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A general Markov-Switching autoregressive conditional mean model, valued in the set of nonnegative numbers, is considered. The conditional distribution of this model is a finite mixture of nonnegative distributions whose conditional mean follows a GARCH-like dynamics with parameters depending on the state of a Markov chain. Three different variants of the model are examined depending on how the lagged-values of the mixing variable are integrated into the conditional mean equation. The model includes, in particular, Markov mixture versions of various well-known nonnegative time series models such as the autoregressive conditional duration (ACD) model, the integer-valued GARCH (INGARCH) model, and the Beta observation driven model. Under contraction in mean conditions, it is shown that the three variants of the model are stationary and ergodic when the stochastic order and the mean order of the mixing distributions are equal. The proposed conditions match those already known for Markov-switching GARCH models. We also give conditions for finite marginal moments. Applications to various mixture and Markov mixture count, duration and proportion models are provided.

In this note, some point of views on virtual ages are presented in terms of the discussion paper written by Finkelstein and Cha, which include generalized stochastic order‐based virtual ages, system‐level virtual ages, virtual ages in Weibull distribution and repair degrees with virtual ages. Finally, some possible future researches on virtual ages are described.

We study a two‐stage decision problem, namely, the allocation and deployment of reserved inventories (RIs) in a supply network with random demand surges. The demand surge follows a time‐dependent stochastic process and our objective is to minimize the expected total unmet demand in the presence of positive transshipment lead times. We first solve the optimal deployment problem given that the demand surges have occurred at some locations. We show that the optimal deployment policy is a “nested” policy with respect to the shadow price at each location, where a shadow price represents the marginal reduction of the expected total unmet demand due to a marginal increase of RIs. Specifically, locations with higher shadow prices have higher priority in inventory allocation. We then consider the optimal allocation problem in the pre‐positioning stage. We show that under certain conditions the optimal allocation is increasing in the total amount of RIs. We introduce a new stochastic order for distributions defined on sets called the first‐order stochastic dominance and use it to show that the expected total unmet demand is higher when one of the following is true: the demand surges tend to occur simultaneously at more locations, the post‐surge delivery takes a longer time, more demand arrives earlier, or the demand has a higher volatility.

Receiver operating characteristic (ROC) curve is commonly used to evaluate and compare the accuracy of classification methods or markers. Estimating ROC curves has been an important problem in various fields including biometric recognition and diagnostic medicine. In real applications, classification markers are often developed under two or more ordered conditions, such that a natural stochastic ordering exists among the observations. Incorporating such a stochastic ordering into estimation can improve statistical efficiency (Davidov and Herman, 2012). In addition, clustered and correlated data arise when multiple measurements are gleaned from the same subject, making estimation of ROC curves complicated due to within-cluster correlations. In this article, we propose to model the ROC curve using a weighted empirical process to jointly account for the order constraint and within-cluster correlation structure. The algebraic properties of resulting summary statistics of the ROC curve such as its area and partial area are also studied. The algebraic expressions reduce to the ones by Davidov and Herman (2012) for independent observations. We derive asymptotic properties of the proposed order-restricted estimators and show that they have smaller mean-squared errors than the existing estimators. Simulation studies also demonstrate better performance of the newly proposed estimators over existing methods for finite samples. The proposed method is further exemplified with the fingerprint matching data from the National Institute of Standards and Technology Special Database 4.

In this paper, we consider a coherent system with n independent and identically distributed components under the condition that the system is monitored at time instances t1 and t2 (t1

In application of diagnostic accuracy, it is possible that a priori information may exist regarding the test score distributions, either between different disease populations for a single test or between multiple correlated tests. Few have considered constrained diagnostic accuracy analysis when the true disease status is binary; almost none when the disease status is ordinal. Motivated by a study on diagnosing endometriosis, we propose an approach to estimating diagnostic accuracy measures that can incorporate different stochastic order constraints on the test scores when an ordinal true disease status is in consideration. We show that the Dirichlet process mixture provides a convenient framework to both flexibly model the test score distributions and embed the a priori ordering constraints. We also utilize the Dirichlet process mixture to model the correlation between multiple tests. In taking a Bayesian perspective to inference, we develop an efficient Markov chain Monte Carlo algorithm to sample from the posterior distribution and provide posterior estimates of the receiver operating characteristic surfaces and the associated summary measures. The proposed approach is evaluated with extensive simulation studies, and is demonstrated with an application to the endometriosis study.

In this paper, we discuss stochastic comparisons of lifetimes of series and parallel systems with heterogeneous exponentiated gamma components. The results established here are developed in two directions. First, when a system possibly has different shape and scale parameters and the matrix of those different parameters following the chain majorization order, we study the reversed hazard rate order of parallel systems and the usual stochastic order of series systems. Next, by using the concept of vector majorization, we establish the usual stochastic order of series systems and the reversed hazard rate order of parallel systems.

We extend the results of Luo et al. (2015, Biometrics 71, 139-145) regarding the alternative hypotheses for the win ratio from hazard orders to the upper quadrant stochastic order on the plane. This extension substantially widens the range of alternatives against which the win ratio is known to be consistent. It incorporates alternatives induced by simple and popular copula models that are left out by the characterization of Luo et al. (2015). We also discuss how our results may be generalized to win ratios in multivariate and stratified settings.

In engineering applications, many reliability systems can be modeled as k‐out‐of‐n systems with components having random weights. Before putting such kind of system into a working state, it is of great significance for a system designer to find out the optimal assembly of the random weights to the components. In this article, we investigate the performance levels of k‐out‐of‐n systems with random weights. Optimal assembly policies are obtained by maximizing the total capacity according to different criteria, including the usual stochastic order, the increasing convex [concave] order, and the expectation order. Based on the optimal assembly strategy derived by maximizing the expected total capacity, we further investigate stochastic properties of the resulting weighted system with respect to the vector of expectations of random weights. Numerical examples are provided to highlight our theoretical findings as well.

The stochastic error distance (SED) introduced by Diebold and Shin () ranks forecast models by divergence between distributions of the errors of the actual and perfect forecast models. The basic SED is defined by the variation distance and provides a representation of the mean absolute error, but by basing ranking on the entire error distribution and divergence, the SED moves beyond the traditional forecast evaluations. First, we establish connections between ranking forecast models by the SED, error entropy and some partial orderings of distributions. Then, we introduce the notion of excess error for forecast errors of magnitudes larger than a tolerance threshold and give the SED representation of the mean excess error (MEE). As a function of the threshold, the MEE is a local risk measure. With the distribution of the absolute error as a prior for the threshold, its Bayes risk is the entropy functional of the survival function, which is a known measure in the information theory and reliability. Notions and results are illustrated using various distributions for the error. The empirical versions of SED, MEE and its Bayes risk are compared with the mean squared error in ranking regression and autoregressive integrated moving average models for forecasting bond risk premia.

We survey effect measures for models for ordinal categorical data that can be simpler to interpret than the model parameters. For describing the effect of an explanatory variable while adjusting for other explanatory variables, we present probability‐based measures, including a measure of relative size and partial effect measures based on instantaneous rates of change. We also discuss summary measures of predictive power that are analogs of R‐squared and multiple correlation for quantitative response variables. We illustrate the measures for an example and provide R code for implementing them.

This paper studies 2 different versions of the general residual lifetime at random times in the context of statistical dependence between the system lifetime and the random time. We prove that the class of residual lifetimes with a semiindependent random age is equivalent to the class of residual lifetimes at independent random times. Additionally, we conduct stochastic comparisons of the 2 versions of the general residual lifetime at random times. Several numerical examples are presented to illustrate the ordering results.

OBJECTIVE Functioning is an important outcome in hand osteoarthritis (OA). The heterogeneity of patient-reported outcome measures (PROMs) available challenges the direct comparability of information collected by these instruments. This study aimed to examine whether it is possible to achieve metric equivalence of PROMs commonly used to measure functioning in people with hand OA. METHODS A secondary analysis of data from 253 persons who participated in the Vienna Hand Osteoarthritis Cohort Study was conducted applying the Rasch measurement model. Participants completed the Health Assessment Questionnaire, the Australian/Canadian Index for Hand Osteoarthritis, the Functional Index for Hand Osteoarthritis (FIHOA), and the Cochin Scale. The assumptions of stochastic ordering, local independence, unidimensionality, and invariance were tested for each scale independently before the scales were co-calibrated onto a common metric. RESULTS Except for the FIHOA, all measures indicated issues of local dependency in the initial analyses. After accommodating those with testlets, all scales satisfied the assumptions of the Rasch model (χ(2) > 0.05). Marginal misfit in 2 items was found in the FIHOA, but this did not disturb person estimates. As the 4 scales form a common metric that satisfies the assumptions of the Rasch model, the scores can be equated. CONCLUSION The scores of the 4 hand function measures can be transformed to a common 0-100 metric, such that scores can be interchanged. A user-friendly transformation table enables clinicians and researchers to have access to the common metric by simply adding up the total score for each instrument and identifying the corresponding transformed score on the common metric.

Summary We consider simple ordinal model-based probability effect measures for comparing distributions of two groups, adjusted for explanatory variables. An “ordinal superiority” measure summarizes the probability that an observation from one distribution falls above an independent observation from the other distribution, adjusted for explanatory variables in a model. The measure applies directly to normal linear models and to a normal latent variable model for ordinal response variables. It equals Φ(β/2) for the corresponding ordinal model that applies a probit link function to cumulative multinomial probabilities, for standard normal cdf Φ and effect β that is the coefficient of the group indicator variable. For the more general latent variable model for ordinal responses that corresponds to a linear model with other possible error distributions and corresponding link functions for cumulative multinomial probabilities, the ordinal superiority measure equals exp(β)/[1+exp(β)] with the log–log link and equals approximately exp(β/2)/[1+exp(β/2)] with the logit link, where β is the group effect. Another ordinal superiority measure generalizes the difference of proportions from binary to ordinal responses. We also present related measures directly for ordinal models for the observed response that need not assume corresponding latent response models. We present confidence intervals for the measures and illustrate with an example.

We consider time-homogeneous Markov chains with state space Eki¾?{0,1,',k} and initial distribution concentrated on the state 0. For pairs of such Markov chains, we study the Stochastic Tail Order and the stochastic order in the usual sense between the respective first passage times in the state k. On this purpose, we will develop a method based on a specific relation between two stochastic matrices on the state space Ek. Our method provides comparisons that are simpler and more refined than those obtained by the analysis based on the spectral gaps. Copyright © 2016 John Wiley & Sons, Ltd.

Based on the physical randomization of completely randomized experiments, in a recent article in Statistics in Medicine, Rigdon and Hudgens propose two approaches to obtaining exact confidence intervals for the average causal effect on a binary outcome. They construct the first confidence interval by combining, with the Bonferroni adjustment, the prediction sets for treatment effects among treatment and control groups, and the second one by inverting a series of randomization tests. With sample size n, their second approach requires performing O(n4 )randomization tests. We demonstrate that the physical randomization also justifies other ways to constructing exact confidence intervals that are more computationally efficient. By exploiting recent advances in hypergeometric confidence intervals and the stochastic order information of randomization tests, we propose approaches that either do not need to invoke Monte Carlo or require performing at most O(n2) randomization tests. We provide technical details and R code in the Supporting Information.

Molecularly targeted agent (MTA) combination therapy is in the early stages of development. When using a fixed dose of one agent in combinations of MTAs, toxicity and efficacy do not necessarily increase with an increasing dose of the other agent. Thus, in dose-finding trials for combinations of MTAs, interest may lie in identifying the optimal biological dose combinations (OBDCs), defined as the lowest dose combinations (in a certain sense) that are safe and have the highest efficacy level meeting a prespecified target. The limited existing designs for these trials use parametric dose–efficacy and dose–toxicity models. Motivated by a phase I/II clinical trial of a combination of two MTAs in patients with pancreatic, endometrial, or colorectal cancer, we propose Bayesian dose-finding designs to identify the OBDCs without parametric model assumptions. The proposed approach is based only on partial stochastic ordering assumptions for the effects of the combined MTAs and uses isotonic regression to estimate partially stochastically ordered marginal posterior distributions of the efficacy and toxicity probabilities. We demonstrate that our proposed method appropriately accounts for the partial ordering constraints, including potential plateaus on the dose–response surfaces, and is computationally efficient. We develop a dose-combination-finding algorithm to identify the OBDCs. We use simulations to compare the proposed designs with an alternative design based on Bayesian isotonic regression transformation and a design based on parametric change-point dose–toxicity and dose–efficacy models and demonstrate desirable operating characteristics of the proposed designs. © 2014 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

The cumulative Kullback-Leibler information has been proposed recently as a suitable extension of Kullback-Leibler information to the cumulative distribution function. In this paper, we obtain various results on such a measure, with reference to its relation with other information measures and notions of reliability theory. We also provide some lower and upper bounds. A dynamic version of the cumulative Kullback-Leibler information is then proposed for past lifetimes. Furthermore, we investigate its monotonicity property, which is related to some new concepts of relative aging. Moreover, we propose an application to the failure of nanocomponents. Finally, in order to provide an application in image analysis, we introduce the empirical cumulative Kullback-Leibler information and prove an asymptotic result. Copyright © 2015John Wiley & Sons, Ltd.

We consider items that are incepted into operation having already a random initial age and define the corresponding remaining lifetime. We show that these lifetimes are identically distributed when the age distribution is equal to the equilibrium distribution of the renewal theory. Then we develop the population studies approach to the problem and generalize the setting in terms of stationary and stable populations of items. We obtain new stochastic comparisons for the corresponding population ages and remaining lifetimes that can be useful in applications. Copyright © 2014 John Wiley & Sons, Ltd.

Most of the research, on the study of the reliability properties of technical systems, assume that the components of the system operate independently. However, in real life situation, it is more reasonable to assume that there is dependency among the components of the system. In this article, we give sufficient conditions based on the signature and the joint distribution of component lifetimes to obtain stochastic ordering results for coherent and mixed systems with exchangeable components. Some stochastic orders on dynamic (or conditional) signature of coherent systems are also provided. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 549–556, 2014

This study investigated differential item functioning (DIF) mechanisms in the context of differential testlet effects across subgroups. Specifically, we investigated DIF manifestations when the stochastic ordering assumption on the nuisance dimension in a testlet does not hold. DIF hypotheses were formulated analytically using a parametric marginal item response function approach and compared with empirical DIF results from a unidimensional item response theory approach. The comparisons were made in terms of type of DIF (uniform or non-uniform) and direction (whether the focal or reference group was advantaged). In general, the DIF hypotheses were supported by the empirical results, showing the usefulness of the parametric approach in explaining DIF mechanisms. Both analytical predictions of DIF and the empirical results provide insights into conditions where a particular type of DIF becomes dominant in a specific DIF direction, which is useful for the study of DIF causes.

Summary The comparison of two or more ordered experimental groups based on multivariate data is common in a variety of applications such as toxicology, clinical trials and drug development, to name just a few. In this article, we develop a nonparametric methodology for analyzing such data. In particular we propose a global K sample nonparametric test for order among vector valued outcomes. The testing procedure can also be used in a post-hoc fashion to answer questions about the ordering of subgroups and/or single outcomes within any subset of experimental groups. Such a methodology does not currently exist. In contrast with standard methodology such as multivariate analysis of variance (MANOVA), and its nonparametric analogues, we do not assume that the groups differ only by a location parameter or that the components of the response vector have the same marginal distributions between and across groups, that is, we allow for the shape of the distribution to change across groups. We emphasize that our test compares the outcome distributions, not just their mean tendencies, and explicitly incorporates and exploits the order constraints. Consequently it is more powerful than the existing unordered tests. The methodology is illustrated using genotoxicity data where the effect of hydrogen peroxide exposure on damage to DNA is evaluated using a comet assay.

The allocation of redundancies in a system to optimize the reliability of system performance is an interesting problem in reliability engineering and system security. In this article, we focus on the optimal allocation of two exponentially distributed active (standby) redundancies in a two-component series system using the tool of stochastic ordering. For the case of active redundancy, stochastic comparisons are carried out in terms of the likelihood ratio and reversed hazard rate orders. For the case of standby redundancy, a likelihood ratio ordering result is developed. The results established here generalize and strengthen corresponding results in the recent literature. In addition, several numerical examples are used to explicate the results. (c) 2013 Wiley Periodicals, Inc.

In this paper, we consider a repairable system with minimal repairs whose number of repairs is a positive random variable with a given probability vector. Some preservation theorems and aging properties of repairable systems are established. Under the condition that at time t the system is working, a new random variable for the residual lifetime of the system is proposed. Some stochastic ordering results among the lifetimes and residual lifetimes of two systems are obtained. Similar results for coherent systems with independent components and exchangeable components were obtained in the previous literature. Copyright © 2013 John Wiley & Sons, Ltd.

When the goal of a study is to compare two groups on an ordinal categorical scale, a large number of inferential methods are available. Most methods are designed to detect a location effect, such as by focusing a single-degree-of-freedom test on an effect parameter. Often, rather than merely summarizing by a P-value to describe the evidence against a null hypothesis, it is of interest to consider whether a stronger conclusion can be made. For example, can we conclude that the population distributions are stochastically ordered? For parameter space regions described by order restrictions, frequentist methods are not well designed for significance testing. For example, a frequentist P-value for testing identical distributions against an alternative of stochastically ordered distributions can be very small even when the sample distributions give clear evidence that the distributions do not have the ordering property. The Bayesian approach seems better equipped to handle such questions. We discuss this in the context of stochastic ordering and other types of ordinal odds ratio structure, for the two-group comparison and for more general contexts. For Dirichlet priors, simple simulations provide posterior probabilities of particular ordinal odds ratio structures. Copyright © 2013 John Wiley & Sons, Ltd.

A framework involving independent competing risks permits observing failures due to a specific cause and failures due to a competing cause, which constitute survival times from the cause of primary interest. Is observing more failures more informative than observing survivals? Intuitively, due to the definitiveness of failures, the answer seems to be the former. However, it has been shown before that this intuition holds when estimating the mean but not the failure rate of the exponential model with a gamma prior distribution for the failure rate. In this article, we address this question at a more general level. We show that for a certain class of distributions failures can be more informative than survivals for prediction of life length and vice versa for some others. We also show that for a large class of lifetime models, failure is less informative than survival for estimating the proportional hazards parameter with gamma, Jeffreys, and uniform priors. We further show that, for this class of lifetime models, on average, failure is more informative than survival for parameter estimation and for prediction. These results imply that the inferential purpose and properties of the lifetime distribution are germane for conducting life tests. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013

In this paper, we consider a repairable system in which two types of failures can occur on each failure. One is a minor failure that can be corrected with minimal repair, whereas the other type is a catastrophic failure that destroys the system. The total number of failures until the catastrophic failure is a positive random variable with a given probability vector. It is assumed that there is some partial information about the failure status of the system, and then various properties of the conditional probability of the system failure are studied. Mixture representations of the reliability function for the system in terms of the reliability function of the residual lifetimes of record values are obtained. Some stochastic properties of the conditional probabilities and the residual lifetimes of two systems are finally discussed. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we obtain ordering properties for coherent systems with possibly dependent identically distributed components. These results are based on a representation of the system reliability function as a distorted function of the common component reliability function. So, the results included in this paper can also be applied to general distorted distributions. The main advantage of these results is that they are distribution-free with respect to the common component distribution. Moreover, they can be applied to systems with component lifetimes having a non-exchangeable joint distribution. Copyright © 2012 John Wiley & Sons, Ltd.

Researchers routinely adopt composite endpoints in multicenter randomized trials designed to evaluate the effect of experimental interventions in cardiovascular disease, diabetes, and cancer. Despite their widespread use, relatively little attention has been paid to the statistical properties of estimators of treatment effect based on composite endpoints. We consider this here in the context of multivariate models for time to event data in which copula functions link marginal distributions with a proportional hazards structure. We then examine the asymptotic and empirical properties of the estimator of treatment effect arising from a Cox regression model for the time to the first event. We point out that even when the treatment effect is the same for the component events, the limiting value of the estimator based on the composite endpoint is usually inconsistent for this common value. We find that in this context the limiting value is determined by the degree of association between the events, the stochastic ordering of events, and the censoring distribution. Within the framework adopted, marginal methods for the analysis of multivariate failure time data yield consistent estimators of treatment effect and are therefore preferred. We illustrate the methods by application to a recent asthma study. Copyright © 2012 John Wiley & Sons, Ltd.

This paper proposes, implements and investigates a new non-parametric two-sample test for detecting stochastic dominance. We pose the question of detecting the stochastic dominance in a non-standard way. This is motivated by existing evidence showing that standard formulations and pertaining procedures may lead to serious errors in inference. The procedure that we introduce matches testing and model selection. More precisely, we reparametrize the testing problem in terms of Fourier coefficients of well-known comparison densities. Next, the estimated Fourier coefficients are used to form a kind of signed smooth rank statistic. In such a setting, the number of Fourier coefficients incorporated into the statistic is a smoothing parameter. We determine this parameter via some flexible selection rule. We establish the asymptotic properties of the new test under null and alternative hypotheses. The finite sample performance of the new solution is demonstrated through Monte Carlo studies and an application to a set of survival times.

This paper presents a framework for comparing bivariate distributions according to their degree of regression dependence. We introduce the general concept of a regression dependence order (RDO). In addition, we deflne a new nonparametric measure of regression dependence and study its properties. Beside being monotone in the new RDOs, the measure takes on its extreme values precisely at independence and almost sure functional dependence, respectively. A consistent nonparametric estimator of the new measure is constructed and its asymptotic properties are investigated. Finally, the flnite sample properties of the estimate are studied by means of small simulation study. 1. Introduction and motivation. There is an extensive body of literature on the problem of ordering and measuring the dependence of two random variables. Almost all of the research in this area is concerned with the concept of positive dependence. Orders of positive dependence were considered by many authors, e.g., Lehmann [16], Esary et al. [7] and Schriever [24]; see also Scarsini and Shaked [23] for a detailed survey. Axiomatic approaches to orders and measures of positive dependence were introduced by Kimeldorf and Sampson [14] and Scarsini [22], respectively. The abundance of notions of positive dependence contrasts, however, with the silence concerning regression dependence, with the exception of the work of Da , browska [3, 4] and the measure suggested by Hall [12]. This paper presents a new approach to the problem of ordering and measuring regression dependence in the bivariate case. The terms \order" and \ordering" are used in the sense of a preorder, i.e., a re∞exive and transitive relation. We drop the requirement of antisymmetry in order to allow for an arbitrary functional form of the regression. For convenience, an order for random variables and the corresponding relations for distributions and distribution functions are used synonymously. Also, we do not strictly dis

In this paper, we show that some ageing classes of a random time T related to the mean residual life are preserved by the discrete random count variable N(T), where {N(t) : t ⩾0}is a renewal process independent from T under suitable conditions. In the particular case of the Poisson process, we extend the results to more reliability classes. We also consider real examples of N(T) and apply the results to queuing systems. Copyright © 2011 John Wiley & Sons, Ltd.

In applications that require cost efficiency, sample sizes are typically small so that the problem of empty strata may often occur in judgment poststratification (JPS), an important variant of balanced ranked set sampling. In this article, we consider estimation of population cumulative distribution functions (CDF) from JPS samples with empty strata. In the literature, the standard and restricted CDF estimators (Stokes and Sager, 1988, Journal of the American Statistical Association 83, 374381; Frey and Ozturk, 2011, Annals of the Institute of Statistical Mathematics, to appear) do not perform well when simply ignoring empty strata. In this article, we show that the original isotonized estimator (Ozturk, 2007, Journal of Nonparametric Statistics 19, 131-144) can handle empty strata automatically through two methods, MinMax and MaxMin. However, blindly using them can result in undesirable results in either tail of the CDF. We thoroughly examine MinMax and MaxMin and find interesting results about their behaviors and performance in the presence of empty strata. Motivated by these results, we propose modified isotonized estimators to improve estimation efficiency. Through simulation and empirical studies, we show that our estimators work well in different regions of the CDF, and also improve the overall performance of estimating the whole function.

We give necessary and sufficient conditions based on signatures to obtain distribution-free stochastic ordering properties for coherent systems with exchangeable components. Specifically, we consider the stochastic, the hazard (failure) rate, the reversed hazard rate, and the likelihood ratio orders. We apply these results to obtain stochastic ordering properties for all the coherent systems with five or less exchangeable components. Our results extend some preceding results. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

In this article, we study reliability properties of m-consecutive-k-out-of-n: F systems with exchangeable components. We deduce exact formulae and recurrence relations for the signature of the system. Closed form expressions for the survival function and the lifetime distribution as a mixture of the distribution of order statistics are established as well. These representations facilitate the computation of several reliability characteristics of the system for a given exchangeable joint distribution or survival function. Finally, we provide signature-based stochastic ordering results for the system's lifetime and investigate the IFR preservation property under the formulation of m-consecutive-k-out-of-n: F systems. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

A generalization of the Gerber–Shiu function proposed by (Cheung et al., Scand. Actuarial J., in press, 2010) is used to derive some ordering properties for certain ruin-related quantities in a Sparre Andersen type risk model. Additional bounds and/or refinements can be obtained by further assuming that the claim size and the interclaim time distributions possess certain reliability properties. Finally, numerical examples are considered to compare the exact solution to the bounds. Copyright © 2010 John Wiley & Sons, Ltd.

Information delays exist in an inventory system when it takes time to collect, process, validate, and transmit inventory/demand data. A general framework is developed in this paper to describe information flows in an inventory system with information delays. We characterize the sufficient statistics for making optimal decisions. When the ordering cost is linear, the optimality of a state-dependent base-stock policy is established even when information flows are allowed to cross over time. Additional insights into the problem are obtained via a comparison between our models and the models with stochastic order lead times. We also show that inventory can substitute for information and vice versa.

In this paper, we provide a stochastic ordering of the Studentized range statistics under a balanced one-way ANOVA model. Based on this result we show that, when restricted to the multiple comparisons with the best, the Newman-Keuls (NK) procedure strongly controls experimentwise error rate for a sequence of null hypotheses regarding the number of largest treatment means. In other words, the NK procedure provides an upper confidence bound for the number of best treatments.

Many assessment instruments used in the evaluation of toxicity, safety, pain, or disease progression consider multiple ordinal endpoints to fully capture the presence and severity of treatment effects. Contingency tables underlying these correlated responses are often sparse and imbalanced, rendering asymptotic results unreliable or model fitting prohibitively complex without overly simplistic assumptions on the marginal and joint distribution. Instead of a modeling approach, we look at stochastic order and marginal inhomogeneity as an expression or manifestation of a treatment effect under much weaker assumptions. Often, endpoints are grouped together into physiological domains or by the body function they describe. We derive tests based on these subgroups, which might supplement or replace the individual endpoint analysis because they are more powerful. The permutation or bootstrap distribution is used throughout to obtain global, subgroup, and individual significance levels as they naturally incorporate the correlation among endpoints. We provide a theorem that establishes a connection between marginal homogeneity and the stronger exchangeability assumption under the permutation approach. Multiplicity adjustments for the individual endpoints are obtained via stepdown procedures, while subgroup significance levels are adjusted via the full closed testing procedure. The proposed methodology is illustrated using a collection of 25 correlated ordinal endpoints, grouped into six domains, to evaluate toxicity of a chemical compound. © 2008, The International Biometric Society.

Following a review of the basic ideas in structural reliability, including signature-based representation and preservation theorems for systems whose components have independent and identically distributed (i.i.d.) lifetimes, extensions that apply to the comparison of coherent systems of different sizes, and stochastic mixtures of them, are obtained. It is then shown that these results may be extended to vectors of exchangeable random lifetimes. In particular, for arbitrary systems of sizes m < n with exchangeable component lifetimes, it is shown that the distribution of an m-component system's lifetime can be written as a mixture of the distributions of k-out-of-n systems. When the system has n components, the vector of coefficients in this mixture representation is precisely the signature of the system defined in Samaniego, IEEE Trans Reliabil R–34 (1985) 69–72. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008

For the analysis of square contingency tables with ordered categories, Caussinus (1965) considered the quasi-symmetry (QS) model, Goodman (1979) considered the diagonals-parameter symmetry (DPS) model, and Agresti (1983) considered the linear diagonals-parameter symmetry (LDPS) model. These models show the structures of symmetry for cell probabilities. Tomizawa (1993) proposed another DPS model which has a similar multiplicative form for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (>i) or above. This paper proposes another LDPS and QS models that have the corresponding similar multiplicative forms for cumulative probabilities instead of cell probabilities. Special cases of the proposed models include symmetry. Two kinds of unaided distance vision data and endometrial cancer data are analyzed using these models. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study a class of new scheduling problems which involve types of teamwork tasks. Each teamwork task consists of several components, and requires a team of processors to complete, with each team member to process a particular component of the task. Once the processor completes its work on the task, it will be available immediately to work on the next task regardless of whether the other components of the last task have been completed or not. Thus, the processors in a team neither have to start, nor have to finish, at the same time as they process a task. A task is completed only when all of its components have been processed. The problem is to find an optimal schedule to process all tasks, under a given objective measure. We consider both deterministic and stochastic models. For the deterministic model, we find that the optimal schedule exhibits the pattern that all processors must adopt the same sequence to process the tasks, even under a general objective function GC = F(f1(C1), f2(C2), … , fn(Cn)), where fi(Ci) is a general, nondecreasing function of the completion time Ci of task i. We show that the optimal sequence to minimize the maximum cost MC = max fi(Ci) can be derived by a simple rule if there exists an order f1(t) ≤ … ≤ fn(t) for all t between the functions {fi(t)}. We further show that the optimal sequence to minimize the total cost TC = ∑ fi(Ci) can be constructed by a dynamic programming algorithm. For the stochastic model, we study three optimization criteria: (A) almost sure minimization; (B) stochastic ordering; and (C) expected cost minimization. For criterion (A), we show that the results for the corresponding deterministic model can be easily generalized. However, stochastic problems with criteria (B) and (C) become quite difficult. Conditions under which the optimal solutions can be found for these two criteria are derived. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004

Summary. Considering the choice between two experiments E and F, an experimenter may choose one or the other depending on the optimality criteria. However, sometimes by using the observations from E, he or she can do at least as well as by using the observations from F for every decision problem, and therefore for every inference problem as well. When that happens, it is said that experiment E is ‘always better than’F or equivalently that E is ‘sufficient for’F. The paper provides a simple explanation of what is meant by this phrase and presents a variety of situations in which one experiment E is known to be always better than an alternative F. In addition, simplifying connections between various results are also revealed. Even though these issues are important to the design of statistical experiments and to the concept of statistical information, the literature reviewed here has largely failed in communicating its results across to many researchers in these areas. One of the objectives is to fill that gap, by stressing the implications of the results, while omitting most of the technicalities that are required in their proofs.

A number of conventional measures of risk as real-valued functions on the space of positive random variables are considered: the expected shortfall, the mean excess over the threshold, the stop-loss and some others. Ordering of risks, based on these measures and the distances between corresponding distribution functions, are described. The perturbed measures, describing the effect of changing environment, are discussed. These measures are defined by the accelerated life and proportional hazards models widely used in reliability and survival analysis. The case of a random environment is of a prime interest in the paper. The main result states that if, for instance, the stochastic environment is ‘neutral in expectation’ with respect to the baseline one, the distance between the corresponding distribution functions can be still sufficiently large. Copyright © 2003 John Wiley & Sons, Ltd.