Abstract : In this paper a large class of multivariate densities and frequency functions including the multivariate Poisson distribution and the compound multivariate Poisson distribution, are shown to have the decreasing in transposition property introduced by Hollander, Proschan, and Sethuraman (1977, Ann. Statist. 5, 722-733). Applications relevant to reliability are given, including applications to shock models, cumulation of damage, and component down times. (Author)
Hardy, Littlewood and Pólya [5] introduced the partial ordering of majorization among n-dimensional real vectors. Many well-known inequalities can be recast as the statement that certain functions are increasing with respect to this ordering. Such functions are said to be Schur-convex. An important result in the theory of majorization is the Schur–Ostrowski theorem, which characterizes Schur-convex functions. The concept of majorization has been extended to elements of $L_1 (0,1)$ by Ryff [10]. A functional on $L_1 (0,1)$ that is increasing with respect to the ordering of majorization is said to be Schur-convex. In this paper, we prove an analogue of the Schur–Ostrowski condition that characterizes Schur-convex functionals in terms of their Gâteaux differentials. We also introduce another partial ordering in $L_1 (0,1)$ called unrestricted majorization. This partial ordering is similar to majorization but does not involve the use of decreasing rearrangements. We establish a characterization of nondecreasing functionals on $L_1 (0,1)$ with respect to the partial ordering of unrestricted majorization through another analogue of the Schur–Ostrowski condition.
Abstract In this paper we consider the problem of inference from accelerated life tests in which a common parametric family of life distributions at the different stress levels is not specified in advance. By assuming a time transformation function, which is a version of the familiar inverse power law, we give a procedure for testing hypotheses that the unknown distributions are specified members of a location-scale family. The behavior of the test procedure is indicated by some simulations. We illustrate the methodology with some real-life data.
A sequence of independent random variables $\{X_1,X_2,\ldots\}$ is called a $B-$harmonic Bernoulli sequence if $P(X_i=1)=1-P(X_i=0) = 1/(i+B)\ i=1,2,\ldots$, with $B\ge 0$. For $k\ge 1$, the count variable $Z_k$ is the number of occurrences of the $k$-string $(1,\protect\underbrace{0,\ldots,0}_{k-1},1)$\vadjust{\vspace*{-2pt}} in the Bernoulli sequence\ldots\$. This paper gives the joint distribution $P_B$ of the count vector ${\bf Z} = (Z_1,Z_2,\ldots)$ of strings of all lengths in a $B-$harmonic Bernoulli sequence. This distribution can be described as follows. There is random variable $V$ with a Beta$(B,1)$ distribution, and given $V=v$, the conditional distribution of ${\bf Z}$ is that of independent Poissons with intensities $(1 -v),\ (1 - v^2)/2,\ (1-v^3)/3, \ldots$. Around 1996, Persi Diaconis stated and proved that when $B=0$, the distribution of $Z_1$ is Poisson with intensity $1$. Emery gave an alternative proof a few months later. For the case $B=0$, it was also recognized that $Z_1,Z_2,\ldots,Z_n$ are independent Poissons with intensities $1, \frac{1}{2},\ldots, \frac{1}{n}$. Proofs up until this time made use of hard combinational techniques. A few years later, Joffe et al, obtained the marginal distribution of $Z_1$ as a Beta-Poisson mixture when $B\geq 0$. Their proof recognizes an underlying inhomogeneous Markov chain and uses moment generating functions. In this note, we give a compact expression for the joint factorial moment of\break $(Z_1,\ldots,Z_N)$ which leads to the joint distribution given above. One might feel that if $Z_1$ is large, it will exhaust the number of $1$'s in the Bernoulli sequence $(X_1,X_2,\ldots)$ and this in turn would favor smaller values for $Z_2$ and introduce some negative dependence. We show that, on the contrary, the joint distribution of ${\bf Z}$ is positively associated or possesses the FKG property.
Arrangement increasing and Schur functions play a central role in establishing stochastic inequalities in several areas of statistics and reliability.The role of a module in the failure of a system measures the importance of the module.We define the role to be the probability that this module is among the modules that failed before the failure of the system.A system is called a second order r-out-of-& system if it is a r-out-of-ib system based on k modules, without common components, and where each module is an α,-out-of-n, system.For such systems, we show that the role of a module is an arrangement increasing or Schur function of parameters that describe the system.These results allow us to compare the role of a module under different values of the parameters of the system.
We consider several classical notions of partial orderings among life distributions which have been used to describe ageing properties and tail domination. We show that if a distribution G dominates another distribution F in one of these partial orderings introduced here, and if two moments of G agree with those of F , including the moment that describes this partial ordering, then G = F. This leads to a characterization of the exponential distribution among HNBUE and HNWUE life distribution classes, and thus extends the results of Basu and Bhattacharjee (1984) and rectifies an error in that paper.
The construction and analysis of repair models is an important area in reliability. A commonly used model is the minimal repair model. Under this model, repair restores the state of the system to its level prior to failure. Kijima introduced repair models that could be classified as "better-than-minimal." Under Kijima's models, the system, upon repair, is functionally the same as a working system of lesser age which has never experienced failure. In this paper, we present a new approach to the modeling of better-than-minimal repair models. Using this approach, we construct a general repair model that contains Kijima's models as special cases. We also study the problem of estimating the distribution of the time to first failure of a system maintained by general repair. We make use of counting processes to show strong consistency of the estimator and prove results on weak convergence. Finally, we derive a Hall-Wellner type asymptotic confidence band for the distribution of the time to first failure of the system.
Abstract : The Markov Chain Chain Monte Carlo (MCMC) method, which is a special case of the Gibbs sampler, is a very powerful method to simulate from complicated distributions arising in many contexts, including image analysis, computational Bayesian analysis, and so on. Existing results that ensure that this method will converge involve conditions which are difficult to verify in practice, and most practitioners, convinced that their particular problem will not be pathological and give up verifying altogether. This paper gives a new set of sufficient conditions which are easy to verify in most applications.
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