Covariance analysis and associated spectra for classes of nonstationary processes

In 1951, Cramér introduced a class of nonstationary processes. This broad class of processes contains the important harmonizable and stationary classes of processes. The Cramér class can have additional structure imposed upon it through Cesàro summability considerations. These refined Cramér classes, termed (c,p)-summable Cramér, have recently been considered by Swift (in: M.M. Rao (Ed.), Real and Stochastic Analysis: Recent Advances, CRC Press, Boca Raton, FL, 1997, p. 303). In this paper, the relationship between the (c,p)-summable Cramér classes and the (KF,p) classes of processes introduced by Rao in 1985 is considered. The (KF,p) classes of processes are a generalization of the class of processes considered by Kampé de Feriet and Frenkiel. A continuity theorem for the (KF,p) classes is obtained. This result yields a spectral representation for the (KF,p) classes. Some (KF,p) class processes are shown to arise as the solution to a difference equation obtained from a linear model of a noisy communication channel.     

Is a pure jump process fitting the high frequency data better than a jump-diffusion process?

Two families of processes: pure jump processes and jump-diffusion processes are widely used in literatures. Recently, empirical findings demonstrate that the underlying processes of high frequency data sets are pure-jump processes of infinite variation in many situations. Statistical tests are also proposed to make the empirical findings theoretically grounded. In this paper, we extend the work of Jing et al. (2012) in two aspects: (1) the jump process in the null hypothesis and the alternative hypothesis could be different; (2) the null hypothesis covers more flexible processes which are more relevant in finance when considering models for asset prices or nominal interest rates. Theoretically, the test is proven to be very powerful and can control the type I error probabilities well under the nominal level.     

Non-ergodic martingale estimating functions and related asymptotics

Well-known estimation methods such as conditional least squares, quasilikelihood and maximum likelihood (ML) can be unified via a single framework of martingale estimating functions (MEFs). Asymptotic distributions of estimates for ergodic processes use constant norm (e.g. square root of the sample size) for asymptotic normality. For certain non-ergodic-type applications, however, such as explosive autoregression and super-critical branching processes, one needs a random norm in order to get normal limit distributions. In this paper, we are concerned with non-ergodic processes and investigate limit distributions for a broad class of MEFs. Asymptotic optimality (within a certain class of non-ergodic MEFs) of the ML estimate is deduced via establishing a convolution theorem using a random norm. Applications to non-ergodic autoregressive processes, generalized autoregressive conditional heteroscedastic-type processes, and super-critical branching processes are discussed. Asymptotic optimality in terms of the maximum random limiting power regarding large sample tests is briefly discussed.     

Bayesian spatio‐temporal models based on discrete convolutions

Abstract: The authors consider a class of models for spatio‐temporal processes based on convolving independent processes with a discrete kernel that is represented by a lower triangular matrix. They study two families of models. In the first one, spatial Gaussian processes with isotropic correlations are convoluted with a kernel that provides temporal dependencies. In the second family, AR(p) processes are convoluted with a kernel providing spatial interactions. The covariance structures associated with these two families are quite rich. Their covariance functions that are stationary and separable in space and time as well as time dependent nonseparable and nonisotropic ones.     

Prediction in a mixed Poisson cluster model

Motivated by insurance applications, a mixed Poisson cluster model is considered, where the cluster center process is a mixed Poisson process and descendant processes are additive processes. Each point of the center process represents a claim’s reported time and descendant processes are interpreted as processes of the corresponding payments or number of payments. In this study, we focus on the process aggregating all separate claim’s payment processes. Given the past observations, we study prediction of future increments and their mean-squared errors, also revealing the dependency between future increments from non-reported (IBNR) claims and the past available information. In the existing literature, they are independent since models were considered with a purely Poissonian center process. We derive computationally reasonable expressions for predictors and their variances.     

Hidden Second‐order Stationary Spatial Point Processes

In the existing statistical literature, the almost default choice for inference on inhomogeneous point processes is the most well‐known model class for inhomogeneous point processes: reweighted second‐order stationary processes. In particular, the K‐function related to this type of inhomogeneity is presented as the inhomogeneous K‐function. In the present paper, we put a number of inhomogeneous model classes (including the class of reweighted second‐order stationary processes) into the common general framework of hidden second‐order stationary processes, allowing for a transfer of statistical inference procedures for second‐order stationary processes based on summary statistics to each of these model classes for inhomogeneous point processes. In particular, a general method to test the hypothesis that a given point pattern can be ascribed to a specific inhomogeneous model class is developed. Using the new theoretical framework, we reanalyse three inhomogeneous point patterns that have earlier been analysed in the statistical literature and show that the conclusions concerning an appropriate model class must be revised for some of the point patterns.     

Estimation for Non-Negative Lévy-Driven CARMA Processes

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a useful and very general class of stationary, nonnegative continuous-time processes that have been used, in particular, for the modeling of stochastic volatility. Brockwell, Davis, and Yang (2007) derived efficient estimates of the parameters of a nonnegative Lévy-driven CAR(1) process and showed how the realization of the underlying Lévy process can be estimated from closely-spaced observations of the process itself. In this article we show how the ideas of that article can be generalized to higher order CARMA processes with nonnegative kernel, the key idea being the decomposition of the CARMA process into a sum of dependent Ornstein–Uhlenbeck processes.     

Rates of convergence of autocorrelation estimates for periodically correlated autoregressive Hilbertian processes

Autoregressive Hilbertian (ARH) processes are of great importance in the analysis of functional time series data and estimation of the autocorrelation operators attracts the attention of various researchers. In this paper, we study estimators of the autocorrelation operators of periodically correlated autoregressive Hilbertian processes of order one (PCARH(1)), which is an extension of ARH(1) processes. The estimation method is based on the spectral decomposition of the covariance operator and considers two main cases: known and unknown eigenvectors. We show the consistency in the mean integrated quadratic sense of the estimators of the autocorrelation operators and present upper bounds for the corresponding rates.     

CARMA(p,q) generalized random processes

There is now a vast literature on the theory and applications of generalized random processes, pioneered by Itô (1953), Gel’fand (1955) and Yaglom (1957). In this note we make use of the theory of generalized random processes as defined in the book of Gel’fand and Vilenkin (1964) to extend the definition of continuous-time ARMA(p,q  ) processes to allow q≥p$q\ge p$, in which case the processes do not exist in the classical sense. The resulting CARMA generalized random processes provide a framework within which it is possible to study derivatives of CARMA processes of arbitrarily high order.     

ASYMPTOTIC POSTERIOR NORMALITY FOR STOCHASTIC PROCESSES: A NON-ERGODIC EXAMPLE

Sweeting & Adekola (1987) presented a fairly general set of conditions, covering a wide class of problems, that establish asymptotic posterior normality for stochastic processes. In this paper, this theory is illustrated via a non-ergodic example, namely, the class of non-homogeneous birth processes.     

An Economic Design of the Bounded Reset Chart Using the Integral Controller for Batch-Oriented Processes

In batch-oriented processes, the quality of the process tends to deteriorate over time. A bounded reset chart (BRC) using the integral controller is proposed to adjust such processes effectively. The effectiveness of the BRC procedure is evaluated in terms of the average cost per unit interval (ACU). Optimal values for the reset limit, considering various process and cost parameters, are studied to minimize the ACU. The BRC procedure is shown to be effective for adjusted processes with deterioration over time compared to the traditional adjustment procedure where resetting is not considered.     

Integrated OU Processes and Non-Gaussian OU-based Stochastic Volatility Models

Abstract. In this paper, we study the detailed distributional properties of integrated non-Gaussian Ornstein–Uhlenbeck (intOU) processes. Both exact and approximate results are given. We emphasize the study of the tail behaviour of the intOU process. Our results have many potential applications in financial economics, as OU processes are used as models of instantaneous variance in stochastic volatility (SV) models. In this case, an intOU process can be regarded as a model of integrated variance. Hence, the tail behaviour of the intOU process will determine the tail behaviour of returns generated by SV models.     

The sample autocorrelation function of I(1) processes

In case of a random walk the theoretical autocorrelations tend to one asymptotically. The sample autocorrelations, however, may decline rather fast even with large samples. We will explain this observation by deriving the asymptotic distribution that turns out to be closely related to the Dickey-Fuller (1979) distribution. Moreover we discuss the behaviour of the sample autocorrelations of integrated MA(1) and AR(1) processes. In order to prove our results we consider more general I(1) processes and apply the functional central limit theorem injected to time series analysis by Phillips (1987). We obtain unit root tests that are based on autocorrelation estimators of higher lags. We discuss their finite sample behaviour experimentally.     

Average outgoing quality of CSP-C continuous sampling plan under short run production processes

A CSP-C continuous sampling plan is a new single-level continuous sampling procedure developed by Govindaraju & Kandasamy (2000) by incorporating the concept of acceptance number to the CSP-1 plan for the application of continuous production processes. In this new plan, the sampling inspection phase is characterized by a maximum allowable number of non-conforming units, c, and a constant sampling rate, f. Govindaraju & Kandasamy (2000) derived the performance measures such as average outgoing quality (AOQ), average fraction inspected (AFI) etc, of the CSP-C plan using a Markov chain model for long run production processes. Yang (1983) has observed that the AOQ and AFI, being long run average measures, are not satisfactory measures of performance for short run production processes. Hence, formulas are derived in this paper, using the renewal theory approach enabling one to compute AOQ and AFI for both long run and short run production processes. Numerical illustrations are also given. By simulation, the accuracy of the short run measures is studied.     

Serial dependence and regression of Poisson INARMA models

Time series of counts occur in many fields of practice, with the Poisson distribution as a popular choice for the marginal process distribution. A great variety of serial dependence structures of stationary count processes can be modelled by the INARMA family. In this article, we propose a new approach to the INMA(q) family in general, including previously known results as special cases. In the particular case of Poisson marginals, we will derive new results concerning regression properties and the serial dependence structure of INAR(1) and INMA(q) models. Finally, we present explicit expressions for the distribution of jumps in such processes.     

Orthogonal series density estimation in a disaggregation scheme

In this paper we consider long-memory processes obtained by aggregation of independent random parameter AR(1) processes. We propose an estimator of the density of the underlying random parameter. This estimator is based on the expansion of the density function on the basis of Gegenbauer polynomials. Rate of convergence to zero of the mean integrated square error (MISE) and of the uniform error are obtained. The results are illustrated by Monte-Carlo simulations.     

Nonparametric estimation of the mean function of a stochastic process with missing observations

In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right- censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.     

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.     

Testing the Null Hypothesis of Nonstationary Long Memory Against the Alternative Hypothesis of a Nonlinear Ergodic Model

Interest in the interface of nonstationarity and nonlinearity has been increasing in the econometric literature. This paper provides a formal method of testing for nonstationary long memory against the alternative of a particular form of nonlinear ergodic processes; namely, exponential smooth transition autoregressive processes. In this regard, the current paper provides a significant generalization to existing unit root tests by allowing the null hypothesis to encompass a much larger class of nonstationary processes. The asymptotic theory associated with the proposed Wald statistic is derived, and Monte Carlo simulation results confirm that the Wald statistics have reasonably correct size and good power in small samples. In an application to real interest rates and the Yen real exchange rates, we find that the tests are able to distinguish between these competing processes in most cases, supporting the long-run Purchasing Power Parity (PPP) and Fisher hypotheses. But, there are a few cases in which long memory and nonlinear ergodic processes display similar characteristics and are thus confused with each other in small samples.     

First order autoregressive time series with negative binomial and geometric marginals

In this paper we present first order autoregressive (AR(1)) time series with negative binomial and geometric marginals. These processes are the discrete analogues of the gamma and exponential processes introduced by Sim (1990). Many properties of the processes discussed here, such as autocorrelation, regression and joint distributions, are studied.