Formulas for stationary planar fibre processes III – intersections with fibre systems

In this paper we investigate the distribution of the periodogram, respectively, the periodogram matrix for stationary random sequences. These .distributions are consid¬ered in the case of a fixed frequency as well as in the case of a finite number of frequencies for Gaussian sequences and for sequences of independent random variables. The exact distribution is obtained in the case of a fixed frequence for one-dimensional GAUSsian sequences. Asymptotic expansions, respectively, the rate of convergence to the asymptotic distribution are given in the case mentioned above     

Stereological formulas for stationary surface processes

In the paper surface processes are considered, i.e. sets of surfaces distributed at random in the space. For translation invariant surface processes the PALM distribution of the direction of the normal in a typical surface point is defined and related to two roses of intersection. Fibre and point processes are studied which result by intersections with planes and lines respectively. For these processes stereological formulas are proved which connect some quantities with such of the surface process. As a special ease motion invariant, i.e. translation invariant and isotropic, surface processes are treated, Furthermore, weighted surface processes are 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.     

Bayesian analysis of non-negative ar(2) processes

A non-negative AR(2) process with exponentially distributed white noise is investigated in the paper. It is assumed that the autoregressive parameters are random variables with a vague prior density. They can be esto,ated by their posterior expectations. Explicit formulas for these estimators are derived and their strong consistency is proved. An approximation to the estimators is proposed which is easier for calculation. The results are illustrated in a simulation study     

Exact Likelihood Equations for Autoregression Models with Multivariate Elliptically Contoured Distributions

Abstract

The multivariate elliptically contoured distributions provide a viable framework for modeling time-series data. It includes the multivariate normal, power exponential, t, and Cauchy distributions as special cases. For multivariate elliptically contoured autoregressive models, we derive the exact likelihood equations for the model parameters. They are closely related to the Yule-Walker equations and involve simple function of the data. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the simulation data.     

Estimation of the Memory Parameters of the Fractionally Integrated Separable Spatial Autoregressive (FISSAR(1, 1)) Model: A Simulation Study

In this article, we implement the Regression Method for estimating (d ₁, d ₂) of the FISSAR(1, 1) model. It is also possible to estimate d ₁ and d ₂ by Whittle's method. We also compute the estimated bias, standard error, and root mean square error by a simulation study. A comparison was made between the Regression Method of estimating d ₁ and d ₂ to that of the Whittle's method. It was found in this simulation study that the Regression Method of estimation was better when compare with the Whittle's estimator, in the sense that it had smaller root mean square errors (RMSE) values.     

Adaptive Test for Periodicity in Self-Exciting Threshold Autoregressive Models

This article is devoted to the study of the periodicity testing problem in a self-exciting threshold autoregressive (SETAR) model. The local asymptotic normality (LAN) property is shown via the adapted sufficient conditions due to Swensen (1985 Swensen , A. R. ( 1985 ). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend . Journal of Multivariate Analysis 16 : 54 – 70 .[Crossref], [Web of Science ®] , [Google Scholar]). Moreover, the LAN of the central sequence is established. First, we consider the case where the innovation density is specified and we obtain a parametric local asymptotic test. Second, we construct an adaptive test in the case where this density is unspecified but symmetric. The performances of these established tests are shown via simulation studies.     

Estimation of Time-Varying Long Memory Parameter Using Wavelet Method

Stationary long memory processes have been extensively studied over the past decades. When we deal with financial, economic, or environmental data, seasonality and time-varying long-range dependence can often be observed and thus some kind of non-stationarity exists. To take into account this phenomenon, we propose a new class of stochastic processes: locally stationary k-factor Gegenbauer process. We present a procedure to estimate consistently the time-varying parameters by applying discrete wavelet packet transform. The robustness of the algorithm is investigated through a simulation study. And we apply our methods on Nikkei Stock Average 225 (NSA 225) index series.     

Infinite Variation Tempered Stable Ornstein–Uhlenbeck Processes with Discrete Observations

We investigate transition law between consecutive observations of Ornstein–Uhlenbeck processes of infinite variation with tempered stable stationary distribution. Thanks to the Markov autoregressive structure, the transition law can be written in the exact sense as a convolution of three random components; a compound Poisson distribution and two independent tempered stable distributions, one with stability index in (0, 1) and the other with index in (1, 2). We discuss simulation techniques for those three random elements. With the exact transition law and proposed simulation techniques, sample paths simulation proves significantly more efficient, relative to the known approximative technique based on infinite shot noise series representation of tempered stable Lévy processes.     

Storage Life Prediction for a High-Performance Capacitor Using Multi-Phase Wiener Degradation Model

Capacitance is a critical performance characteristic of high-voltage-pulse capacitor which is used to store and discharge electrical energy rapidly. The capacitors usually are stored for a long period of time before put into use. Experimental result and engineering experience indicate that the capacitance increases with storage time and will eventually exceed the failure threshold, which means that the capacitor may fail during storage. This is a typical mode of degradation failure for long storage products. Further, the capacitance degradation path can be extrapolated in several stages based on the shifting characteristics. That is, the capacitance increases slowly or fluctuates in the initial storage stage that lasts about three months. Then it increases sharply in the middle stage which lasts about four months. After the two stages, the capacitor enters into the third stage in which capacitance increases constantly. This degradation phenomenon motivates us to study the storage life prediction method based on multi-phase degradation path model. The storage performance degradation mechanism of high-voltage-pulse capacitor was investigated, which provides the physical basis for multi-phase Wiener degradation model. Identification procedure for the transition points in the degradation path was proposed using maximum likelihood principle (MLP). The result of Kruskal-Wallis test which is the method to test whether two populations are consistent or not in statistics showed that the transition points are statistically effective. Other parameters in the multi-phase degradation model are estimated with maximum likelihood estimation (MLE) after the transition points have been specified. The multi-phase Inverse Gaussian (IG) distribution for storage life was deduced for the capacitor, and the point and interval estimation procedure for reliable storage life are constructed with bootstrap method. The efficiency and effectiveness of the proposed multi-phase degradation model is compared with storage life prediction under single-phase condition.