Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.     

Stochastic comparisons of capital allocations with applications

This paper studies capital allocation problems using a general loss function. Stochastic comparisons are conducted for general loss functions in several scenarios: independent and identically distributed risks; independent but non-identically distributed risks; comonotonic risks. Applications in optimal capital allocations and policy limits allocations are discussed as well.     

On maximum order statistics from heterogeneous geometric variables

Let X ₁,X ₂ be independent geometric random variables with parameters p ₁,p ₂, respectively, and Y ₁,Y ₂ be i.i.d. geometric random variables with common parameter p. It is shown that X _2:2, the maximum order statistic from X ₁,X ₂, is larger than Y _2:2, the second order statistic from Y ₁,Y ₂, in terms of the hazard rate order [usual stochastic order] if and only if $p\geq \tilde{p}$ , where $\tilde{p}=(p_{1}p_{2})^{\frac{1}{2}}$ is the geometric mean of (p ₁,p ₂). This result answers an open problem proposed recently by Mao and Hu (Probab. Eng. Inf. Sci. 24:245–262, 2010) for the case when n=2.     

SOME COMPARISONS BETWEEN GENERALIZED ORDER STATISTICS

Some stochastic comparisons of generalized order statistics under the right spread order,the location independent riskier order and the total time transform order are investigated in this paper.The underlying distributions and parameters on which generalized order statistics are based are also surveyed to obtain the conditions for increasing the expectations of spacings between the first two generalized order statistics and between the last two generalized order statistics.     

On the skewness of order statistics with applications

Order statistics from heterogeneous samples have been extensively studied in the literature. However, most of the work focused on the effect of heterogeneity on the magnitude or dispersion of order statistics. In this paper, we study the skewness of order statistics from heterogeneous samples according to star ordering. The main results extend the corresponding results in Kochar and Xu (J. Appl. Probab. 46:342–352, 2009; J. Appl. Probab. 48:271–284, 2011). Examples and applications are highlighted as well.     

Unimodality of uniform generalized order statistics,with applications to mean bounds

We prove that uniform generalized order statistics are unimodal for an arbitrary choice of model parameters. The result is applied to establish optimal lower and upper bounds on the expectations of generalized order statistics based on nonnegative samples in the population mean unit of measurement. The bounds are attained by two-point distributions.     

Stochastic properties of INID progressive Type-II censored order statistics

Let X_1:n≤X_2:n≤?≤X_n:n denote the order statistics of random variables X₁,X₂,…,X_n which are independent but not necessarily identically distributed (INID), and let K₁,K₂ be two integer-valued random variables, independent of {X₁,…,X_n}, such that 1≤K₁≤K₂≤n. It is shown that if K₁ has a log-concave probability function and SI(K₂|K₁) then RTI(X_K2:n|X_K1:n), and if K₂ has a log-concave probability function and SI(K₁|K₂) then LTD(X_K1:n|X_K2:n), where SI, RTI and LTD are three notions of bivariate positive dependence. Based on these, we obtain that RTI and LTD whenever 1≤i<j≤m, where are progressive Type-II censored order statistics from INID random variables {X₁,…,X_n}. Furthermore, one result concerning the likelihood ratio ordering of the progressive Type-II censored order statistics is also given.     

Stochastic equivalence of ordered random variables with applications in reliability theory

This paper presents some conditions for stochastic equality of (univariate or multivariate) random variables under various stochastic orderings. The main results provide generalizations of several known results. Applications to stochastic equivalence and characterization problems associated with multivariate NBU (NBUE) distributions are also included.     

Sequential order statistics with an order statistics prior

In the model of sequential order statistics, prior distributions are considered for the model parameters, which, for example, describe increasing load put on remaining components. Gamma priors are examined as well as priors out of a class of extended truncated Erlang distributions (ETED), which is introduced along with some properties. The choice of independent priors in both set-ups leads to respective independent, conjugate posterior distributions for the model parameters of sequential order statistics. Since, in practical applications, the model parameters will often be increasingly ordered, a multivariate prior is applied being the joint distribution of common ETED-order statistics. Whatever baseline distribution of the sequential order statistics is chosen, the joint posterior distribution turns out to be a Weinman multivariate exponential distribution. Posterior moments are given explicitly, and HPD credible sets for the model parameters are stated.     

Stochastic comparisons in multivariate mixed model of proportional reversed hazard rate with applications

This paper studies the multivariate mixed proportional reversed hazard rate model having dependent mixing variables. Stochastic comparison as well as aging properties in this model are investigated, and stochastic monotone properties of the population vector with respect to the mixing vector are also discussed. Moreover, MTP₂ dependence among the mixing vectors is proved to imply the increasingness of the reversed hazard rate with respect to the baseline one. Finally, some interesting applications are presented as well.     

Stochastic comparisons of lifetimes of series and parallel systems with dependent and heterogeneous components

This work considers stochastic comparisons of lifetimes of series and parallel systems with dependent and heterogeneous components having lifetimes following the proportional odds (PO) model. The joint distribution of component lifetimes is modeled by Archimedean survival copula. We discuss some potential applications of our findings in system reliability and actuarial science.     

The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let _ηt${\eta }_t$ be a Poisson point process of intensity t≥1$t\ge 1$ on some state space Y$\mathbb{Y}$ and let f$f$ be a non-negative symmetric function on Y^k${\mathbb{Y}}^k$ for some k≥1$k\ge 1$. Applying f$f$ to all k$k$-tuples of distinct points of _ηt${\eta }_t$ generates a point process _ξt${\xi }_t$ on the positive real half-axis. The scaling limit of _ξt${\xi }_t$ as t$t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m$m$-th smallest point of _ξt${\xi }_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k$k$-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.     

The Fourier transform of order statistics with applications to Lorentz spaces

We present a formula for the Fourier transforms of order statistics in ℝ ⁿ showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in ℝ ⁿ . Fora ₁≥...≥a _n≥0 andq>0, denote by ℓ _w,q ⁿ then-dimensional Lorentz space with the norm ‖(x ₁,...,x _n)‖=(a ₁(x ₁ ^* ) ^q +...+a _n(x _n ^* ) ^q )^1/q , where (x ₁ ^* ,...,x _n ^* ) is the non-increasing permutation of the numbers |x ₁|,...,|x _n|. We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces intoL _q [10] to prove that, forn≥3 andq≤1, the space ℓ _w,q ⁿ is isometric to a subspace ofL _q if and only if the numbersa ₁,...,a _n form an arithmetic progression. Forq>1, all the numbersa _i must be equal so that ℓ _w,q ⁿ = ℓ _q ⁿ . Consequently, the Lorentz function spaceL _w,q(0, 1) is isometric to a subspace ofL _q if and only ifeither 0<q<∞ and the weightw is a constant function (so thatL _w,q=L_q),or q≤1 andw(t) is a decreasing linear function. Finally, we relate our results to the theory of positive definite functions. Both authors were supported in part by the NSF Workshop in Linear Analysis and Probability held at Texas A&M University in August 1993. The work was done during the first author’s visit to Texas A&M University.     

Mixture representation for order statistics from INID progressive censoring and its applications

In this paper, a mixture representation for the joint distribution function of progressively Type-II censored order statistics from heterogeneous distributions is established. Applications of this representation to stochastic orderings and inequalities are then illustrated.     

Joint asymptotic multinormality for a class of rank order statistics in multivariate paired comparisons

The joint asymptotic multinormality of certain linear signed-rank statistics introduced by Shane and Puri (1969) is established for the nonidentically distributed case; moreover, the usual restriction forbidding constant score generating functions is dropped. In addition, sufficient conditions more general than those of Shane and Puri are given for the convergence of certain dispersion matrices, and these conditions guarantee the asymptotic independence of the statistics under consideration.     

Quality of service statistics over heterogeneous networks: Analysis and applications

Heterogeneous wireless/wired networks and ubiquitous environments are gaining ever more attention by research community. To properly control and manage such puzzles a deep knowledge of quality of service parameters is needed and, therefore, a complete and robust performance assessment is necessary. This paper deals with a performance evaluation and measurement of a number of heterogeneous end-to-end paths taking into account a wide range of statistics. To study the behavior of QoS parameters, an active measurement approach has been introduced for the analysis of properties we called (i) concise statistics (mean, standard deviation, inter quantile range, minimum, maximum, and median) and (ii) detailed statistics (Probability Density Function, Auto-correlation Function, Entropy, Complementary Cumulative Distribution Function, and Bivariate Probability Density Function). We show how, thanks to this view on QoS statistics, a more complete understanding of QoS parameters behavior is possible. In addition, we show how the measured statistics can be fruitfully used in the context of network control and management. More precisely, we present two proof of concepts regarding frameworks for QoS-based anomaly detection and for QoS-based identification of network elements.