Strong near-epoch dependence

We introduce a new class of dependent sequences of random variables, which is a subclass of near-epoch dependent sequences, but can also be approximated by mixing sequences. For this kind of sequences of random variables, we call them strong near-epoch dependent sequences, ap-order,p > 2, (maximum) moment inequality is established under weaker dependence sizes.     

A CENTRAL LIMIT THEOREM FOR STRONG NEAR-EPOCH DEPENDENT RANDOM VARIABLES

In this paper, a central limit theorem for strong near-epoch dependent sequences of random variables introduced in [9] is showed. Under the same moments condition, the authors essentially weaken the "size" requirement mentioned in other papers about near epoch dependence.     

The invariance principle for fractionally integrated processes with strong near-epoch dependent innovations

In this paper, we show the invariance principle for the partial sum processes of fractionally integrated processes, otherwise known as I(d + m) processes, where |d| < 1/2 and m is a nonnegative integer, with strong near-epoch dependent innovations. The results are applied to the test of unit root. The conditions given improve previous results in the literature concerning fractionally integrated processes.     

Asymptotically (in)dependent multivariate maxima of moving maxima processes

Smith and Weissman introduced a M4 class of processes which are very flexible models for temporally dependent multivariate extreme value processes. However all variables in these M4 models are asymptotically dependent and what this paper does is to extend this M4 class in a number of ways to produce classes of models which are also asymptotically independent. We shall study properties of the proposed models. In particular, asymptotic dependence indexes, coefficients of tail dependence, and extremal indexes are derived for each case.     

Bayesian uncertainty management in temporal dependence of extremes

Both marginal and dependence features must be described when modelling the extremes of a stationary time series. There are standard approaches to marginal modelling, but long- and short-range dependence of extremes may both appear. In applications, an assumption of long-range independence often seems reasonable, but short-range dependence, i.e., the clustering of extremes, needs attention. The extremal index 0 < ?? ≤ 1 is a natural limiting measure of clustering, but for wide classes of dependent processes, including all stationary Gaussian processes, it cannot distinguish dependent processes from independent processes with ?? = 1. Eastoe and Tawn (Biometrika 99, 43–55 2012) exploit methods from multivariate extremes to treat the subasymptotic extremal dependence structure of stationary time series, covering both 0 < ?? < 1 and ?? = 1, through the introduction of a threshold-based extremal index. Inference for their dependence models uses an inefficient stepwise procedure that has various weaknesses and has no reliable assessment of uncertainty. We overcome these issues using a Bayesian semiparametric approach. Simulations and the analysis of a UK daily river flow time series show that the new approach provides improved efficiency for estimating properties of functionals of clusters.     

Central limit theorems for dependent variables,II

Summary Two families of measures of the dependence between two random variables (rv's) are introduced. They include the strong-mixing ‘distance’. Two Central Limit Theorems (CLT's) are proved for dependent samples or processes where the dependence of the ‘past’ is not too strong. Tightness of the empirical process is shown to hold under conditions involving only the four-dimensional marginals of the sample.     

Asymptotic models and inference for extremes of spatio-temporal data

Recently there has been a lot of effort to model extremes of spatially dependent data. These efforts seem to be divided into two distinct groups: the study of max-stable processes, together with the development of statistical models within this framework; the use of more pragmatic, flexible models using Bayesian hierarchical models (BHM) and simulation based inference techniques. Each modeling strategy has its strong and weak points. While max-stable models capture the local behavior of spatial extremes correctly, hierarchical models based on the conditional independence assumption, lack the asymptotic arguments the max-stable models enjoy. On the other hand, they are very flexible in allowing the introduction of physical plausibility into the model. When the objective of the data analysis is to estimate return levels or kriging of extreme values in space, capturing the correct dependence structure between the extremes is crucial and max-stable processes are better suited for these purposes. However when the primary interest is to explain the sources of variation in extreme events Bayesian hierarchical modeling is a very flexible tool due to the ease with which random effects are incorporated in the model. In this paper we model a data set on Portuguese wildfires to show the flexibility of BHM in incorporating spatial dependencies acting at different resolutions.     

Modelling Joint Behaviour of Asset Prices Using Stochastic Correlation

Association or interdependence of two stock prices is analyzed, and selection criteria for a suitable model developed in the present paper. The association is generated by stochastic correlation, given by a stochastic differential equation (SDE), creating interdependent Wiener processes. These, in turn, drive the SDEs in the Heston model for stock prices. To choose from possible stochastic correlation models, two goodness-of-fit procedures are proposed based on the copula of Wiener increments. One uses the confidence domain for the centered Kendall function, and the other relies on strong and weak tail dependence. The constant correlation model and two different stochastic correlation models, given by Jacobi and hyperbolic tangent transformation of Ornstein-Uhlenbeck (HtanOU) processes, are compared by analyzing daily close prices for Apple and Microsoft stocks. The constant correlation, i.e., the Gaussian copula model, is unanimously rejected by the methods, but all other two are acceptable at a 95% confidence level. The analysis also reveals that even for Wiener processes, stochastic correlation can create tail dependence, unlike constant correlation, which results in multivariate normal distributions and hence zero tail dependence. Hence models with stochastic correlation are suitable to describe more dangerous situations in terms of correlation risk.

     

Extremal behavior of pMAX processes

The well-known M4 processes of Smith and Weissman are very flexible models for asymptotically dependent multivariate data. Extended M4 of Heffernan et al. allows to also account for asymptotic independence. In this paper we introduce a more general multivariate model comprising asymptotic dependence and independence, which has the extended M4 class as a particular case. We study properties of the proposed model. In particular, we compute the multivariate extremal index, tail dependence and extremal coefficients.     

Degradation modeling of 2 fatigue‐crack growth characteristics based on inverse Gaussian processes: A case study

Most modern products that are highly reliable are complex in their inner and outer structures. This situation indicates quality characterization by the interaction of multiple performance characteristics, which motivates the utilization of robust reliability models to obtain robust estimates. It is paramount to obtaining substantial information about a product's life cycle; therefore, when multiple performance characteristics are dependent, it is important to find models that address the joint distribution of performance degradation of such. In this paper, a reliability model for products with 2 fatigue‐crack growth characteristics related to 2 degradation processes is developed. The proposed model considers the dependence among degradation processes by using copula functions considering the marginal degradation processes as inverse Gaussian processes. The statistical inference is performed by using a Bayesian approach to estimate the parameters of the joint bivariate model. A time‐scale transformation is considered to assure monotone paths of the degradation trajectories. The comparison results of the reliability analysis, under both dependent and independent assumptions, are reported with the implementation of the proposed modeling in a case study, which consists of the crack propagation data of 2 terminals of an electronic device.     

Nonparametric Density Estimation for a Long-Range Dependent Linear Process

We estimate the marginal density function of a long-range dependent linear process by the kernel estimator. We assume the innovations are i.i.d. Then it is known that the term of the sample mean is dominant in the MISE of the kernel density estimator when the dependence is beyond some level which depends on the bandwidth and that the MISE has asymptotically the same form as for i.i.d. observations when the dependence is below the level. We call the latter the case where the dependence is not very strong and focus on it in this paper. We show that the asymptotic distribution of the kernel density estimator is the same as for i.i.d. observations and the effect of long-range dependence does not appear. In addition we describe some results for weakly dependent linear processes.     

Tail dependence of random variables from ARCH and heavy-tailed bilinear models

Discussed in this paper is the dependent structure in the tails of distributions of random variables from some heavy-tailed stationary nonlinear time series. One class of models discussed is the first-order autoregressive conditional heteroscedastic (ARCH) process introduced by Engle (1982). The other class is the simple first-order bilinear models driven by heavy-tailed innovations. We give some explicit formulas for the asymptotic values of conditional probabilities used for measuring the tail dependence between two random variables from these models. Our results have significant meanings in finance.     

多维随机过程首中时的强正相依性

研究多维随机过程X^-(t)的首中时的SPD相依结构，拓展了Ebrahimi等关于POD（Positively Orthant Dependent)和作者关于SPOD（Strongly Positively Orthant Dependent)的某些结果。刻划SPD的另一特征，还给出最大无穷可分过程首中时之间的SPD性质及其首中时向量（r1(U1),r2(U2))联合分布的下界（其中Ui是增Borel集，i=1,2)。     

Properties of uniform consistency of the kernel estimators of density and regression functions under dependence assumptions

The paper contains exponential inequalities for dependent random variables. As a measure of dependence we use φand ρ-mixing coefficients, the last one being based on the maximal coefficient of correlation. These results allow us to study the problem of uniform strong convergence for the kernel estimators of a density and for a kernel predictor for stochastic processes. Our uniform consistency theorems extend some known results     

Conditionally Specified Space-Time Models for Multivariate Processes

This article proposes a class of conditionally specified models for the analysis of multivariate space-time processes. Such models are useful in situations where there is sparse spatial coverage of one of the processes and much more dense coverage of the other process(es). The dependence structure across processes and over space, and time is completely specified through a neighborhood structure. These models are applicable to both point and block sources; for example, multiple pollutant monitors (point sources) or several county-level exposures (block sources). We introduce several computational tricks that are integral for model fitting, give some simple sufficient and necessary conditions for the space-time covariance matrix to be positive definite, and implement a Gibbs sampler, using Hybrid MC steps, to sample from the posterior distribution of the parameters. Model fit is assessed via the DIC. Predictive accuracy, over both time and space, is assessed both relatively and absolutely via mean squared prediction error and coverage probabilities. As an illustration of these models, we fit them to particulate matter and ozone data collected in the Los Angeles, CA, area in 1995 over a three-month period. In these data, the spatial coverage of particulate matter was sparse relative to that of ozone.     

Multivariate Stochastic Orders Induced by Case-Control Sampling

It is shown that retrospective sampling induces stochastic order relations in case-control studies. More specifically if the regression function is increasing and the covariates are positively dependent, then the covariates for cases are larger, with respect to some multivariate stochastic order, than the covariates of the controls. Strong dependence concepts yield strong multivariate stochastic orders. Conversely, different multivariate stochastic orders imply different monotonicity properties on the regression function. The results carry over to marginal models, transformed models and to problems involving confounders. The results set forth a new theoretical foundation for the analysis of case-control studies.     

Binary Markov Mesh Models and Symmetric Markov Random Fields: Some Results on their Equivalence

In this article, we focus on statistical models for binary data on a regular two-dimensional lattice. We study two classes of models, the Markov mesh models (MMMs) based on causal-like, asymmetric spatial dependence, and symmetric Markov random fields (SMFs) based on noncausal-like, symmetric spatial dependence. Building on results of Enting (1977), we give sufficient conditions for the asymmetrically defined binary MMMs (of third order) to be equivalent to a symmetrically defined binary SMF. Although not every binary SMF can be written as a binary MMM, our results show that many can. For such SMFs, their joint distribution can be written in closed form and their realizations can be simulated with just one pass through the lattice. An important consequence of the latter observation is that there are nontrivial spatial processes for which exact probabilities can be used to benchmark the performance of Markov-chain-Monte-Carlo and other algorithms.     

A higher order Markov model for analyzing covariate dependence

During the recent past, there has been a renewed interest in Markov chain for its attractive properties for analyzing real life data emerging from time series or longitudinal data in various fields. The models were proposed for fitting first or higher order Markov chains. However, there is a serious lack of realistic methods for linking covariate dependence with transition probabilities in order to analyze the factors associated with such transitions especially for higher order Markov chains. L.R. Muenz and L.V. Rubinstein [Markov models for covariate dependence of binary sequences, Biometrics 41 (1985) 91–101] employed logistic regression models to analyze the transition probabilities for a first order Markov model. The methodology is still far from generalization in terms of formulating a model for higher order Markov chains. In this study, it is aimed to provide a comprehensive covariate-dependent Markov model for higher order. The proposed model generalizes the estimation procedure for Markov models for any order. The proposed models and inference procedures are simple and the covariate dependence of the transition probabilities of any order can be examined without making the underlying model complex. An example from rainfall data is illustrated in this paper that shows the utility of the proposed model for analyzing complex real life problems. The application of the proposed method indicates that the higher order covariate dependent Markov models can be conveniently employed in a very useful manner and the results can provide in-depth insights to both the researchers and policymakers to resolve complex problems of underlying factors attributing to different types of transitions, reverse transitions and repeated transitions. The estimation and test procedures can be employed for any order of Markov model without making the theory and interpretation difficult for the common users.     

Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters

Most modeling efforts involve multiple physical or biological processes. All physical or biological processes take time to complete. Therefore, multiple time delays occur naturally and shall be considered in more advanced modeling efforts. Carefully formulated models of such natural processes often involve multiple delays and delay dependent parameters. However, a general and practical theory for the stability analysis of models with more than one discrete delay and delay dependent parameters is nonexistent. The main purpose of this paper is to present a practical geometric method to study the stability switching properties of a general transcendental equation which may result from a stability analysis of a model with two discrete time delays and delay dependent parameters that dependent only on one of the time delay. In addition to simple and illustrative examples, we present a detailed application of our method to the study of a two discrete delay SIR model.     

Discrete-time host–parasitoid models with Allee effects: Density dependence versus parasitism

We present two general discrete-time host–parasitoid models with Allee effects on the host. In the first model, it is assumed that parasitism occurs prior to density dependence, while in the second model we assume that density dependence operates first followed by parasitism. It is shown that both models have similar asymptotic behaviour. The parasitoid population will definitely go extinct if the maximal growth rate of the host population is less than or equal to one, independent of whether density dependence or parasitism occurs first. The fate of the population is initial condition dependent if this maximal growth rate exceeds one. In particular, there exists a host population threshold, the Allee threshold, below which the host population goes extinct and so does the parasitoid. This threshold is the same for both models. Numerical examples with different functions are simulated to illustrate our analytical results.