The Scale Invariant Wigner Spectrum Estimation of Gaussian Locally Self-Similar Processes

We study locally self-similar processes (LSSPs) in Silverman’s sense. By deriving the minimum mean-square optimal kernel within Cohen’s class counterpart of time–frequency representations, we obtain an optimal estimation for the scale invariant Wigner spectrum (SIWS) of Gaussian LSSPs. The class of estimators is completely characterized in terms of kernels, so the optimal kernel minimizes the mean-square error of the estimation. We obtain the SIWS estimation for two cases: global and local, where in the local case, the kernel is allowed to vary with time and frequency. We also introduce two generalizations of LSSPs: the locally self-similar chirp process and the multicomponent LSSP, and obtain their optimal kernels. Finally, the performance and accuracy of the estimation is studied via simulation.     

Characterization of discrete scale invariant Markov sequences

ABSTRACT

Some special sampling of discrete scale invariant (DSI) processes are presented to provide a multi-dimensional self-similar process in correspondence. By imposing Markov property we show that the covariance functions of such Markov DSI sequences are characterized by variance, and covariance of adjacent samples in the first scale interval. We also provide a theoretical method for estimating spectral density matrix of corresponding multi-dimensional self-similar Markov process. Some examples such as simple Brownian motion (sBm) with drift and scale invariant autoregressive model are presented and these properties are investigated. We present two new method to estimate Hurst parameter of DSI processes and apply them to some sBm and also to the SP500 indices for some period which has DSI property. We compare our estimates with the maximum-likelihood and rescaled range (R/S) method which are applied to the corresponding multi-dimensional self-similar processes.     

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.     

New compactly supported spatiotemporal covariance functions from SPDEs

Potential theory and Dirichlet’s priciple constitute the basic elements of the well-known classical theory of Markov processes and Dirichlet forms. This paper presents new classes of fractional spatiotemporal covariance models, based on the theory of non-local Dirichlet forms, characterizing the fundamental solution, Green kernel, of Dirichlet boundary value problems for fractional pseudodifferential operators. The elements of the associated Gaussian random field family have compactly supported non-separable spatiotemporal covariance kernels admitting a parametric representation. Indeed, such covariance kernels are not self-similar but can display local self-similarity, interpolating regular and fractal local behavior in space and time. The associated local fractional exponents are estimated from the empirical log-wavelet variogram. Numerical examples are simulated for illustrating the properties of the space–time covariance model class introduced.     

A Sequential Multi-Hypothesis Test for the Mean Function of a Discrete-Time Gaussian Process

Abstract

A sequential multi-hypothesis test for the mean function of a discrete-time Gaussian process with known covariance kernel is developed. It is obtained by applying the Bechhofer-Kiefer-Sobel generalized sequential probability ratio test GSPRT, and its properties are studied analytically. Selected applications to i.i.d. normal random variables, observation in a time series AR(1) model, and Wiener processes are given.     

A remark on self-similar processes with stationary increments

The upper bound of the parameter of self-similar processes with stationary increments is given in terms of the moment condition.     

Optimal estimation in functional linear regression for sparse noise‐contaminated data

In this article, we propose a novel approach to fit a functional linear regression in which both the response and the predictor are functions. We consider the case where the response and the predictor processes are both sparsely sampled at random time points and are contaminated with random errors. In addition, the random times are allowed to be different for the measurements of the predictor and the response functions. The aforementioned situation often occurs in longitudinal data settings. To estimate the covariance and the cross‐covariance functions, we use a regularization method over a reproducing kernel Hilbert space. The estimate of the cross‐covariance function is used to obtain estimates of the regression coefficient function and of the functional singular components. We derive the convergence rates of the proposed cross‐covariance, the regression coefficient, and the singular component function estimators. Furthermore, we show that, under some regularity conditions, the estimator of the coefficient function has a minimax optimal rate. We conduct a simulation study and demonstrate merits of the proposed method by comparing it to some other existing methods in the literature. We illustrate the method by an example of an application to a real‐world air quality dataset. The Canadian Journal of Statistics 47: 524–559; 2019 © 2019 Statistical Society of Canada     

Innovative methods for modeling of scale invariant processes

This paper presents some innovative methods for modeling discrete scale invariant (DSI) processes and evaluation of corresponding parameters. For the case where the absolute values of the increments of DSI processes are in general increasing, we consider some moving sample variance of the increments and present some heuristic algorithm to characterize successive scale intervals. This enables us to estimate scale parameter of such DSI processes. To present some superior structure for the modeling of DSI processes, we consider the possibility that the variations inside the prescribed scale intervals show some further self-similar behavior. Such consideration enables us to provide more efficient estimators for Hurst parameters. We also present two competitive estimation methods for the Hurst parameters of self-similar processes with stationary increments and prove their efficiency. Using simulated samples of some simple fractional Brownian motion, we show that our estimators of Hurst parameter are more efficient as compared with the celebrated methods of convex rearrangement and quadratic variation. Finally we apply the proposed methods to evaluate DSI behavior of the S&P500 indices in some period.     

债券定价的离散时间仿射模型群

一、引言传统的债券定价方法—未来现金流量贴现法 ,是由美国的威廉姆斯 (Ｗｉｌｌｉａｍｓ .ＪｏｈｎＨｅｎｒｙ)根据现值理论推导而来的 ,曾被广大投资者用来作为衡量债券投资价值的方法。然而 ,该模型由于贴现率的选取没有确定的标准 ,具有比较大的随意性 ,因而导致所计算的债券价格也表现出较大的随意性 ,逐渐暴露其不足之处。随着利率期限结构理论的不断发展 ,债券定价方法也相应地获得了很大的进展。尤其是最近十几年来出现了利率期限结构的随机过程无套利分析方法 ,该方法认为利率期限结构和债券价格同某些随机因素 (即状态变量 )相…     

Continuous processes derived from the solution of generalized Langevin equation: theoretical properties and estimation

ABSTRACT

In this paper we present a class of continuous-time processes arising from the solution of the generalized Langevin equation and show some of its properties. We define the theoretical and empirical codifference as a measure of dependence for stochastic processes. As an alternative dependence measure we also consider the spectral covariance. These dependence measures replace the autocovariance function when it is not well defined. Results for the theoretical codifference and theoretical spectral covariance functions for the mentioned process are presented. The maximum likelihood estimation procedure is proposed to estimate the parameters of the process arising from the classical Langevin equation, i.e. the Ornstein–Uhlenbeck process, and of the so-called Cosine process. We also present a simulation study for particular processes arising from this class showing the generation, and the theoretical and empirical counterpart for both codifference and spectral covariance measures.     

Self-similarity index estimation via wavelets for locally self-similar processes

Many naturally occurring phenomena can be effectively modeled using self-similar processes. In such applications, accurate estimation of the scaling exponent is vital, since it is this index which characterizes the nature of the self-similarity. Although estimation of the scaling exponent has been extensively studied, previous work has generally assumed that this parameter is constant. Such an assumption may be unrealistic in settings where it is evident that the nature of the self-similarity changes as the phenomenon evolves. For such applications, the scaling exponent must be allowed to vary as a function of time, and a procedure must be available which provides a statistical characterization of this progression. In what follows, we propose and describe such a procedure. Our method uses wavelets to construct local estimates of time-varying scaling exponents for locally self-similar processes. We establish a consistency result for these estimates. We investigate the effectiveness of our procedure in a simulation study, and demonstrate its applicability in the analyses of a hydrological and a geophysical time series, each of which exhibit locally self-similar behavior.     

Non‐stationary Cross‐Covariance Models for Multivariate Processes on a Globe

Abstract. In geophysical and environmental problems, it is common to have multiple variables of interest measured at the same location and time. These multiple variables typically have dependence over space (and/or time). As a consequence, there is a growing interest in developing models for multivariate spatial processes, in particular, the cross‐covariance models. On the other hand, many data sets these days cover a large portion of the Earth such as satellite data, which require valid covariance models on a globe. We present a class of parametric covariance models for multivariate processes on a globe. The covariance models are flexible in capturing non‐stationarity in the data yet computationally feasible and require moderate numbers of parameters. We apply our covariance model to surface temperature and precipitation data from an NCAR climate model output. We compare our model to the multivariate version of the Matérn cross‐covariance function and models based on coregionalization and demonstrate the superior performance of our model in terms of AIC (and/or maximum loglikelihood values) and predictive skill. We also present some challenges in modelling the cross‐covariance structure of the temperature and precipitation data. Based on the fitted results using full data, we give the estimated cross‐correlation structure between the two variables.     

On the estimation of the spectral density for continuous spatial processes

In this work, we study the asymptotic properties of smoothed nonparametric kernel spectral density estimators for the spatial spectral density. We consider the case of continuous stationary spatial processes under a shrinking asymptotic framework. Expressions for the bias and the covariance structure are obtained and the implications for the edge effect bias of the choice of the kernel, bandwidth and spacing parameter in the design are also discussed, both for tapered and untapered estimates. Results are illustrated with a simulation study.     

Continuous-Time Stochastic Processes with Cyclical Long-Range Dependence

This paper introduces continuous‐time random processes whose spectral density is unbounded at some non‐zero frequencies. The discretized versions of these processes have asymptotic properties similar to those of discrete‐time Gegenbauer processes. The paper presents some properties of the covariance function and spectral density as well as a theory of statistical estimation of the mean and covariance function of such processes. Some directions for further generalizations of the results are indicated.     

Posterior consistency of logistic Gaussian process priors in density estimation

We establish weak and strong posterior consistency of Gaussian process priors studied by Lenk [1988. The logistic normal distribution for Bayesian, nonparametric, predictive densities. J. Amer. Statist. Assoc. 83 (402), 509–516] for density estimation. Weak consistency is related to the support of a Gaussian process in the sup-norm topology which is explicitly identified for many covariance kernels. In fact we show that this support is the space of all continuous functions when the usual covariance kernels are chosen and an appropriate prior is used on the smoothing parameters of the covariance kernel. We then show that a large class of Gaussian process priors achieve weak as well as strong posterior consistency (under some regularity conditions) at true densities that are either continuous or piecewise continuous.     

On the estimators and tests for the semiparametric hazards regression model

In the accelerated hazards regression model with censored data, estimation of the covariance matrices of the regression parameters is difficult, since it involves the unknown baseline hazard function and its derivative. This paper provides simple but reliable procedures that yield asymptotically normal estimators whose covariance matrices can be easily estimated. A class of weight functions are introduced to result in the estimators whose asymptotic covariance matrices do not involve the derivative of the unknown hazard function. Based on the estimators obtained from different weight functions, some goodness-of-fit tests are constructed to check the adequacy of the accelerated hazards regression model. Numerical simulations show that the estimators and tests perform well. The procedures are illustrated in the real world example of leukemia cancer. For the leukemia cancer data, the issue of interest is a comparison of two groups of patients that had two different kinds of bone marrow transplants. It is found that the difference of the two groups are well described by a time-scale change in hazard functions, i.e., the accelerated hazards model.     

Spatial sampling design for parameter estimation of the covariance function

We study the spatial optimal sampling design for covariance parameter estimation. The spatial process is modeled as a Gaussian random field and maximum likelihood (ML) is used to estimate the covariance parameters. We use the log determinant of the inverse Fisher information matrix as the design criterion and run simulations to investigate the relationship between the inverse Fisher information matrix and the covariance matrix of the ML estimates. A simulated annealing algorithm is developed to search for an optimal design among all possible designs on a fine grid. Since the design criterion depends on the unknown parameters, we define relative efficiency of a design and consider minimax and Bayesian criteria to find designs that are robust for a range of parameter values. Simulation results are presented for the Matérn class of covariance functions.     

A note on the non-negativity of continuous-time ARMA and GARCH processes

A general approach for modeling the volatility process in continuous-time is based on the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a necessary condition for the kernel to be non-negative, and propose a numerical method for checking the non-negativity of a kernel function. These results can be lifted to solving a similar problem with another approach to modeling volatility via the COntinuous-time Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes.     

A class of correlated weighted Poisson processes

In this paper, we propose new classes of correlated Poisson processes and correlated weighted Poisson processes on the interval [0,1], which generalize the class of weighted Poisson processes defined by Balakrishnan and Kozubowski (2008), by incorporating a dependence structure between the standard uniform variables used in the construction. In this manner, we obtain another process that we refer to as correlated weighted Poisson process. Various properties of this process such as marginal and joint distributions, stationarity of the increments, moments, and the covariance function, are studied. The results are then illustrated through some examples, which include processes with length-biased Poisson, exponentially weighted Poisson, negative binomial, and COM-Poisson distributions.     

On multivariate smoothed bootstrap consistency

This paper deals with the convergence in Mallows metric for classical multivariate kernel distribution function estimators. We prove the convergence in Mallows metric of a locally orientated kernel smooth estimator belonging to the class of sample smoothing estimators. The consistency follows for the smoothed bootstrap for regular functions of the marginal means. Two simple simulation studies show how the smoothed versions of the bootstrap give better results than the classical technique.