Occupancy numbers for dynamic heterogeneous populations: Estimate of particles lifetimes

A dynamic model of a heterogeneous population is studied. Particles belonging to a population are divided, at every time t, into a finite number of classes according to their types and the partition changes over time. The role of the occupancy numbers, namely the cardinality of each class, is highlighted. The relationship between the stochastic process of occupancy numbers and the process of particle types is analyzed. The main goal of this paper is the estimation of the lifetime of each particle at a given time t, when the observed data are the history of the process of the number of dead particles up to t. Furthermore, a discrete time approximation of the filter is given.     

Extremal limit theorems for observations separated by random power law waiting times

This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.     

A note on replacement policy comparisons from NBUC lifetime of the unit

Age and block replacement policies are commonly used in order to reduce the number of in-service failures when the systems are functioning indefinitely. In reliability theory, the lifetime of a system can be modeled by means of the NBUC aging class that is characterized throughout comparisons of the residual lives in the sense of the icx order. The purpose of this paper is to establish stochastic comparisons between the age (block) replacement policy and a renewal process with no planned replacements when the lifetime of the unit is NBUC. Supported by Ministerio de Ciencia y Tecnología under grant BFM2000-0362     

Construction of non-Gaussian random fields with any given correlation structure

A positive definite function can be thought of as the covariance function of a Gaussian random field, according to the celebrated Kolmogorov existence theorem. A question of great theoretical and practical interest is: how could one construct a non-Gaussian random field with the given positive definite function as its covariance function? In this paper we demonstrate a novel and simple method for constructing many such non-Gaussian random fields, with the corresponding finite-dimensional distributions identified. Also, we show how to construct a non-Gaussian random field with a given negative definite function as its variogram.     

Convex rearrangements,generalized Lorenz curves,and correlated Gaussian data

We propose a statistical index for measuring the fluctuations of a stochastic process ξ$\xi$. This index is based on the generalized Lorenz curves and (modified) Gini indices of econometric theory.     

Non-linear models 1

This is the first application of a new method for testing stationary random point processes. Consider the class of all stationary ergodic point processes on the real line with arbitrary dependences among the inter–point distances (spacing).The hypothesis is :The observed process φ is a homogeneous Poisson process or more (resp.less) regular than a Poisson process.The sample is the vector of the first n points t₁, …,t_n.There is a close relation between our method for testing and queueing theory: For finding an appropriate test statistic, we observe the behaviour of a single server queue with the input φ.A table of critical values is given.     

Geostochastic calculus

A stochastic calculus for a family of continuous measure-valued Markov processes is developed. Such processes arise naturally in the construction of stochastic models of spatially distributed populations. The stochastic calculus is a tool whereby a class of density-dependent models can be studied in terms of the multiplicative measure diffusion process. In this paper the stochastic integral is introduced in the space-time setting and a Cameron-Martin-Girsanov theorem is established.     

Some observations on the KMT dyadic scheme

We point out some useful properties of the KMT dyadic scheme, which was used to construct a Brownian bridge approximation to the uniform empirical process.     

Testing for changes in the covariance structure of linear processes

We consider several procedures to detect changes in the mean or the covariance structure of a linear process. The tests are based on the weighted CUSUM process. The limit distributions of the test statistics are derived under the no change null hypothesis. We develop new strong and weak approximations for the sample mean as well as the sample correlations of linear processes. A small Monte Carlo simulation illustrates the applicability of our results.     

On Vervaat and Vervaat-error-type processes for partial sums and renewals

We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space D[0,1]$D[0,1]$. On the other hand, we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the deviations of these processes, the so-called Vervaat-error-type processes.     

Transient probability functions of finite birth–death processes with catastrophes

A representation of the transient probability functions of finite birth–death processes (with or without catastrophes) as a linear combination of exponential functions is derived using a recursive, Cayley–Hamilton approach. This method of solution allows practitioners to solve for these transient probability functions by reducing the problem to three calculations: determining eigenvalues of the Q$Q$-matrix, raising the Q$Q$-matrix to an integer power and solving a system of linear equations. The approach avoids Laplace transforms and permits solution of a particular transition probability function from state i$i$ to j$j$ without determining all such functions.     

A generalized drop-the-loser rule for multi-treatment clinical trials

Urn models are popular for response adaptive designs in clinical studies. Among different urn models, Ivanova's drop-the-loser rule is capable of producing superior adaptive treatment allocation schemes. Ivanova [2003. A play-the-winner-type urn model with reduced variability. Metrika 58, 1–13] obtained the asymptotic normality only for two treatments. Recently, Zhang et al. [2007. Generalized drop-the-loser urn for clinical trials with delayed responses. Statist. Sinica, in press] extended the drop-the-loser rule to tackle more general circumstances. However, their discussion is also limited to only two treatments. In this paper, the drop-the-loser rule is generalized to multi-treatment clinical trials, and delayed responses are allowed. Moreover, the rule can be used to target any desired pre-specified allocation proportion. Asymptotic properties, including strong consistency and asymptotic normality, are also established for general multi-treatment cases.     

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 volatility modelling in continuous time with general marginal distributions: Inference,prediction and model selection

We compare results for stochastic volatility models where the underlying volatility process having generalized inverse Gaussian (GIG) and tempered stable marginal laws. We use a continuous time stochastic volatility model where the volatility follows an Ornstein–Uhlenbeck stochastic differential equation driven by a Lévy process. A model for long-range dependence is also considered, its merit and practical relevance discussed. We find that the full GIG and a special case, the inverse gamma, marginal distributions accurately fit real data. Inference is carried out in a Bayesian framework, with computation using Markov chain Monte Carlo (MCMC). We develop an MCMC algorithm that can be used for a general marginal model.     

The tube method for the moment index in projection pursuit

The projection pursuit index defined by a sum of squares of the third and the fourth sample cumulants is known as the moment index proposed by Jones and Sibson [1987. What is projection pursuit? J. Roy. Statist. Soc. Ser. A 150, 1–36]. The limiting distribution of the maximum of the moment index under the null hypothesis that the population is multivariate normal is shown to be the maximum of a Gaussian random field with a finite Karhunen–Loève expansion. An approximate formula for tail probability of the maximum, which corresponds to the p-value, is given by virtue of the tube method through determining Weyl's invariants of all degrees and the critical radius of the index manifold of the Gaussian random field.     

On convergence of properly weighted samples to the target distribution

We consider importance sampling as well as other properly weighted samples with respect to a target distribution π$\pi$ from a different point of view. By considering the associated weights as sojourn times until the next jump, we define appropriate jump processes. When the original sample sequence forms an ergodic Markov chain, the associated jump process is an ergodic semi-Markov process with stationary distribution π$\pi$. In this respect, properly weighted samples behave very similarly to standard Markov chain Monte Carlo (MCMC) schemes in that they exhibit convergence to the target distribution as well. Indeed, some standard MCMC procedures like the Metropolis–Hastings algorithm are included in this context. Moreover, when the samples are independent and the mean weight is bounded above, we describe a slight modification in order to achieve exact (weighted) samples from the target distribution.     

Stochastic comparisons and properties of conditional generalized order statistics

Generalized order statistics introduced by Kamps [1995. A concept of generalized order statistics. J. Statist. Plann. Inference 48, 1–23] provides a unified approach to a variety of models concerning ordered random variables. This paper carries out stochastic comparisons of conditional generalized order statistics in terms of the likelihood ratio order and the hazard rate order from two samples, and establishes some stochastic monotonicity properties. The main results strengthen and generalize the corresponding results established recently in the literature. Finally, some applications of the main results are given as well.     

Estimating the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences

Discrete one-dimensional scan statistics can be viewed as extremes of 1-dependent stationary sequences. A result of Haiman [1999. First passage time for some stationary processes. Stochastic Process. Appl. 80, 231–248] provides approximations of the distributions of extremes of 1-dependent stationary sequences together with sharp bounds for the corresponding errors. We apply this result to scan statistics generated by Bernoulli r.v.'s and to the charge problem.     

Champs aléatoires gaussiens

A limit theorem concerning quadratic variation in Gaussian random fields is established. The theorem is analogous to a result of Gladyshev (1961).