A generalized Hollander-Proschan type test for NBUE alternatives

In this note we develop a family of test statistics for testing exponentiality against NBUE alternatives. The asymptotic distribution of the test statistics is derived. The test statistics are shown to be asymptotically normal and consistent. This family of test statistics includes the test proposed by Hollander and Proschan (1975) as a special case. Efficiency studies have also been done.     

Reliability of a system under two types of failures using a Markovian arrival process

We consider a system subject to external and internal failures. The operational time has a phase-type distribution (PH-distribution). Failures arrive following a Markovian arrival process (MAP). Some failures require the replacement of the system, and others a minimal repair. This model extends previous papers with arrivals governed by PH-renewal processes.     

A multivariate extension of Sarhan and Balakrishnan’s bivariate distribution and its ageing and dependence properties

A new class of bivariate distributions (NBD) was recently introduced by Sarhan and Balakrishnan [A.M. Sarhan, N. Balakrishnan, A new class of bivariate distributions and its mixture, J. Multivariate Anal. 98 (2007) 1508-1527]. In this note, we give the joint survival function of a multivariate extension of the NBD, which is not an absolutely continuous multivariate distribution, and its marginal and extreme order statistics distributions are also derived. The multivariate ageing and dependence properties of the proposed n-dimensional distribution are also discussed, and then we analyze the stochastic ageing of its marginals and its minimum and maximum order statistics.     

Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations

The concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression for this extension as a difference of weighted means of the structure function values. We then derive a formula for the computation of the coefficients of these weighted means in the special case of independent continuous lifetimes. Finally, we interpret this extended concept of signature through a natural least squares approximation problem.     

Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments

Let F$F$ be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by F$F$. The expansion is based on an expansion for the right Wiener–Hopf factor which we derive first. An application to ruin probabilities is developed.     

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.     

Smooth depth contours characterize the underlying distribution

The Tukey depth is an innovative concept in multivariate data analysis. It can be utilized to extend the univariate order concept and advantages to a multivariate setting. While it is still an open question as to whether the depth contours uniquely determine the underlying distribution, some positive answers have been provided. We extend these results to distributions with smooth depth contours, with elliptically symmetric distributions as special cases. The key ingredient of our proofs is the well-known Cramér-Wold theorem.     

Successive direction extraction for estimating the central subspace in a multiple-index regression

In this paper we propose a dimension reduction method for estimating the directions in a multiple-index regression based on information extraction. This extends the recent work of Yin and Cook [X. Yin, R.D. Cook, Direction estimation in single-index regression, Biometrika 92 (2005) 371-384] who introduced the method and used it to estimate the direction in a single-index regression. While a formal extension seems conceptually straightforward, there is a fundamentally new aspect of our extension: We are able to show that, under the assumption of elliptical predictors, the estimation of multiple-index regressions can be decomposed into successive single-index estimation problems. This significantly reduces the computational complexity, because the nonparametric procedure involves only a one-dimensional search at each stage. In addition, we developed a permutation test to assist in estimating the dimension of a multiple-index regression.     

Stationary-excess operator and convex stochastic orders

The present paper aims to point out how the stationary-excess operator and its iterates transform s-convex stochastic orders and the associated moment spaces. This allows us to propose a new unified method on constructing s-convex extrema for distributions that are known to be t-monotone. Both discrete and continuous cases are investigated. Several extremal distributions under monotonicity conditions are derived. They are illustrated with some applications in insurance.     

Global connectivity and optimization on an infinite step distribution

This paper provides an example of a 2-dimensional distribution that does not satisfy Chow?s bracket generating condition, but is horizontally connected. We prove the global connectivity by non-holonomic geodesics. Explicit solutions for geodesics are obtained.     

Asymptotic normality of test statistics under alternative hypotheses

The aim of this paper is to present a framework for asymptotic analysis of likelihood ratio and minimum discrepancy test statistics. First order asymptotics are presented in a general framework under minimal regularity conditions and for not necessarily nested models. In particular, these asymptotics give sufficient and in a sense necessary conditions for asymptotic normality of test statistics under alternative hypotheses. Second order asymptotics, and their implications for bias corrections, are also discussed in a somewhat informal manner. As an example, asymptotics of test statistics in the analysis of covariance structures are discussed in detail.     

Estimating the inter-arrival time density of Markov renewal processes under structural assumptions on the transition distribution

We consider a stationary Markov renewal process whose inter-arrival time density depends multiplicatively on the distance between the past and present state of the embedded chain. This is appropriate when the jump size is governed by influences that accumulate over time. Then we can construct an estimator for the inter-arrival time density that has the parametric rate of convergence. The estimator is a local von Mises statistic. The result carries over to the corresponding semi-Markov process.     

Asymptotic theory for the multidimensional random on-line nearest-neighbour graph

The on-line nearest-neighbour graph on a sequence of n$n$ uniform random points in (0,1)^d${(0,1)}^d$ (d∈N$d\in \mathbb{N}$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this graph, with weight exponent α∈(0,d/2]$\alpha \in (0,d/2]$, we prove O(max{n^1−(2α/d),logn})$O(max\{n^{1-(2\alpha /d)},logn\})$ upper bounds on the variance. On the other hand, we give an n→∞$n\to \infty$ large-sample convergence result for the total power-weighted edge-length when α>d/2$\alpha >d/2$. We prove corresponding results when the underlying point set is a Poisson process of intensity n$n$.     

On skew-Hadamard matrices

Skew-Hadamard matrices are of special interest due to their use, among others, in constructing orthogonal designs. In this paper, we give a survey on the existence and equivalence of skew-Hadamard matrices. In addition, we present some new skew-Hadamard matrices of order 52 and improve the known lower bound on the number of the skew-Hadamard matrices of this order.     

Asymptotic robustness of standard errors in multilevel structural equation models

Data in social and behavioral sciences are often hierarchically organized. Multilevel statistical methodology was developed to analyze such data. Most of the procedures for analyzing multilevel data are derived from maximum likelihood based on the normal distribution assumption. Standard errors for parameter estimates in these procedures are obtained from the corresponding information matrix. Because practical data typically contain heterogeneous marginal skewnesses and kurtoses, this paper studies how nonnormally distributed data affect the standard errors of parameter estimates in a two-level structural equation model. Specifically, we study how skewness and kurtosis in one level affect standard errors of parameter estimates within its level and outside its level. We also show that, parallel to asymptotic robustness theory in conventional factor analysis, conditions exist for asymptotic robustness of standard errors in a multilevel factor analysis model.     

An explicit example of a maximal 3-cyclically monotone operator with bizarre properties

Subdifferential operators of proper convex lower semicontinuous functions and, more generally, maximal monotone operators are ubiquitous in optimization and nonsmooth analysis. In between these two classes of operators are the maximal n$n$-cyclically monotone operators. These operators were carefully studied by Asplund, who obtained a complete characterization within the class of positive semidefinite (not necessarily symmetric) matrices, and by Voisei, who presented extension theorems à la Minty.     

On signature-based expressions of system reliability

The concept of signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. This concept proved to be useful in the analysis of theoretical behaviors of systems. In particular, it provides an interesting signature-based representation of the system reliability in terms of reliabilities of k-out-of-n systems. In the non-i.i.d. case, we show that, at any time, this representation still holds true for every coherent system if and only if the component states are exchangeable. We also discuss conditions for obtaining an alternative representation of the system reliability in which the signature is replaced by its non-i.i.d. extension. Finally, we discuss conditions for the system reliability to have both representations.     

The optimal value and optimal solutions of the proximal average of convex functions

The proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection. This paper analyzes the dependence of the optimal value and the minimizers of the proximal average on the weighting parameter. Concavity of the optimal value is established and implies further regularity properties of the optimal value. Boundedness, outer semicontinuity, single-valuedness, continuity, and Lipschitz continuity of the minimizer mapping are concluded under various assumptions. Sharp minimizers are given further attention. Several examples are given to illustrate our results.     

On the weighted multivariate Wilcoxon rank regression estimate

Zhou (2010) introduced a multivariate Wilcoxon regression estimate which possesses some nice properties: computational ease, asymptotic normality and high efficiency. However, it is sensitive to the leverage points. To circumvent this problem, we propose a weighted multivariate Wilcoxon regression estimate. Under some regularity conditions, the asymptotic normality is established. We further study the robustness of the proposed estimate through the influence function. By properly choosing the weight functions, our results show that the corresponding estimate can have bounded influence function on both response and covariates.