Characterizations of exponentiality within the HNBUE class and related tests

A harmonic new better than used in expectation (HNBUE) variable is a random variable which is dominated by an exponential distribution in the convex stochastic order. We use a recently obtained condition on stochastic equality under convex domination to derive characterizations of the exponential distribution and bounds for HNBUE variables based on the mean values of the order statistics of the variable. We apply the results to generate discrepancy measures to test if a random variable is exponential against the alternative that is HNBUE, but not exponential.     

The Harris Extended Exponential Distribution

A new lifetime distribution is proposed and studied. The Harris extended exponential is obtained from a mixture of the exponential and Harris distributions, which arises from a branching process. Several structural properties of the new distribution are discussed, including moments, generating function and order statistics. The new distribution can model data with increasing or decreasing failure rate. The shape of the hazard rate function is controlled by one of the added parameters in an uncomplicated manner. An application to a real dataset illustrates the usefulness of the new distribution.     

Multivariate Matsumoto–Yor property is rather restrictive

Matsumoto and Yor [2001. An analogue of Pitman's 2M-X$2M-X$ theorem for exponential Wiener functionals. Part II: the role of the GIG laws. Nagoya Math. J. 162, 65–86] discovered an interesting invariance property of a product of the generalized inverse Gaussian (GIG) and the gamma distributions. For univariate random variables or symmetric positive definite random matrices it is a characteristic property for this pair of distributions. It appears that for random vectors the Matsumoto–Yor property characterizes only very special families of multivariate GIG and gamma distributions: components of the respective random vectors are grouped into independent subvectors, each subvector having linearly dependent components. This complements the version of the multivariate Matsumoto–Yor property on trees and related characterization obtained in Massam and Weso?owski [2004. The Matsumoto–Yor property on trees. Bernoulli 10, 685–700].     

Nonparametric estimation of the conditional distribution function in a semiparametric censorship model

In this paper we propose a new nonparametric estimator of the conditional distribution function under a semiparametric censorship model. We establish an asymptotic representation of the estimator as a sum of iid random variables, balanced by some kernel weights. This representation is used for obtaining large sample results such as the rate of uniform convergence of the estimator, or its limit distributional law. We prove that the new estimator outperforms the conditional Kaplan–Meier estimator for censored data, in the sense that it exhibits lower asymptotic variance. Illustration through real data analysis is provided.     

Nonparametric estimation of conditional expectation

Denote the integer lattice points in the N  -dimensional Euclidean space by Z^N${\mathbb{Z}}^N$ and assume that (_Xi,_Yi)$(X_{\mathbf{i}},Y_{\mathbf{i}})$, i∈Z^N$\mathbf{i}\in {\mathbb{Z}}^N$ is a mixing random field. Estimators of the conditional expectation r(x)=E[_Yi|_Xi=x]$r(\mathbf{x})=\mathrm{E}[Y_{\mathbf{i}}|X_{\mathbf{i}}=\mathbf{x}]$ by nearest neighbor methods are established and investigated. The main analytical result of this study is that, under general mixing assumptions, the estimators considered are asymptotically normal. Many difficulties arise since points in higher dimensional space N?2$N?2$ cannot be linearly ordered. Our result applies to many situations where parametric methods cannot be adopted with confidence.     

IG-symmetry and R-symmetry: Interrelations and applications to the inverse Gaussian theory

The two parameter inverse Gaussian (IG) distribution is often more appropriate and convenient for modelling and analysis of nonnegative right skewed data than the better known and now ubiquitous Gaussian distribution. Its convenience stems from its analytic simplicity and the striking similarities of its methodologies with those employed with the normal theory models. These, known as the G–IG analogies, include the concepts and measures of IG-symmetry, IG-skewness and IG-kurtosis, the IG-analogues of the corresponding classical notions and measures. The new IG-associated entities, although well defined and mathematically transparent, are intuitively and conceptually opaque. In this paper, we first elaborate the importance of the IG distribution and of the G–IG analogies. Then we consider the IG-related root-reciprocal IG (RRIG) distribution and introduce a physically transparent, conceptually clear notion of reciprocal symmetry (R-symmetry) and use it to explain the IG-symmetry. We study the moments and mixture properties of the R-symmetric distributions and the relationship of R-symmetry with IG-symmetry and note that RRIG distribution provides a link, in addition to Tweedie's Laplace transform link, between the Gaussian and inverse Gaussian distributions. We also give a structural characterization of the unimodal R-symmetric distributions. This work further expands the long list of G–IG analogies. Several applications including product convolution, monotonicity of power functions, peakedness and monotone limit theorems of R-symmetry are outlined.     

Flexible univariate and multivariate models based on hidden truncation

A broad spectrum of flexible univariate and multivariate models can be constructed by using a hidden truncation paradigm. Such models can be viewed as being characterized by a basic marginal density, a family of conditional densities and a specified hidden truncation point, or points. The resulting class of distributions includes the basic marginal density as a special case (or as a limiting case), but also includes an array of models that may unexpectedly include many well known densities. Most of the well known skew-normal models (developed from the seed distribution popularized by Azzalini [(1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12(2), 171–178]) can be viewed as being products of such a hidden truncation construction. However, the many hidden truncation models with non-normal component densities undoubtedly deserve further attention.     

A generalization of the multivariate slash distribution

In this paper, we propose a generalization of the multivariate slash distribution and investigate some of its properties. We show that the new distribution belongs to the elliptically contoured distributions family, and can have heavier tails than the multivariate slash distribution. Therefore, this generalization of the multivariate slash distribution can be considered as an alternative heavy-tailed distribution for modeling data sets in a variety of settings. We apply the generalized multivariate slash distribution to two real data sets to provide some illustrative examples.     

Comparing relative importance of the same component in a series system in two environments

The relative performance of a component of a series system in two different environments is considered. The conditional probability of the failure of the system due to the failure of the specified component given that the system failed before time t is regarded as a measure of relative importance of the component to the system. A U-statistic test for checking the equality of the relative importance of the component to the system in two different environments against the alternative that the relative importance is smaller in one of the environments, is proposed. Some simulation results for estimating the power of the test are reported. The proposed test is applied to one real data set and it is seen that a different aspect of the data is brought out by this comparison than that by the comparisons of the absolute importance functions such as the subsurvival functions, considered in earlier studies.     

Characterizations of geometric distribution and discrete IFR (DFR) distributions using order statistics

Let X be a discrete random variable the set of possible values (finite or infinite) of which can be arranged as an increasing sequence of real numbers a₁<a₂<a₃<…. In particular, a_i could be equal to i for all i. Let X_1n≦X_2n≦?≦X_nn denote the order statistics in a random sample of size n drawn from the distribution of X, where n is a fixed integer ≧2. Then, we show that for some arbitrary fixed k(2≦k≦n), independence of the event {X_kn=X_1n} and X_1n is equivalent to X being either degenerate or geometric. We also show that the montonicity in i of P{X_kn = X_1n | X_1n = a_i} is equivalent to X having the IFR (DFR) property. Let a_i = i and $\text{G(i) = P(X≧i), i = 1, 2, …}$. We prove that the independence of {X_2n ? X_1n ∈B} and X_1n for all i is equivalent to X being geometric, where B = {m} (B = {m,m+1,…}), provided G(i) = q^i?1, 1≦i≦m+2 (1≦i≦m+1), where 0<q<1.     

Some properties of the generalized TTT transform

Recently Li and Shaked [2007. A general family of univariate stochastic orders. J. Statist. Plann. Inference 137, 3601–3610] introduced the generalized total time on test (GTTT) transform with respect to a given function ?$?$. In this paper we study some properties of it which are related with stochastic orderings. A concept of Lehmann and Rojo [1992. Invariant directional orderings. Ann. Statist. 20, 2100–2110] is applied to a new setting and the GTTT transform is used to define invariance properties and distances of some stochastic orders. Iterations of the GTTT transforms are also studied and their relations with exponential mixtures of gamma distributions are established.     

Stochastic comparisons of multivariate frailty models

A multivariate frailty model in which survival function depends on baseline distributions of components and the frailty random variable is considered. Since misspecification in choice of frailty distribution and/or baseline distribution may affect the distribution of multivariate frailty model, using theory of stochastic orders, we compare multivariate frailty models arising from different choices of frailty distribution.     

Ordering conditional lifetimes of coherent systems

Consider a system of n components that has the property that there exists a number r  (r<n)$(r<n)$, such that if it is known that at most r components have failed, the system is still functioning with probability 1. Suppose that such a system is equipped with a warning light that comes up at the time of the failure of the rth component. The system is still working then, and we are interested in its residual life. In this paper we obtain some results which stochastically compare the residual lives of such systems with the same type, or with different types, of components. Some applications are given. In particular, we derive upper and lower bounds on the expected residual lives of such systems given that the warning light has not come up yet, and given that the component hazard rate functions are bounded from below or from above by a known constant.     

Negative dependence in frailty models

This note investigates the negative dependence in frailty models. First, we show that the frailty variable and the overall population variable are negatively likelihood ratio dependent and derive an upper bound for the survival function of the population with higher frailty. Secondly, we prove that the DFR property and the logconvex hazard rate of the baseline variable imply the DLR property of the population variable. Finally, we further prove that the likelihood ratio order, hazard rate order and reversed hazard rate order between two frailty variables imply the likelihood ratio order, reversed hazard rate order, and hazard rate order between the corresponding overall population variables, respectively.     

Characterizations based on record values and order statistics

Consider a sequence of independent and identically distributed random variables {_Xi,i?1}$\{X_i,i?1\}$ with a common absolutely continuous distribution function F  . Let _X1:n?_X2:n???_Xn:n$X_{1:n}?X_{2:n}???X_{n:n}$ be the order statistics of {_X1,_X2,…,_Xn}$\{X_1,X_2,\dots ,X_n\}$ and {_Yl,l?1}$\{Y_l,l?1\}$ be the sequence of record values generated by {_Xi,i?1}$\{X_i,i?1\}$. In this work, the conditional distribution of _Yl$Y_l$ given _Xn:n$X_{n:n}$ is established. Some characterizations of F   based on record values and _Xn:n$X_{n:n}$ are then given.     

On the uth geometric conditional quantile

Motivated by Chaudhuri's work [1996. On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91, 862–872] on unconditional geometric quantiles, we explore the asymptotic properties of sample geometric conditional quantiles, defined through kernel functions, in high-dimensional spaces. We establish a Bahadur-type linear representation for the geometric conditional quantile estimator and obtain the convergence rate for the corresponding remainder term. From this, asymptotic normality including bias on the estimated geometric conditional quantile is derived. Based on these results, we propose confidence ellipsoids for multivariate conditional quantiles. The methodology is illustrated via data analysis and a Monte Carlo study.     

Ordering conditional general coherent systems with exchangeable components

This paper investigates some ordering properties of the residual lives and the inactivity times of coherent systems with dependent exchangeable absolutely continuous components, based on the stochastically ordered signatures between systems, extending the results of Li and Zhang [2008. Some stochastic comparisons of conditional coherent systems. Applied Stochastic Models in Business and Industry 24, 541–549] for the case of independent and identically distributed components.