (P = 0.009 and 0.018, respectively); however, after adjustment, these differences became nonsignificant (all P values > 0.05). For patients ≤ 2 years of age (n = 833), the adjusted 1‐year patient and allograft survivals were also similar (all P values > 0.05). In conclusion, in the current era of pediatric liver transplantation, WLT recipients have better immediate postoperative survivals. By 1 year, adjusted patient and allograft survivals are similar, regardless of the allograft type. Liver Transpl 14:1125–1132, 2008.

Least median of squares filtering of locally optimal point matches for compressible flow image registration

Compressible flow based image registration operates under the assumption that the mass of the imaged material is conserved from one image to the next. Depending on how the mass conservation assumption is modeled, the performance of existing compressible flow methods is limited by factors such as image quality, noise, large magnitude voxel displacements, and computational requirements. The Least Median of Squares Filtered Compressible Flow (LFC) method introduced here is based on a localized, nonlinear least squares, compressible flow model that describes the displacement of a single voxel that lends itself to a simple grid search (block matching) optimization strategy. Spatially inaccurate grid search point matches, corresponding to erroneous local minimizers of the nonlinear compressible flow model, are removed by a novel filtering approach based on least median of squares fitting and the forward search outlier detection method. The spatial accuracy of the method is measured using ten thoracic CT image sets and large samples of expert determined landmarks (available at www.dir-lab.com). The LFC method produces an average error within the intra-observer error on eight of the ten cases, indicating that the method is capable of achieving a high spatial accuracy for thoracic CT registration.

On nonparametric maximum likelihood estimation of a distribution uniformly stochastically smaller than a standard

The class of distributions F which are uniformly stochastically smaller than a known standard G arises naturally when life testing experiments are conducted in an environment more severe or stressful than the environment in which the system has been pretested. The problem of estimating a distribution in this class via nonparametric maximum likelihood is considered. Under the assumption that the standard lifetime distribution G is continuous and strictly increasing, the NPMLE of F is derived in closed form, and its inconsistency is demonstrated.

Invariant Directional Orderings

Statistical concepts of order permeate the theory and practice of statistics. The present paper is concerned with a large class of directional orderings of univariate distributions. (What do we mean by saying that a random variable Y is larger than another random variable X?) Attention is restricted to preorders that are invariant under monotone transformations; this includes orderings such as monotone likelihood ratio, hazard ordering, and stochastic ordering. Simple characterizations of these orderings are obtained in terms of a maximal invariant. It is shown how such invariant preorderings can be used to generate concepts of Y2 being further to the right of X2 than Y1 is of X1.

On Estimating a Survival Curve Subject to a Uniform Stochastic Ordering Constraint

Abstract If F and G are cumulative distribution functions on [0, ∞) governing the lifetimes of specific systems under study, and if F and G are their corresponding survival functions, then F is said to be uniformly stochastically smaller than G, denoted by F 0}). When F and G are absolutely continuous, F (+) G is equivalent to the assumption that the corresponding failure rates are ordered. The applicability of the notion of uniform stochastic ordering in reliability and life testing is discussed. Given that a random sample X 1, …, X n of lifetimes has been obtained from F, where F is assumed to satisfy the uniform stochastic ordering constraint F (+) G), where G is fixed and known, the problem of estimating F is addressed. It has been shown elsewhere that the method of nonparametric maximum likelihood estimation fails to provide consistent estimators in this type of ...

On the weak convergence of certain estimators of stochastically ordered survival functions

Let F and G be distribution functions with corresponding survival functions . When is known and for all x, the estimator where represents the empirical survival function, has been considered by Rojo and Ma. In the two-sample problem when both are unknown with for all x, Lo and Rojo and Ma have considered the estimators and where represent the empirical survival functions based on random samples from F and G of sizes n and m respectively. Here, the weak convergence of is considered. It is shown, under mild conditions on F and G, that and converge weakly when for all x. However, when for some x0, weak convergence doesnot hold. Similar results are obtained in the case of censored data by replacing the empirical survival functions by their Kaplan-Meier counterparts. The estimators are illustrated with some examples from engineering and the biomedical sciences.

On Tail Categorization of Probability Laws

Abstract A classification scheme for probability laws by tail behavior is proposed. It circumvents the smoothness conditions usually imposed on the probability laws by current classification schemes and yields a complete characterization of the probability distributions belonging to each category. That is, a distribution function F has a long, medium, or short tail, depending on whether F (ln x) is slowly varying, regularly varying, or rapidly varying. A clear and concise connection with the limiting behavior of extreme spacings is also established and the connection with the existence of moments and moment generating functions is noted. Closure properties of the classification scheme under reliability operations are also considered.