Asymptotic inference for stochastic processes

This is a survey of some aspects of large-sample inference for stochastic processes. A unified framework is used to study the asymptotic properties of tests and estimators parameters in discrete-time, continuous-time jump-type, and diffusion processes. Two broad families of processes, viz, ergodic and non-ergodic type are introduced and the qualitative differences in the asymptotic results for the two families are discussed and illustrated with several examples. Some results on estimation and testing via Bayesian, nonparametric, and sequential methods are also surveyed briefly.     

Asymptotic theory of conditional inference for stochastic processes

This paper develops an approach to conditional inference for nonergodic stochastic process models by considering asymptotic properties. The context for most of the analysis is that of Le Cam's local asymptotic theory: in particular, the locally asymptotically mixed normal (LAMN) situation. An attempt has been made to evaluate local asymptotic properties of global procedures.     

Multistage weighted least squares estimation of ARCH processes in the stable and unstable cases

For an ARCH model, we propose a multistage weighted least squares (WLS) estimate which consists of repeated WLS procedures until the corresponding asymptotic variance equals that of the quasi-maximum likelihood estimate (QMLE). At every stage, the current estimate is of a WLS type weighted by the squared conditional variance evaluated at the estimate of the previous stage. Initially, the weighting parameter is any fixed and known value in the parameter space. The procedure provides, without any moment requirement, an asymptotically Gaussian estimate having the same asymptotic distribution as the QMLE even in the unstable case. Apart from the initialization stage, two additional stages are required in the stable case to obtain the same asymptotic distribution as the QMLE, while in the unstable case only one stage is enough. So in all, the proposed procedure involves three stages WLS in the stable case and two stages WLS in the unstable case.     

Asymptotic behavior of integral functionals of diffusion processes with periodic coefficients

A central limit theorem is proved for functionals of integral form and with its help is established the local asymptotic normality of the logarithm of the likelihood ratio for diffusion processes with periodic coefficients.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 245–252, February, 1992.     

Asymptotic behavior of trajectories for some nonautonomous,almost periodic processes

In this paper, asymptotics are studied for some almost periodic processes on a complete metric space (X, d): (1) It is shown that any precompact positive trajectory of a contractive periodic process is asymptotically almost periodic as t → +∞. This property does not hold for general almost periodic contractive processes. (2) A compactness result is obtained for weakly almost periodic complete trajectories of some (possibly nonlinear) processes in a uniformly convex Banach space. (3) The existence of almost periodic trajectories is studied for “affine” processes in a uniformly convex Banach space. These results are applicable to some evolution equations of the form $\text{du}\text{dt}\text{+ A (t) u(t) ? f(t)}$, where $\text{?(t)}$ is almost periodic: R → V uniformly convex Banach space and A(t) is a periodic, time-dependent, m-accretive operator in V.     

Asymptotic inference for AR(1) processes with (nonnormal) stable errors

This is the first of several papers in which we consider problems related to the asymptotic distribution of the least squares estimate of the parameter γ in theAR(1) model $$X_k = \gamma X_{k - 1} + \varepsilon _k , k = 1,...,n,$$ where ε_k are independent identically distributed (i.i.d.) random variables in the domain of attraction of a stable law. In §1 we give a summary in the case ε_k is in the domain of attraction of the normal distribution. In §2 we consider errors in the domain of attraction of a (nonnormal) stable distribution. In §3 we prove a result in the case of the completely asymmetric stable distribution with α=β=1.     

Asymptotic optimal inference for multivariate branching-Markov processes via martingale estimating functions and mixed normality

Multivariate tree-indexed Markov processes are discussed with applications. A Galton-Watson super-critical branching process is used to model the random tree-indexed process. Martingale estimating functions are used as a basic framework to discuss asymptotic properties and optimality of estimators and tests. The limit distributions of the estimators turn out to be mixtures of normals rather than normal. Also, the non-null limit distributions of standard test statistics such as Wald, Rao’s score, and likelihood ratio statistics are shown to have mixtures of non-central chi-square distributions. The models discussed in this paper belong to the local asymptotic mixed normal family. Consequently, non-standard limit results are obtained.     

Asymptotic behavior of unstable ARMA processes with application to least squares estimates of their parameters

A time series x(t), t≥1, is said to be an unstable ARMA process if x(t) satisfies an unstableARMA model such asx(t)=a_1x(t-1) a_2x(t-2) … a_8x(t-s) w(t)where w(t) is a stationary ARMA process; and the characteristic polynomial A(z)=1-a_1z-a_2z~2-…-a_3z~3 has all roots on the unit circle. Asymptotic behavior of sum form 1 to n (x~2(t)) will be studied by showing somerates of divergence of sum form 1 to n (x~2(t)). This kind of properties Will be used for getting the rates of convergenceof least squares estimates of parameters a_1, a_2,…, a_?     

Asymptotic inference for continuous-time Markov chains

Summary This paper deals with asymptotic optimal inference in a time-continuous ergodic Markov chain with countable state space, based on observation of the process up to timet. Let the infinitesimal generator depend on an unknown parameter. Under weak assumptions on the parametrization, we show local asymptotic normality for the statistical model ast. As a consequence, limit distributions of sequences of competing estimators for the unknown parameter are more spread out than a specified normal distribution.     

Inductive inference of ultimately periodic sequences

An algorithm is presented for inductive inference of ultimately periodic sequences. It can be implemented in linear time and space.Dedicated to Peter Naur on the occasion of his 60th birthday     

All periodic minimizers are unstable

We consider the periodic action functional associated to some Lagrangian verifying the Legendre convexity condition and show that all minimizers are unstable. Supported by D.G.I. MTM2005-03483, Ministerio de Educación y Ciencia, Spain.     

Isolated periodic minima are unstable

A classical result, studied, among others, by Carathéodory [C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, Chelsea, New York, 1989], says that, at least generically, periodic minimizers are hyperbolic, and consequently, unstable as solutions of the associated Euler–Lagrange equation. A new version of this fact, also valid in the nonhyperbolic case, is given.     

Queue inference from periodic reporting data

We introduce new algorithms for queue inference problems involving periodic reporting data. We assume that in each of several consecutive time periods, new customer arrivals follow a Poisson process with period-specific arrival rates, but only the number of departing customers during each period is observed. Rather than exploiting relations with order statistics, our algorithms rely on direct recursions involving Poisson probabilities. We illustrate with several examples, some of which are not typically viewed as queueing applications.     

Statistical inference in stochastic processes

(Probability: Pure and Applied, Series of Textbooks and Reference Books, vol. 6), edited by N. U. Prabhuand I. W. Basawa, Marcel Dekker, New York (1991), 288 pp. $89.75 (USA and Canda), $107.50 (All other countries). ISBN 0-8247-8417-0     

Asymptotic distinguishing of autoregressive processes

We study the behavior of the probability of errors of the Neumann-Pearson criterion under various null and alternative hypotheses by the results of observations of autoregressive processes.     

Asymptotic distinction of counting processes

A canonical representation is obtained for the logarithm of the likelihood ratio. Limit theorems describing its asymptotic behavior are proved. Using these theorems, we study the rate of decrease of the probability of an error of the second-kind in the Neyman-Pearson test.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 972–979, July, 1993.