Stochastic order relations among parallel systems from Weibull distributions

In this article, we focus on stochastic orders to compare the magnitudes of two parallel systems from Weibull distributions when one set of scale parameters majorizes the other. The new results obtained here extend some of those proved by Dykstra et al. (1997) and Joo and Mi (2010) from exponential to Weibull distributions. Also, we present some results for parallel systems from multiple-outlier Weibull models.Comment: 14 pages, 3 figures. Journal of Applied Probability (2015

On stochastic comparisons of largest order statistics in the scale model

Let $X_{\lambda _{1}},X_{\lambda _{2}},\ldots ,X_{\lambda _{n}}$ be independent nonnegative random variables with $X_{\lambda _{i}}\sim F(\lambda _{i}t)$, $i=1,\ldots ,n$, where $\lambda _{i}>0$, $i=1,\ldots ,n$ and $F$ is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic $X_{n:n}^{\lambda }$ is smaller than another one $X_{n:n}^{\theta }$ according to likelihood ratio ordering. Furthermore, we apply these results when $F$ is a generalized gamma distribution which includes Weibull, gamma and exponential random variables as special cases

On Residual Lifetimes of k-out-of-n Systems With Nonidentical Components

In this article, mixture representations of survival functions of residual lifetimes of k-out-of-n systems are obtained when the components are independent but not necessarily identically distributed. Then we stochastically compare the residual lifetimes of k-out-of-n systems in one- and two-sample problems. In particular, the results extend some results in Li and Zhao [14], Khaledi and Shaked [13], Sadegh [17], Gurler and Bairamov [7] and Navarro, Balakrishnan, and Samaniego [16]. Applications in the proportional hazard rates model are presented as well

Dependence Among Order Statistics for Time-transformed Exponential Models

Let (X1, . . . ,Xn) be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions and with identical marginals. Let for 1 ≤ i ≤ n, Xi:n denote the corresponding ith order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is proved that for m = 2, . . . , n, the dependence of X2:m on X1:m is more than that of X2 on X1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that (X1:m,X2:m) is more concordant than (X1,X2). It will be interesting to examine whether these results can be extended to other exchangeable models

Some Unified Results on Comparing Linear Combinations of Independent Gamma Random Variables

In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9]

Dependence Among Order Statistics for time-transformed exponential models

Let X1, ..., Xn be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions. Let for 1 i n, Xi:n denote the corresponding ith-order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is in particular proved that for m = 2, ..., n, the dependence of X2:m on X1:m is more than that of X2 on X1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that X1:m, X2:mº is more concordant than X1, X2. It will be interesting to examine whether these results can be extended to other exchangeable models

Dependence Among Spacings

In this paper, we study the dependence properties of spacings. It is proved that if X1,..., Xn are exchangeable random variables which are TP2 in pairs and their joint density is log-convex in each argument, then the spacings are MTP2 dependent. Next, we consider the case of independent but nonhomogeneous exponential random variables. It is shown that in this case, in general, the spacings are not MTP2 dependent. However, in the case of a single outlier when all except one parameters are equal, the spacings are shown to be MTP2 dependent and, hence, they are associated. A consequence of this result is that in this case, the variances of the order statistics are increasing. It is also proved that in the case of the multiple-outliers model, all consecutive pairs of spacings are TP2 dependent

Data from: Statistical Inference for Multimodal Travel Time Reliability

Travel time reliability is a key metric of interest to practitioners and researchers because it affects travel choice and the economic competitiveness of urban areas. This research focuses on three travel time reliability metrics – buffer index, modified buffer index, and the relative width of travel time distributions. The key novel contributions of this research include using the multivariate delta method to prove that the sampling distributions of the three travel time reliability metrics are asymptotically normal. The asymptotic standard error for the three reliability metrics is derived. The asymptotic normality and the standard error result are used to arrive at a formula for the confidence interval. The derivations are non-parametric since they do not impose any shape requirement on travel time distributions. A case study based on a highway corridor in Portland, OR, is utilized to estimate confidence intervals for the three travel time reliability metrics for several travel time distributions with different shapes and levels of skewness. The performance of the proposed method is compared against several bootstrapping-based confidence intervals with favorable performance. Finally, upper-tailed, lower-tailed, and two-tailed one-sample hypothesis testing procedures are developed, and numerical tests show a positive performance and high statistical power for sample sizes that can be readily obtained

Statistical Inference for Multimodal Travel Time Reliability

Travel time reliability is a key metric of interest to practitioners and researchers because it affects travel choice and the economic competitiveness of urban areas. This research focuses on three travel time reliability metrics – buffer index, modified buffer index, and the relative width of travel time distributions. The key novel contributions of this research include using the multivariate delta method to prove that the sampling distributions of the three travel time reliability metrics are asymptotically normal. The asymptotic standard error for the three reliability metrics is derived. The asymptotic normality and the standard error result are used to arrive at a formula for the confidence interval. The derivations are non-parametric since they do not impose any shape requirement on travel time distributions. A case study based on a highway corridor in Portland, OR, is utilized to estimate confidence intervals for the three travel time reliability metrics for several travel time distributions with different shapes and levels of skewness. The performance of the proposed method is compared against several bootstrapping-based confidence intervals with favorable performance. Finally, upper-tailed, lower-tailed, and two-tailed one-sample hypothesis testing procedures are developed, and numerical tests show a positive performance and high statistical power for sample sizes that can be readily obtained

Estimation of a Monotone Mean Residual Life

In survival analysis and in the analysis of life tables an important biometric function of interest is the life expectancy at age x,M(x), defined by M(x)=E[X?x|X\u3ex], where X is a lifetime. M is called the mean residual life function. In many applications it is reasonable to assume that M is decreasing (DMRL) or increasing (IMRL); we write decreasing (increasing) for nonincreasing (non-decreasing). There is some literature on empirical estimators of M and their properties. Although tests for a monotone M are discussed in the literature, we are not aware of any estimators of M under these order restrictions. In this paper we initiate a study of such estimation. Our projection type estimators are shown to be strongly uniformly consistent on compact intervals, and they are shown to be asymptotically root-n equivalent in probability to the (unrestricted) empirical estimator when M is strictly monotone. Thus the monotonicity is obtained free of charge , at least in the aymptotic sense. We also consider the nonparametric maximum likelihood estimators. They do not exist for the IMRL case. They do exist for the DMRL case, but we have found the solutions to be too complex to be evaluated efficiently