Weak approximations for quantile processes of stationary sequences

ABSTRACT By studying the deviations between uniform empirical and quantile processes (the so-called Bahadur-Kiefer representations) of a stationary sequence in properly weighted sup-norm metrics, we find a general approach to obtaining weighted results for uniform quantile processes of stationary sequences. Consequently we are able to obtain weak convergence for weighted uniform quantile processes of stationary mixing and associated sequences. Further, by studying the sup-norm distance of a general quantile process from its corresponding uniform quantile process, we find that information at the two end points of the uniform quantile process can be so utilized that this weighted sup-norm distance converges in probability to zero under the so-called Csörgõ-Révész conditions. This enables us to obtain weak convergence for weighted general quantile processes of stationary mixing and associated sequences.     

Asymptotic linearity and limit distributions,approximations

Linear and quadratic forms as well as other low degree polynomials play an important role in statistical inference. Asymptotic results and limit distributions are obtained for a class of statistics depending on μ+X$\mu +\mathbf{X}$, with X any random vector and μ$\mu$ non-random vector with ∥μ∥→+∞$\parallel \mu \parallel \to +\infty$. This class contain the polynomials in μ+X$\mu +\mathbf{X}$. An application to the case of normal X is presented. This application includes a new central limit theorem which is connected with the increase of non-centrality for samples of fixed size. Moreover upper bounds for the suprema of the differences between exact and approximate distributions and their quantiles are obtained.     

Asymptotic properties of randomly indexed sequences of random variables

Conditions are given for a randomly indexed sequence of random variables to converge weakly. The key concept employed is the so-called generalized Anscombe condition. The results give a method of determining sequential stopping rules, which have the required accuracy of estimation of an unknown parameter in the case when the observations are not necessarily independent and identically distributed.     

Weighted empirical spacings processes

We give a complete asymptotic characterization of the supremum of weighted empirical spacings processes.     

Saddlepoint approximations for subordinator processes

ABSTRACT

We develop the saddlepoint approximations in obtaining the transition functions for general subordinator processes. We derive explicit expressions of the first- and second-order approximations. Specifically, we consider some particular classes of subordinators including the Poisson processes, the Gamma processes, the α-stable subordinators, and the Poisson random integrals. We test this technique on the Poisson and Gamma processes, which have closed-form transition functions. Outcomes show that the approximate expressions are consistent with the true transition functions. We then use this method to predict transition density functions for the α-stable subordinator processes. Finally, we calculate approximated transition densities for some Poisson random integrations. Numerical analysis shows the perfect ability of the saddlepoint approximations to predict the transition densities of the α-stable processes and the Poisson random integrations.     

Strong approximations for time-varying infinite-server queues with non-renewal arrival and service processes

In real stochastic systems, the arrival and service processes may not be renewal processes. For example, in many telecommunication systems such as internet traffic where data traffic is bursty, the sequence of inter-arrival times and service times are often correlated and dependent. One way to model this non-renewal behavior is to use Markovian Arrival Processes (MAPs) and Markovian Service Processes (MSPs). MAPs and MSPs allow for inter-arrival and service times to be dependent, while providing the analytical tractability of simple Markov processes. To this end, we prove fluid and diffusion limits for MAP_t/MSP_t/∞ queues by constructing a new Poisson process representation for the queueing dynamics and leveraging strong approximations for Poisson processes. As a result, the fluid and diffusion limit theorems illuminate how the dependence structure of the arrival or service processes can affect the sample path behavior of the queueing process. Finally, our Poisson representation for MAPs and MSPs is useful for simulation purposes and may be of independent interest.     

On the asymptotic independence of numbers of observations near order statistics

In this paper, we consider the numbers of observations in two-sided neighbourhoods of the kth and (n?r)th order statistics from a sample of size n and show that they are asymptotically independent as n→∞. We also establish a result that generalizes all the existing results regarding the asymptotic independence of numbers of observations in the left and right neighbourhoods of order statistics. Finally, we consider the limiting joint behaviour of numbers of observations in the neighbourhoods of s central order statistics and establish that they are asymptotically independent.     

Formulas for the second-order analysis of marked point processes

Following Ripley, for the second moment measure of stationary isotropic marked point processes in R^destimators are given. Two examples are considered.     

Asymptotic results for random coefficient bifurcating autoregressive processes

The purpose of this paper is to study the asymptotic behaviour of the weighted least-squares estimators of the unknown parameters of random coefficient bifurcating autoregressive processes. Under suitable assumptions on the immigration and the inheritance, we establish the almost sure convergence of our estimators, as well as a quadratic strong law and central limit theorems. Our study mostly relies on limit theorems for vector-valued martingales.     

Decomposition for multivariate extremal processes

The probability distribution of an extremal process in R^d with independent max-increments is completely determined by its distribution function. The df of an extremal process is similar to the cdf of a random vector. It is a monotone function on (0, ∞) × R^d with values in the interval [0,1]. On the other hand the probability distribution of an extremal process is a probability measure on the space of sample functions. That is the space of all increasing right continuous functions y: (0, ∞) → R^d with the topology of weak convergence. A sequence of extremal processes converges in law if the probability distributions converge weakly. This is shown to be equivalent to weak convergence of the df's.

An extremal process Y: [0, ∞) → R^d is generated by a point process on the space [0, ∞) × [-∞, ∞)^d and has a decomposition Y = X v Z as the maximum of two independent extremal processes with the same lower curve as the original process. The process X is the continuous part and Z contains the fixed discontinuities of the process Y. For a real valued extremal process the decomposition is unique: for a multivariate extremal process uniqueness breaks down due to blotting.     

Dependent Lindeberg central limit theorem for the fidis of empirical processes of cluster functionals

Drees H. and Rootzén H. [Limit theorems for empirical processes of cluster functionals (EPCF). Ann Stat. 2010;38(4):2145–2186] have proven central limit theorems (CLTs) for EPCF built from β-mixing processes. However, this family of β-mixing processes is quite restrictive. We expand some of those results, for the finite-dimensional marginal distributions (fidis), to a more general dependent processes family, known as weakly dependent processes in the sense of Doukhan P. and Louhichi S. [A new weak dependence condition and applications to moment inequalities. Stoch. Proc. Appl. 1999;84:313–342]. In this context, the CLT for the fidis of EPCF is sufficient in some applications. For instance, we prove the convergence without mixing conditions of the extremogram estimator, including a small example with simulation of the extremogram of a weakly dependent random process but nonmixing, in order to confirm the efficacy of our result.     

Allocating observations in a test for the ratio of scales of independent gamma variates

Consider two independent gamma populations with common shape parameter α. Let ρ denote the ratio of their scale parameters, and consider the problem of testing the null hypothesis ρ = 1 against the alternative ρ = 1 + r , where r >0 . A procedure, not depending on α , that maximizes the exact slope at this alternative is proposed.     

Asymptotic inference in poverty indices: an empirical processes approach

We derive an empirical poverty index containing most of those proposed in the literature. Then, we study its asymptotic behavior by using empirical processes theory. From the results obtained, we derive a Wald-type test for comparing a vector of theoretical poverty indices to a vector of reference indices. We study the test statistic under the null hypothesis of equality, fixed alternatives, and a sequence of local alternatives. A simulation experiment conducted shows that our test performs well for exponential and Pareto data.     

Asymptotic results for empirical and partial-sum processes: A review

Two processes of importance in statistics and probability are the empirical and partial-sum processes. Based on d-dimensional data X₁, … X_a the empirical measure is defined for any A ⊂ R^d by the sample proportion of observations in A. When normalized, F_n yields the empirical process W_n: = n^1/2 (F_n - F), where F denotes the “true” probability measure. To define partial-sum processes, one needs data that are assigned to specified locations (in contrast to the above, where specified unit masses are assigned to random locations). A suitable context for many applications is that of data attached to points of a lattice, say {X_j:j ϵ J^d} where J = {1, 2,…}, for which the partial sums are defined for any A ⊂ R^d by Thus S(A) is the sum of the data contained in A. When normalized, S yields the partial-sum process. This paper provides an overview of asymptotic results for empirical and partial-sum processes, including strong laws and central limit theorems, together with some indications of their inferential implications.     

Central Limit Theorems of Local Polynomial Threshold Estimator for Diffusion Processes with Jumps

Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the non‐parametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the σ ‐field generated by diffusion processes is destroyed, so the classical central limit theorem for martingale difference sequences cannot work. In high‐frequency data, we proved the central limit theorems of local polynomial threshold estimators for the volatility function in diffusion processes with jumps by Jacod's stable convergence theorem. We believe that our proof procedure for local polynomial threshold estimators provides a new method in this field, especially in the local linear case.     

Estimating conditional occupation-time distributions for dependent sequences

Consider a random integer-valued process X(t) on Z+ that satisfies some weak dependence condition. We study the empirical distribution function of the occupation times of such a process and prove convergence to a suitable Gaussian process. An application to the statistical analysis of open and closed sojourn-time distributions for ion channels is provided.     

On an exponential bound for the Kaplan–Meier estimator

We review limit theory and inequalities for the Kaplan–Meier Kaplan and Meier (J Am Stat Assoc 53:457–481, 1958) product limit estimator of a survival function on the whole line . Along the way we provide bounds for the constant in an interesting inequality due to Biotouzé et al. (Ann Inst H Poincaré Probab Stat 35:735–763, 1999), and provide some numerical evidence in support of one of their conjectures. Supported in part by NSF grant DMS-0503822 and by NI-AID grant 2R01 AI291968-04.     

Markov-modulated infinite-server queues driven by a common background process

This paper studies a system with multiple infinite-server queues that are modulated by a common background process. If this background process, being modeled as a finite-state continuous-time Markov chain, is in state j, then the arrival rate into the i-th queue is λ_{i, j}, whereas the service times of customers present in this queue are exponentially distributed with mean μ^{? 1}_{i, j}; at each of the individual queues all customers present are served in parallel (thus reflecting their infinite-server nature).

Three types of results are presented: in the first place (i) we derive differential equations for the probability-generating functions corresponding to the distributions of the transient and stationary numbers of customers (jointly in all queues), then (ii) we set up recursions for the (joint) moments, and finally (iii) we establish a central limit theorem in the asymptotic regime in which the arrival rates as well as the transition rates of the background process are simultaneously growing large.