Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We discuss joint temporal and contemporaneous aggregation of N$N$ independent copies of AR(1) process with random-coefficient a∈[0,1)$a\in [0,1)$ when N$N$ and time scale n$n$ increase at different rate. Assuming that a$a$ has a density, regularly varying at a=1$a=1$ with exponent −1<β<1$-1<\beta <1$, different joint limits of normalized aggregated partial sums are shown to exist when N^1/(1+β)/n$N^{1/(1+\beta )}/n$ tends to (i) ∞$\infty$, (ii) 0$0$, (iii) 0<μ<∞$0<\mu <\infty$. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R)$(0,\infty )\times C\left(\mathbb{R}\right)$ and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).     

Quantile inference for nonstationary processes with infinite variance innovations

Based on the quantile regression,we extend Koenker and Xiao (2004) and Ling and McAleer (2004)'s works from finite-variance innovations to infinite-variance inn...     

Smoothed estimates for models with random coefficients and infinite variance innovations

Infinite variance processes have attracted growing interest in recent years due to its applications in many areas of statistics (see [1] and references therein). For example, ARIMA time-series models with infinite variance innovations are widely used in financial modelling. However, a little attention has been paid to incorporate infinite variance innovations for time-series models with random coefficients introduced by [2]. This paper considers the problem of nonparametric estimation for some time-series models using the smoothed least absolute deviation (SLAD) estimating function approach. We introduce a class of kernels in order to smooth the LAD estimators. It is also shown that this new SLAD estimators are superior than some existing ones.     

Parameter estimation for a misspecified arma model with infinite variance innovations

We consider a general linear model , where the innovations Z_t belong to the domain of attraction of an α-stable law for α<2, so that neither Z_t nor X_t have a finite variance. We do not assume that (X_t) is a standardARMA process of the form φ(B)X_t=ϕ(B)Z_t, but we fit anARMA process of a given order to the data X₁,...,X_n by estimating the coefficients of φ and ϕ. Given that (X_t) is anARMA process, it has been proved that the Whittle estimator is a consistent estimator of the true coefficients of ϕ and φ. Moreover, it then has a heavytailed limit distribution and the rate of convergence is (n/logn)^1/α, which compares favorably with the L² situation with rate . In this note we study the limit properties of the Whittle estimator when the underlying model is not necessarily anARMA process. Under general conditions we show that the Whittle estimate converges in probability. It converges weakly to a distribution which does not have a finite moment of order a and the rate of convergence is again (n/logn)^1/α. We also give an analytic expression for the limit distribution. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process

Let e _t=(e _t1,...e _tp) be a p-dimensional nonnegative strict white noise with finite second moments. Let h _ij(x) be nondecreasing functions from [0,) onto [0,) such that h _ij(x) x for i, j = 1,...,p. Let U = (u _ij) be a p×p matrix with nonnegative elements having all its roots inside the unit circle. Define a process X t=(X _t1,...,X _tp) by for

Limiting mixture distributions for AR(1) model indexed by a branching process

First order autoregressive model indexed by a supercritical Galton–Watson branching process is discussed. Limiting distributions of the least squares estimates are derived both for the stationary and explosive cases. It is shown that a certain random variable inherent in the branching process is acting as a mixing variable in limiting mixture distributions. In particular, with explosive Gaussian case, we obtain a mixture of Cauchy distributions rather than Cauchy.     

Trimmed stable AR(1) processes

In this paper we investigate the distribution of trimmed sums of dependent observations with heavy tails. We consider the case of autoregressive processes of order one with independent innovations in the domain of attraction of a stable law. We show if the $d$ largest (in magnitude) terms are removed from the sample, then the sum of the remaining elements satisfies a functional central limit theorem with random centering provided $d=d\left(n\right)\ge n^{\gamma }$ (for some $\gamma >0$) and $d\left(n\right)/n\to 0$. This result is used to get asymptotics for the widely used CUSUM process in case of dependent heavy tailed observations.     

On AR(1) models with periodic and almost periodic coefficients

This paper is concerned with the existence of bounded solutions to the system of equations X_n=a_nX_n−1+ξ_n, nZ, where ξ_n are uncorrelated constant variance zero mean random variables. We give necessary and sufficient conditions for boundedness in the general case and then specifically for periodic and almost periodic (a_n). This provides the first step in extending the periodic autoregressive models, for which boundedness is equivalent to the stationarity of the blocked vector sequence

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to the almost periodic case.     

Discrete time parametric models with long memory and infinite variance

We study a large class of infinite variance time series that display long memory. They can be represented as linear processes (infinite order moving averages) with coefficients that decay slowly to zero and with innovations that are in the domain of attraction of a stable distribution with index 1 < α < 2 (stable fractional ARIMA is a particular example). Assume that the coefficients of the linear process depend on an unknown parameter vector β which is to be estimated from a series of length n. We show that a Whittle-type estimator β_n for β is consistent (β_n converges to the true value β₀ in probability as n → ∞), and, under some additional conditions, we characterize the limiting distribution of the rescaled differences (n/logn)^1/ga(β_n − β₀).     

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.     

具有AR(1)误差的均值漂移模型的Score检验和似然比检验

讨论了具有AR(1)误差的线性均值漂移模型,研究了自相关性的检验问题,导出了关于误差相关性的Score检验统计量和似然比检验统计量,并把它推广到误差项为AR(1)非线性均值漂移模型.本文还给出了一个数值例子说明检验方法的实用性.     

Jumps in binomial AR(1) processes

We consider the binomial AR(1) model for serially dependent processes of binomial counts. After a review of its definition and known properties, we investigate marginal and serial properties of jumps in such processes. Based on these results, we propose the jumps control chart for monitoring a binomial AR(1) process. We show how to evaluate the performance of this control chart and give design recommendations.     

Inference in binomial AR(1) models

This paper studies inference methods for stationary time series with binomial distributions. Such series describe, for example, the number of rainy days in consecutive weeks. First, we formulate the renewal sequence version of the model that seemingly generates a new class of stationary binomial series. The model is shown to obey an AR(1) recursion in cases where the renewal lifetime has a constant hazard rate past lag one. Explicit asymptotic variances of the parameter estimators in the AR(1) case are derived from conditional least squares methods; likelihood techniques are also considered.     

Properties of spectral covariance for linear processes with infinite variance

We consider a measure of dependence for symmetric α-stable random vectors, which was introduced by the second author in 1976. We demonstrate that this measure of dependence, which we suggest to call the spectral covariance, can be extended to random vectors in the domain of normal attraction of general stable vectors. We investigate the asymptotic of the spectral covariance function for linear stable (Ornstein–Uhlenbeck, log-fractional, linear-fractional) processes with infinite variance and show that, in comparison with the results on the properties of codifference of these processes, obtained two decades ago, the results for the spectral variance are obtained under more general conditions and calculations are simpler.     

On convergence of LAD estimates in autoregression with infinite variance

The least absolute deviation estimates L(N), from N data points, of the autoregressive constants a = (a₁, …, a_q)′ for a stationary autoregressive model, are shown to have the property that N^σ(L(N) ? a) converge to zero in probability, for $\text{σ <}\text{1}\text{α}$, where the disturbances are i.i.d., attracted to a stable law of index α, 1 ≤ α < 2, and satisfy some other conditions.