Models for Extremal Dependence Derived from Skew‐symmetric Families

Skew‐symmetric families of distributions such as the skew‐normal and skew‐t represent supersets of the normal and t distributions, and they exhibit richer classes of extremal behaviour. By defining a non‐stationary skew‐normal process, which allows the easy handling of positive definite, non‐stationary covariance functions, we derive a new family of max‐stable processes – the extremal skew‐t process. This process is a superset of non‐stationary processes that include the stationary extremal‐t processes. We provide the spectral representation and the resulting angular densities of the extremal skew‐t process and illustrate its practical implementation.     

THE CONVERGENCE OF AUTOCORRELATIONS AND AUTOREGRESSIONS1

For a general class of scalar stationary processes, essentially those for which the best linear predictor is the best predictor (in the mean square sense), it is shown that, under fairly minor additional conditions, the sample autocorrelations converge to the true values almost surely and hniformly in the lag, t, at a rate (T^-1log T)^1/2, where T is the sample size. For ARMA processes, if |t|(log T)^a, a < ∞, the rate is the best possible, namely (T^-1log log T)^1/2. In particular the somewhat implausible condition, on the innovations, that E{ε(t)²| F_t-l} is constant is avoided in these results. The theorems are used to discuss autoregressive approximation. When the stationary process is a vector process the condition on the innovation sequence, ε(t), that E{ε(t)ε(t)| F_t-l} be constant, cannot be entirely avoided in relation to autoregressive approximation. This is also discussed.     

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.     

Nonparametric regression estimation in a null recurrent time series

We derive an asymptotic theory of nonparametric estimation for a time series regression model Z_t=f(X_t)+W_t, where {X_t} and {Z_t} are observed nonstationary processes, and {W_t} is an unobserved stationary process. The class of nonstationary processes allowed for {X_t} is a subclass of the class of null recurrent Markov chains. This subclass contains the random walk, unit root processes and nonlinear processes. The process {W_t} is assumed to be linear and stationary.     

On the quantile process based on the autoregressive residuals

Let {X_t} be the stationary AR(p) process satisfying the difference equation X_t=β₁X_t−1 + … + β_pX_t−p+ε_t, where {ε_t} is a sequence of iid random variables with mean zero and finite variance. Motivated by a goodness of fit test on the true errors {ε_t}, we are led to study the asymptotic behavior of the quantile process based on residuals (the residual quantile process). Particularly, we concentrate on the deviations between the residual quantile process and the empirical process based on the true errors. In this asymptotic study, it is shown that the deviations converge to zero in probability uniformly over certain intervals with specific order as sample size increases. Here, these intervals are allowed to vary with the sample size n and converge to the unit interval as n goes to infinity. Then, based on our result and the strong approximation result of Csörgö and Révész (1978), we propose a goodness of fit test statistic of which limiting distribution is the same as of a functional form of a standard Brownian bridge.     

Approximation to two independent Gaussian processes from a unique Lévy process and applications

Abstract

In this article, we construct two families of processes, from a unique Lévy process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As applications of this result, we obtain families of processes that converge in law towards fractional Brownian motion, sub-fractional Brownian motion and bifractional Brownian motion, respectively.     

Nonparametric Bayesian estimation of survival function under random left truncation

Several observational studies give rise to randomly left truncated data. In a nonparametric model for such data X denotes a variable of interest, T denotes the truncation variable and the distributions of both X and T are left unspecified. For this model, the product-limit estimator, which is also the maximum likelihood estimator of the survival curve, has been widely discussed. In this article, a nonparametric Bayes estimator of the survival function based on randomly left truncated data and Dirichlet process prior is presented. Some new results on the mixtures of Dirichlet processes in the context of truncated data are obtained. These results are then used to derive the Bayes estimator of the survival function under squared error loss. The weak convergence of the Bayes estimator is studied. An example using transfusion related AIDS data quoted in Kalbfleisch and Lawless (1989) is considered.     

Non-linear models 1

This is the first application of a new method for testing stationary random point processes. Consider the class of all stationary ergodic point processes on the real line with arbitrary dependences among the inter–point distances (spacing).The hypothesis is :The observed process φ is a homogeneous Poisson process or more (resp.less) regular than a Poisson process.The sample is the vector of the first n points t₁, …,t_n.There is a close relation between our method for testing and queueing theory: For finding an appropriate test statistic, we observe the behaviour of a single server queue with the input φ.A table of critical values is given.     

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.     

Approximate wavelet-based simulation of long memory processes

In this article, we investigate an algorithm for the fast O(N) and approximate simulation of long memory (LM) processes of length N using the discrete wavelet transform. The algorithm generates stationary processes and is based on the notion that we can improve standard wavelet-based simulation schemes by noting that the decorrelation property of wavelet transforms is not perfect for certain LM process. The method involves the simulation of circular autoregressive process of order one. We demonstrate some of the statistical properties of the processes generated, with some focus on four commonly used LM processes. We compare this simulation method with the white noise wavelet simulation scheme of Percival and Walden [Percival, D. and Walden, A., 2000, Wavelet Methods for Time Series Analysis (Cambridge: Cambridge University Press).].     

Optimal designs for approximating the path of a stochastic process

We consider a centered stochastic process {X(t):t ∈ T} with known and continuous covariance function. On the basis of observations X(t₁), …, X(t_n) we approximate the whole path by orthogonal projection and measure the performance of the chosen design d = (t₁, …, t_n)′ by the corresponding mean squared L²-distance. For covariance functions on T² = [0, 1]², which satisfy a generalized Sacks-Ylvisaker regularity condition of order zero, we construct asymptotically optimal sequences of designs. Moreover, we characterize the achievement of a lower error bound, given by Micchelli and Wahba (1981), and study the question of whether this bound can be attained.     

Estimation in dynamic regression with an integrated process

This paper deals with a dynamic regression model y_t = αy_t−1 + βz_t + u_t, where z_t is an integrated process of order one abbreviated as z_t ∼ I(1). Generally speaking, nonstandard asymptotic theory is required to investigate asymptotic properties of statistics related to an integrated process and the asymptotic results are very different from standard ones. There are two distinctive properties in nonstandard asymptotics: the so-called ‘super-consistency’ or T-consistency (where T is a sample size) and the weak convergence to a functional of the Wiener process. In spite of z_t being involved in our model, however, it is shown that our asymptotic results are the same as in the standard asymptotics in classical dynamic regression models, or if the disturbance u_t is serially correlated the OLS estimators of α and β have √T-inconsistency. This is due to the cointegration between y_t−1 and z_t. Although this point was clarified by Park and Phillips (1989) in a general context, we examine this explicitly through our specific model and connect the standard asymptotic theory with the nonstandard one in our case. Furthermore we investigate the limiting properties of other statistics such as t-ratio, the Durbin-Watson test and h-test. We also propose a consistent estimator of α and β by making use of Durbin's 2-step method. Finally, we carry out simulation studies which support our theoretical results.     

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.     

CAPABILITY MEASURES FOR m-DEPENDENT STATIONARY PROCESSES

Process capability indices, providing numerical measures on process potential and process performance, have received substantial research attention. Most research assumes that the process is normally distributed and the process data are independent. In real-world applications such as chemical, soft drinks, or tobacco/cigaratte manufacturing processes, process data are often auto-correlated. In this paper, we consider the capability indices Cp, Cpk, Cpm, Cpmk for strictly m-dependent stationary processes. We investigate the statistical properties of their natural estimators. We derive the asymptotic distributions, and establish confidence intervals so that capability testing can be performed.     

Bootstrap-Based T 2 Multivariate Control Charts

Control charts have been used effectively for years to monitor processes and detect abnormal behaviors. However, most control charts require a specific distribution to establish their control limits. The bootstrap method is a nonparametric technique that does not rely on the assumption of a parametric distribution of the observed data. Although the bootstrap technique has been used to develop univariate control charts to monitor a single process, no effort has been made to integrate the effectiveness of the bootstrap technique with multivariate control charts. In the present study, we propose a bootstrap-based multivariate T ² control chart that can efficiently monitor a process when the distribution of observed data is nonnormal or unknown. A simulation study was conducted to evaluate the performance of the proposed control chart and compare it with a traditional Hotelling's T ² control chart and the kernel density estimation (KDE)-based T ² control chart. The results showed that the proposed chart performed better than the traditional T ² control chart and performed comparably with the KDE-based T ² control chart. Furthermore, we present a case study to demonstrate the applicability of the proposed control chart to real situations.     

Inference for Local Autocorrelations in Locally Stationary Models

For nonstationary processes, the time-varying correlation structure provides useful insights into the underlying model dynamics. We study estimation and inferences for local autocorrelation process in locally stationary time series. Our constructed simultaneous confidence band can be used to address important hypothesis testing problems, such as whether the local autocorrelation process is indeed time-varying and whether the local autocorrelation is zero. In particular, our result provides an important generalization of the R function acf() to locally stationary Gaussian processes. Simulation studies and two empirical applications are developed. For the global temperature series, we find that the local autocorrelations are time-varying and have a “V” shape during 1910–1960. For the S&P 500 index, we conclude that the returns satisfy the efficient-market hypothesis whereas the magnitudes of returns show significant local autocorrelations.     

Skewness Reduction Approach in Multi-Attribute Process Monitoring

Since the product quality of many industrial processes depends upon more than one dependent variable or attribute, they are either multivariate or multi-attribute in nature. Although multivariate statistical process control is receiving increased attention in the literature, little work has been done to deal with multi-attribute processes. In this article, we develop a new methodology to monitor multi-attribute processes. To do this, first we transform multi-attribute data in a way that their marginal probability distributions have almost zero skewness. Then, we estimate the transformed covariance matrix and apply the well-known T ² control chart. In order to illustrate the proposed method and evaluate its performance, we use two simulation experiments and compare the results with the ones from both MNP chart and the χ² control chart.     

Darling–Erdős limit results for change-point detection in panel data

We wish to test the null hypothesis if the means of N panels remain the same during the observation period of length T. A quasi-likelihood argument leads to self-normalized statistics whose limit distribution under the null hypothesis is double exponential. The main results are derived assuming that the each panel is based on independent observations and then extended to linear processes. The proofs are based on an approximation of the sum of squared CUSUM processes using the Skorokhod embedding scheme. A simulation study illustrates that our results can be used in case of small and moderate N and T. We apply our results to detect change in the “corruption index”.     

Berry-Esseen bounds for kernel estimates of stationaryprocesses

In this paper we obtain nonuniform Berry-Esseen bounds of the kernel estimate of stationary process {X_i} by the method of martingale approximation. In particular, by choice of bandwidth b_n, the Berry-Esseen bounds can be n^−2/15. The dependence condition of {X_i} is expressed in terms of physical dependence measures introduced by Wu (2005).