A class of consistent tests for exponentiality based on the empirical Laplace transform

The Laplace transform (t=E[exp(–tX)]) of a random variable with exponential density exp(–x), x0, satisfies the differential equation (+t)(t)+(t=0, t0). We study the behaviour of a class of consistent (omnibus) tests for exponentiality based on a suitably weighted integral of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGBbGaaiikai% qbeU7aSzaajaWaaSbaaSqaaGqaciaa-5gaaeqaaOGaey4kaSIaamiD% aiaacMcacqaHipqEcaWFNaWaaSbaaSqaaiaad6gaaeqaaOGaaiikai% aadshacaGGPaGaey4kaSIaeqiYdK3aaSbaaSqaaiaad6gaaeqaaOGa% aiikaiaadshacaGGPaGaaiyxamaaCaaaleqabaGaaGOmaaaaaaa!4C69!\[[(\hat \lambda _n + t)\psi '_n (t) + \psi _n (t)]^2 \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBgaqcam% aaBaaaleaaieGacaWFUbaabeaaaaa!3A66!\[\hat \lambda _n \] is the maximum-likelihood-estimate of and _n is the empirical Laplace transform, each based on an i.i.d. sample X ₁,...,X _n .     

A minimal repair replacement model with two types of failure and a safety constraint

Consider a system subject to two types of failures. If the failure is of type 1, the system is minimally repaired, and a cost C₁ is incurred. If the failure is of type 2, the system is minimally repaired with probability p and replaced with probability 1−p  . The associated costs are _C2,_m$C_{2\text{,}m}$ and _C2,_r$C_{2\text{,}r}$, respectively. Failures of type 2 are safety critical and to control the risk, management has specified a requirement that the probability of at least one such failure occurring in the interval [0, A] should not exceed a fixed probability limit ω. The problem is to determine an optimal planned replacement time T, minimizing the expected discounted costs under the safety constraint. A cost C_r is incurred whenever a planned replacement is performed. Conditions are established for when the safety constraint affects the optimal replacement time and causes increased costs.     

On risk processes with the Markov property and with independent increments

We study the influence on the underlying counting process of the Markov property and of the property of independent increments for a risk process.     

A New Approach to Poisson Approximation of Simple Point Processes Using Compensators

Consider a simple point process N on the line, and let be its compensator. We use a result of Kallenberg (1990, Probab. Theory Relat. Fields 86, 167–202) to give a new approach to estimate the total variation distance between the distributions of N and that of a Poisson process when has small jump sizes.     

Position dependent and stochastic thinning of point processes

A short probabilistic proof of Kallenberg's theorem [2] on thinning of point processes is given. It is extended to the case where the probability of deletion of a point depends on the position of the point and is itself random. The proof also leads easily to a statement about the rate of convergence in Renyi's theorem on thinning a renewal process.     

Poisson过程分解问题的一个注记

在研究Poisson过程分解问题时,现有文献的证明往往令人费解,本文主要运用极限理论,给出了一个简明易懂的证明.     

Nonparametric hypothesis testing for intensity of the poisson process

We propose a goodness-of-fit test for the hypothesis that the observed Poisson point process has a given periodic intensity function against a nonparametric close alternative of known smoothness. We obtain rate and sharp asymptotics for the errors in the minimax setup.      

A note on residual allocation models

Residual allocation models (RAMs) arise in many subjects including Bayesian statistics, combinatorics, ecology, finance, information theory, machine learning, and population genetics. In this paper, we give a brief review of RAM and presents a few examples where the model arises. An extended discussion will focus a concrete model, the GEM distribution, and its ordered analogue, the Poisson-Dirichlet distribution. The paper concludes with a discussion of the GEM process.     

A martingale characterization of the set-indexed Brownian motion

We characterize a Brownian motion indexed by a semilattice of sets, using the theory of set-indexed martingales: a square integrable continuous set-indexed strong martingale is a Brownian motion if and only if its compensator is deterministic and continuous.Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.Research done while this author was visiting the University of Ottawa. He wishes to thank Professor Ivanoff for her kind hospitality.     

Some test statistics based on the martingale term of the empirical distribution function

Summary It is proved that the martingale term of the empirical distribution function converges weakly to a Gaussian process inD[0, 1]. Some statistics for goodness-of-fit tests based on the martingale term of the empirical distribution function are proposed. Asymptotic distributions of these statistics under the null hypothesis are given. The approximate Bahadur efficiencies of the statistics to the Kolmogorov-Smirnov statistic and to the Cramér-von Mises statistic are also calculated. The Institute of Statistical Mathematics     

A note on convergence in Banach spaces of cotype p

Let B be a separable Banach space. The following is one of the results proved in this paper. The Banach space B is of cotype p if and only if

1. d_n, n 1, has no subsequence converging in probability, and

2. ∑_{n 1}|a_n|^p < ∞ whenever ∑_{n 1}a_nd_n converges almost surely are equivalent for every sequence d_n, n 1, of symmetric independent random elements taking values in B.

A note on applications of the martingale central limit theorem to random permutations

A unified martingale approach is presented for establishing the asymptotic normality of some sequences of random variables. It is applied to the numbers of inversions, rises, and peaks, respectively, as well as the oscillation and the sum of consecutive pair products of a random permutation. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 323–332 (1997)     

A model of imperfect preventive maintenance with dependent failure modes

Consider a system subject to two modes of failures: maintainable and non-maintainable. A failure rate function is related to each failure mode. Whenever the system fails, a minimal repair is performed. Preventive maintenances are performed at integer multiples of a fixed period. The system is replaced when a fixed number of preventive maintenances have been completed. The preventive maintenance is imperfect because it reduces the failure rate of the maintainable failures but does not affect the failure rate of the non-maintainable failures. The two failure modes are dependent in the following way: after each preventive maintenance, the failure rate of the maintainable failures depends on the total of non-maintainable failures since the installation of the system. The problem is to determine an optimal length between successive preventive maintenances and the optimal number of preventive maintenances before the system replacement that minimize the expected cost rate. Optimal preventive maintenance schedules are obtained for non-decreasing failure rates and numerical examples for power law models are given.     

A characterization of certain diffusions on the Sierpinski gasket by the exit distributions

LetK be the Sierpinski gasket with verticesa ₁,a ₂,a ₃ forming an equilateral triangle. Suppose that (X _t ,P ^x ) and (Y _t ,Q ^x ) are diffusions onK with the same hitting probabilities to verticesa ₁,a ₂,a ₃. We show that ifX is an asymptotically one-dimensional diffusion or ap-stream diffusion, thenY is a time-change ofX.Part of this work was done while the author was an Alexander von Humboldt fellow at the Universität des Saarlandes in Saarbrücken, Germany.     

A class of strong limit theorems for countable nonhomogeneous Markov chains on the generalized gambling system

In this paper, we study the limit properties of countable nonhomogeneous Markov chains in the generalized gambling system by means of constructing compatible distributions and martingales. By allowing random selection functions to take values in arbitrary intervals, the concept of random selection is generalized. As corollaries, some strong limit theorems and the asymptotic equipartition property (AEP) theorems for countable nonhomogeneous Markov chains in the generalized gambling system are established. Some results obtained are extended.      

A note on the improved derivative free root-solvers

Using a fixed point relation of the square-root type and the basic fourth-order method, improved methods of fifth and sixth order for the simultaneous determination of simple zeros of a polynomial are obtained. An increase in convergence is achieved without additional numerical operations, which points to high computational efficiency of the accelerated methods. The main aim of this work is the convergence analysis of improved simultaneous methods given under computationally verifiable initial conditions in the spirit of Smale’s point estimation theory.     

A note on arbitrage,approximate arbitrage and the fundamental theorem of asset pricing

We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing by B. Wong and C.C. Heyde [Stochastics 82 (2010), pp. 189–200] in the context of incomplete Itô-process models. We show that their approach can only work in the known case of a complete financial market model and give an explicit counter example.     

A note on the Poisson structures of the space of probability measures

The space of probability measures on a Riemannian manifold is endowed with the Fisher information metric. In [4] T. Friedrich showed that this space admits also Poisson structures {, }. In this note, we give directly another proof for the structure {, } being Poisson. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on asymptotic expansions for sums over a weakly dependent random field with application to the Poisson and Strauss processes

Previous results on Edgeworth expansions for sums over a random field are extended to the case where the strong mixing coefficient depends not only on the distance between two sets of random variables, but also on the size of the two sets. The results are applied to the Poisson and the Strauss point processes, giving rise also to local limit results.     

A note on the total unimodularity of matrices

A necessary and sufficient characterization of totally unimodular matrices is given which is derived from a necessary condition for total unimodularity due to Camion. This characterization is then used in connection with a theorem of Hoffman and Kruskal to provide an elementary proof of the characterization of totally unimodular matrices in terms of forbidden submatrices due to Camion.