On the theory of continuous time series

This paper considers spectral estimation for a zero-mean strictly stationary r-vector valued continuous time series. The case of interest is when some of observations are missing due to some random failure. Spectral estimation procedures are developed in disjoint segments of observations. Expanded finite Fourier transform, modified poriodogram and spectral density statistics are constructed. The theoretical properties of these estimators are developed. Asymptotic distributions are discussed     

On the invertibility of time series models

A generalized definition of invertibility is proposed and applied to linear, non-linear and bilinear models. It is shown that some recently studied non-linear models are not invertible, but conditions for invertibility can be achieved for the other models.     

On the errors-in-variables problem for time series

The usual assumption in the classical errors-in-variables problem of independent measurement errors cannot necessarily be maintained when the data are time series; errors may be strongly serially correlated, possibly containing seasonal effects and trends. When it is possible to identify frequency bands over which the signal-to-noise ratio is large, an approximate solution to the errors-in-variables problem is to omit the remaining frequencies from a time series regression. We draw attention to the danger of “leakage” from the omitted frequencies, and show that the consequent bias can be reduced by means of tapering.     

On the estimation of nonlinear time series models

In the present paper, a framework for parametric estimation in nonlinear time series is developed. Strong consistency and asymptotic normality of minimum Hellinger distance estimates for a determined class of nonlinear models are investigated. The main Interest for these estimates is motivated by their robustness under perturbations as it has been emphazized in Beran [2]. The first part of the paper is devoted to the study of some probabilistic properties which ensure the existence and the optimal properties of the estimates     

On estimating the hidden periodicities in linear time series models

In this paper the tollowing modelX(n)=sum from j=1 to p α_je~(inλj)+ξ_nis considered,where p,λ_1,λ_2,…,λ_p,are constants α=(α_1,α_2,…,α_p) is a random vector and {ξ_n;n=0,±1,±2,…} is a wide-sense stationary sequence with zero means.In [4],theorem about thestrong consistent estimates of λ_1,λ_2.…,λ_p and α are proved under the assumption that α is a constantvector and p and δ are known constants such that0<δ<(?){λ_i-λ_j}.The main purpose of the present paper is to prove theorems on the strong consistent estimates ofparameters p,λ_1,λ_2,…,λ_p and random vector α without knowing p and δ.Numerical examples arealso given to illustrate our method of estimation.     

On categorical time series models with covariates

We study the problem of stationarity and ergodicity for autoregressive multinomial logistic time series models which possibly include a latent process and are defined by a GARCH-type recursive equation. We improve considerably upon the existing conditions about stationarity and ergodicity of those models. Proofs are based on theory developed for chains with complete connections. A useful coupling technique is employed for studying ergodicity of infinite order finite-state stochastic processes which generalize finite-state Markov chains. Furthermore, for the case of finite order Markov chains, we discuss ergodicity properties of a model which includes strongly exogenous but not necessarily bounded covariates.     

Sojourns and extremes of Fourier sums and series with random coefficients

Let X(t) be the trigonometric polynomial Σ^k_j=0a_j(U_tcosjt+V_jsinjt), –∞< t<∞, where the coefficients U_t and V_t are random variables and a_j is real. Suppose that these random variables have a joint distribution which is invariant under all orthogonal transformations of R^2k–2. Then X(t) is stationary but not necessarily Gaussian. Put L_t(u) = Lebesgue measure {s: 0?s?t, X(s) > u}, and M(t) = max{X(s): 0?s?t}. Limit theorems for L_t(u) and $\text{P}\text{(M(t) > u)}$ for u→∞ are obtained under the hypothesis that the distribution of the random norm (Σ^k_j=0(U²_j+V²_j))^{1 2} belongs to the domain of attraction of the extreme value distribution exp{ e^–2}. The results are also extended to the random Fourier series (k=∞).     

On multiplicative seasonal modelling for vector time series

Many time series encountered in real applications display seasonal behavior. In this paper, we consider multiplicative seasonal vectorial autoregressive moving average (SVARMA) models to describe seasonal vector time series. We discuss conditional maximum likelihood estimation of the model parameters, allowing them to satisfy general linear constraints. Having fitted a model, residual autocovariances (or autocorrelations) have been found useful in checking time series models. Consequently, we obtain the asymptotic distributions of the residual autocovariance matrices. As applications of these results, Portmanteau test statistics are proposed and their asymptotic distributions are studied. The finite-sample properties of the test statistics are evaluated using Monte Carlo experiments.     

On pricing derivatives under nonlinear time series models

In this note, we summarize some of our recent works on pricing derivative securities under nonlinear time series models. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On asymptotic properties of the plug-in cepstrum estimator for Gaussian time series

The paper considers probabilistic properties of the plug-in estimator of the cepstrum (Fourier transform of the logarithm of the spectral density) for stationary Gaussian time series. In the asymptotics of increasing observation time under some regularity condition we construct asymptotic expansions for the high-order central mixed moments of the log-periodogram and of the cepstrumestimator. These asymptotic expansions can be useful in signal processing for construction and performance analysis of statistical inference based on the plug-in estimator of the cepstrum.     

A note on the time series which is the product of two stationary time series

The time series […,x_-1y_-1,x₀y₀,x₁y₁,…]> which is the product of two stationary time series x_t and y_t is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.     

On the cusum of squares test for variance change in nonstationary and nonparametric time series models

In this paper we consider the problem of testing for a variance change in nonstationary and nonparametric time series models. The models under consideration are the unstable AR(q) model and the fixed design nonparametric regression model with a strong mixing error process. In order to perform a test, we employ the cusum of squares test introduced by Inclán and Tiao (1994,J. Amer. Statist. Assoc.,89, 913–923). It is shown that the limiting distribution of the test statistic is the sup of a standard Brownian bridge as seen in iid random samples. Simulation results are provided for illustration.     

On convergence of extremes under power normalization

In this note, we discuss two aspects of convergence of extremes under power normalization: convergence of moments and convergence of densities. The moments convergence is established for four p-max-stable laws according to conditions imposed on the considered distributions or on the parameter of the p-max-stable laws. For densities convergence, local uniform convergence of the densities is shown to coincide with some von Mises conditions.     

On some sampling schemes for estimating the parameters of a continuous time series

Summary In estimating the mean of certain stationary processes it is shown that it is better to sample at fixed equi-spaced time intervals than to sample randomly according to a renewal process. On the other hand it is shown that the estimation of autocorrelation is sometimes better accomplished by random sampling.     

Threshold levels in economics and time series

In this paper, we present theorems specifying the critical values for series associated with debts arranged in the order of their duration. The study of the debt crisis leads to a new interpretation of the phase transition to the Bose-condensate state, of the λ-point, and of the Thiess-Landau model.     

On estimation of mean and covariance functions in repeated time series with long-memory errors*

We consider kernel estimation of trend and covariance functions in models typically encountered in functional data analysis (FDA), with the modification that the random curves are perturbed by error processes that exhibit short- or long-range dependence. Uniform convergence of standardized maximal differences between estimated and true (trend and covariance) functions is established. For the covariance function, a transformation based on contrasts is proposed that does not require explicit trend estimation. Improved estimators can be obtained by using higher-order kernels.     

On the rate of convergence of the sum of the sample extremes

Summary Let X ₁, X ₂,..., X _n be independent random variables having a common distribution in the domain of normal attraction of a completely asymmetric stable law with characteristic exponent }(0,1) and support bounded below. Let X _n:n X _n:n ^-1...X _n:1 denote the ordered sample. We obtain the rate of convergence of n ^-1/ (X _n:n +...+X _{n:n-k n+1} ) to the stable limit law as both n and k _n ». As a consequence we obtain a representation of the sum X _n:n +...+X _{n:n-k n+1} .     

Risk forecasting in the context of time series*

Lithuanian Mathematical Journal - We propose an approach for forecasting risk contained in future observations in a time series. We take into account both the shape parameter and the extremal index...