Sharp Negative Bounds On Differences of Small Quantiles and Means of Distributions from Nonparametric Families

For the distributions with decreasing density and decreasing failure rate, and decreasing density and decreasing failure rate on the average, the quantiles are less than the mean if their orders do not exceed fixed levels. We determine the sharp negative upper bounds on the differences of the small quantiles and means for the distributions from the respective families in terms of scale units generated by pth absolute central moments. We show that the bounds amount trivially to zero when p > 1, and are strictly negative for p = 1.     

On Mixture Failure Rates Ordering

Mixtures of increasing failure rate distributions (IFR) can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. This is due to the fact that the mixture failure rate is “bent down” compared with the corresponding unconditional expectation of the baseline failure rate, which was proved previously for some specific cases. We generalize this result and discuss the “weakest populations are dying first” property, which leads to the change in the failure rate shape. We also consider the problem of mixture failure rate ordering for the ordered mixing distributions. Two types of stochastic ordering are analyzed: ordering in the likelihood ratio sense and ordering in variances when the means are equal.     

Projection bounds on expectations of generalized order statistics from DD and DDA families

We present sharp upper mean-variance bounds for expectations of generalized order statistics based on distributions coming from restricted families of distributions. Two families are considered: distributions with decreasing density and with density decreasing on the average. The bounds are derived by application of the projection method.     

Sharp upper mean-variance bounds for trimmed means from restricted families

We apply the method of projection to derive sharp upper mean-variance bounds on trimmed means of order statistics of i.i.d. samples coming from restricted families of probability measures. Two families are considered: distributions with decreasing density and decreasing failure rate. We also present some numerical results.     

Another Proof of Borel's Strong Law of Large Numbers

Let X be a continuous nonnegative random variable with finite first and second moments and a continuous pdf that is positive on the interior of its support. A nonzero limiting density at the origin and a coefficient of variation (CV) greater than 1 are shown to be sufficient conditions for the distribution truncated below at t > 0 to have a variance greater than the variance of the full distribution. Distributions that satisfy these conditions include those with decreasing hazard rates (e.g., the gamma and Weibull distributions with shape parameters less than 1) and the beta distribution with parameter values p and q for which q > p(p + q + 1). The bound T for which truncation at 0 < t < T increases the variance relative to the full distribution is shown to be greater than the (1 — 1/CV)th percentile of the full distribution.     

The Gamma Distribution as a Mixture of Exponential Distributions

A gamma distribution with arbitrary scale parameter θ and shape parameter r < 1 can be represented as a scale mixture of exponential distributions.     

Improved Asymptotics of a Decreasing Mean Residual Life Estimator

The mean residual life of a life distribution, X, with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. Kochar et al. (2000 Kochar, S.C., Mukerjee, H., Samaniego, F.J. (2000). Estimation of a monotone mean residual life. Ann. Stat. 28: 905–921.[Crossref], [Web of Science ®] , [Google Scholar]) provided an estimator of M when it is assumed to be decreasing. They showed that its asymptotic distribution was the same as that of the empirical estimate, but only under very stringent analytic and distributional assumptions. We provide a more general asymptotic theory, and under much weaker conditions. We also provide improved asymptotic confidence bands.     

Some Reliability Properties of the Inactivity Time

In this article, it is shown that, apart from the expected inactivity time, the second-order moment of inactivity time determines the distribution uniquely. We also show that any one partial moment, fractional or integral, of inactivity time uniquely determines a distribution. Different properties of the inactivity time of the components of a parallel system, at the system level, have been studied. Stochastic comparisons between two parallel systems based on this is also studied here.     

On the increasing failure rate criteria

We introduce the t₀ -increasing failure rate (IFR-t₀) class, where the failure rate at age x+t₀is greater then or equal to the failure rate at age x for x≥0. The dual class of t₀- decreasing failure rate (DER-t₀) is defined analogoualy.The relation between the IFR -t₀class and other classes of life distributions is studied. Preservation and nonpreservation properties of the IFR-t₀ and the DFR-t₀ classes under various reliablity operation are presented. The concet of stochastic comparison is utilized to cheracterized the IFR-t ₀ class and to suggest other classes of life distributions for aging.     

Refined solution to upper-bound problem for the expectations of order statistics from decreasing density on the average distributions

Rychlik [Metrika 77, 539–557, 2014] described sharp upper negative bounds for the expectations of low-rank order statistics, centered about the population mean and measured in the mean absolute deviation from the mean units, for the i.i.d. sequences with common distribution possessing decreasing density function on the average. The bounds coincide with the negatives of maximal values of complicated functions on the unit interval. Here, we provide more precise solutions to the maximization problems.     

Estimating the proportion of true null hypotheses, with application to DNA microarray data

Summary.  We consider the problem of estimating the proportion of true null hypotheses, π ₀, in a multiple-hypothesis set-up. The tests are based on observed p -values. We first review published estimators based on the estimator that was suggested by Schweder and Spjøtvoll. Then we derive new estimators based on nonparametric maximum likelihood estimation of the p -value density, restricting to decreasing and convex decreasing densities. The estimators of π ₀ are all derived under the assumption of independent test statistics. Their performance under dependence is investigated in a simulation study. We find that the estimators are relatively robust with respect to the assumption of independence and work well also for test statistics with moderate dependence.     

Some new classes of multivariate survival distribution functions

The concept of the univariate mean remaining life (m.r.l.) function is generalized to the multivariate case. The multivariate mean remaining life (m.m.r.l.) function is utilized to introduce four new classes of multivariate survival distribution functions (s.d.f.'s). Each of these classes is a new generalization of the univariate decreasing mean remaining life (DMRL) class of s.d.f.'s. The duals of these classes are introduced. Some properties, physical interpretation, and relationships among these classes are investigated. Also for each case, the class of s.d.f.'s common in a class and its dual is characterized.     

DETERMINATION OF CHANGE POINTS OF NON-MONOTONIC FAILURE RATES

This paper was motivated by the problem of the determination of the change points of the failure rate of a mixture of two gamma distributions. For certain values of the parameters the existing methods are not applicable since, in this case, there are two turning points of the failure rate. Thus, we extend the results to models having two or more turning points of the failure rates. The extended procedure helps us to identify failure rates of more complex forms. Finally, the mixture gamma case is completely resolved employing theoretical, graphical and numerical techniques wherever necessary.     

Tests of independence for censored bivariate failure time data

Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of chi2-type tests for independence between pairs of failure times after adjusting for covariates. A bivariate accelerated failure time model is proposed for the joint distribution of bivariate failure times while leaving the dependence structures for related failure times completely unspecified. Theoretical properties of the proposed tests are derived and variance estimates of the test statistics are obtained using a resampling technique. Simulation studies show that the proposed tests are appropriate for practical use. Two examples including the study of infection in catheters for patients on dialysis and the diabetic retinopathy study are also given to illustrate the methodology.     

On a redundant repairable system with switching failure: Bayesian approach

System characteristics of a redundant repairable system are studied from a Bayesian viewpoint with different types of priors assumed for the unknown parameters. The system consists of two primary units, one standby unit, and one repair facility which is activated when switching to standby fails. Times to failure and times to repair of the operating units are assumed to follow exponential distributions. When time to failure and time to repair have uncertain parameters, a Bayesian approach is adopted to evaluate system characteristics. Monte Carlo simulation is used to derive the posterior distribution for the mean time to system failure and steady-state availability. Some numerical experiments are performed to illustrate the results derived in this paper.     

A Proportional Odds Beta-Binomial Model for Evaluating the Effect of Treatment in Cross-Over Studies with Baseline Covariates: An Application to Condom Failure Data

Cross-over trials with correlated Bernoulli outcomes are common designs. In condom functionality studies, for example, an indicator of condom failure is reported for each sex act using standard or experimental condoms. Two popular analysis methods for such data are Generalized Estimating Equations and logit-normal random effects models. An alternative random effects model, the beta-binomial, is commonly used in contexts involving only between-cluster effects. The flexibility of the beta distribution and the interpretation of random effects as cluster-specific failure probabilities make it appealing, and we consider an extension of the model to account for within-cluster treatment effects using proportional odds assumptions.     

An overview of tests for exponentiality

Developments since 1960 in goodness-of-fit tests for the one and two parameter exponential models using both complete and censored samples are reviewed. Special attention is given to both the omnibus or general alternative and to specialized alter-natives such as the class of distributions with increasing failure rates. The use of transformations in developing tests is also discussed.     

Copula-based reliability analysis for a parallel system with a cold standby

The traditional reliability models cannot well reflect the effect of performance dependence of subsystems on the reliability of system, and neglect the problems of initial reliability and standby redundancy. In this paper, the reliability of a parallel system with active multicomponents and a single cold-standby unit has been investigated. The simultaneously working components are dependent and the dependence is expressed by a copula function. Based on the theories of conditional probability, the explicit expressions for the reliability and the MTTF of the system, in terms of the copula function and marginal lifetime distributions, are obtained. Let the copula function be the FGM copula and the marginal lifetime distribution be exponential distribution, a system with two parallel dependent units and a single cold-standby unit is taken as an example. The effect of different degrees of dependence among components on system reliability is analyzed, and the system reliability can be expressed as the linear combination of exponential reliability functions with different failure rates. For investigating how the degree of dependence affects the mean lifetime, furthermore, the parallel system with a single cold standby, comprising different number of active components, is also presented. The effectiveness of the modeling method is verified, and the method presented provides a theoretical basis for reliability design of engineering systems and physics of failure.     

Characterizations and ordering properties based on log-odds functions

In this paper, we obtain some general results on characterizations of probability distributions from relationships between conditional moment, failure rate, and log-odds rate functions. We also study stochastic orders and classes based on the log-odds rate function and some relationships with usual stochastic orderings and classes. Some characterizations and ordering properties are obtained by using weighted distributions.