On the Multivariate Two-Sample Problem Using Strong Approximations of Empirical Copula Processes

In this article, we establish optimal rates for the strong approximation of empirical copula processes in ?² by sequences of Gaussian processes. These results are applied to investigate Cramér–von Mises-type statistics.     

Copulas related to piecewise monotone functions of the interval and associated processes

In this work, we derive the copulas related to vectors obtained from the so-called chaotic stochastic processes. These are defined by the iteration of certain piecewise monotone functions of the interval [0, 1] to some initial random variable. We study some of its properties and present some examples. Since often these types of copulas do not have closed formulas, we provide a general approximation method which converges uniformly to the true copula. Our results cover a wide class of processes, including the so-called Manneville–Pomeau processes. The general theory is applied to the parametric estimation in certain chaotic processes. A Monte Carlo simulation study is also presented.     

Bivariate degradation modelling with marginal heterogeneous stochastic processes

In this paper, we consider that the degradation of two performance characteristics of a product can be modelled by stochastic processes and jointly by copula functions, but different stochastic processes govern the behaviour of each performance characteristic (PC) degradation. Different heterogeneous and homogeneous models are presented considering copula functions and different combinations of the most used stochastic processes in degradation analysis as marginal distributions. This is an important aspect to consider because the behaviour of the degradation of each PC may be different in its nature. As the joint distributions of the proposed models result in complex distributions, the estimation of the parameters of interest is performed via MCMC. A simulation study is performed to compare heterogeneous and homogeneous models. In addition, the proposed models are implemented to crack propagation data of two terminals of an electronic device, and some insights are provided about the product reliability under heterogeneous models.     

General tests of independence based on empirical processes indexed by functions

The present paper is mainly concerned with the statistical tests of the independence problem between random vectors. We develop an approach based on general empirical processes indexed by a particular class of functions. We prove two abstract approximation theorems that include some existing results as particular cases. Finally, we characterize the limiting behavior of the Möbius transformation of empirical processes indexed by functions under contiguous sequences of alternatives.     

Evidential inference for diffusion-type processes

This article analyses diffusion-type processes from a new point-of-view. Consider two statistical hypotheses on a diffusion process. We do not use a classical test to reject or accept one hypothesis using the Neyman–Pearson procedure and do not involve Bayesian approach. As an alternative, we propose using a likelihood paradigm to characterizing the statistical evidence in support of these hypotheses. The method is based on evidential inference introduced and described by Royall [Royall R. Statistical evidence: a likelihood paradigm. London: Chapman and Hall; 1997]. In this paper, we extend the theory of Royall to the case when data are observations from a diffusion-type process instead of iid observations. The empirical distribution of likelihood ratio is used to formulate the probability of strong, misleading and weak evidences. Since the strength of evidence can be affected by the sampling characteristics, we present a simulation study that demonstrates these effects. Also we try to control misleading evidence and reduce them by adjusting these characteristics. As an illustration, we apply the method to the Microsoft stock prices.     

Normalizing constants for record values in Archimedean copula processes

In this paper, we detail an asymptotic behavior of record values for some types of Archimedean copula processes, refining the form of the corresponding normalizing constants.     

Weighted empirical spacings processes

We give a complete asymptotic characterization of the supremum of weighted empirical spacings processes.     

Monitoring test for stability of copula parameter in time series

In this paper, we consider a monitoring procedure to detect changes of the copula parameter of strong mixing processes. We propose two monitoring procedures based on the cumulative sums of scores evaluated at consistent copula parameter estimates and their fluctuations. We investigate the asymptotic properties of our monitoring procedures under both the null of no change in the copula parameter and its alternative. We also illustrate a simulation study and a real data analysis.     

The Copula Information Criteria

We derive two types of Akaike information criterion (AIC)‐like model‐selection formulae for the semiparametric pseudo‐maximum likelihood procedure. We first adapt the arguments leading to the original AIC formula, related to empirical estimation of a certain Kullback–Leibler information distance. This gives a significantly different formula compared with the AIC, which we name the copula information criterion. However, we show that such a model‐selection procedure cannot exist for copula models with densities that grow very fast near the edge of the unit cube. This problem affects most popular copula models. We then derive what we call the cross‐validation copula information criterion, which exists under weak conditions and is a first‐order approximation to exact cross validation. This formula is very similar to the standard AIC formula but has slightly different motivation. A brief illustration with real data is given.     

A Strong Invariance Theorem of the Tail Empirical Copula Processes

We study the behavior of bivariate empirical copula process 𝔾 _n (·, ·) on pavements [0, k _n /n]² of [0, 1]², where k _n is a sequence of positive constants fulfilling some conditions. We provide a upper bound for the strong approximation of 𝔾 _n (·, ·) by a Gaussian process when k _n /n↘γ as n → ∞, where 0 ≤ γ ≤1.     

Interacting neighbour point processes: Some models for clustering

We introduce a class of spatial point processesinteracting neighbour point (INP) processes, where the density of the process can be written by means of local interactions between a point and subsets of its neighbourhood but where the processes may not be Ripley-Kelly Markov processes with respect to this neighbourhood. We show that the processes are iterated Markov processes defined by Hayat and Gubner (1996). Furthermore, we pay special attention to a subclass of interacting neighbour processes, where the density belongs to the exponential family and all neighbours of a point affect it simultaneously. A simulation study is presented to show that some simple processes of this subclass can produce clustered patterns of great variety. Finally, an empirical example is given.     

Empirical and rank processes of observations and residuals

Empirical and rank processes of a sample are shown to converge under weak conditions. These results are then extended to analogous empirical and rank processes formed from the residuals of a linear model. Representations of the limiting processes are obtained by means of a special construction. The technicalities are honed to require few regularity conditions.     

Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes

In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck (ROU) processes based on continuous observations. Both the cases with one-sided barrier and two-sided barriers are considered. We derive the explicit formulas for the estimators, and then prove their strong consistency and asymptotic normality. Moreover, the bias and mean square errors are represented in terms of the solutions to some PDEs with homogeneous Neumann boundary conditions. We also illustrate the asymptotic behavior of the estimators through a simulation study.     

Multi-type insurance claim processes with high-dimensional covariates

Multi-type insurance claim processes have attracted considerable research interest in the literature. The existing statistical inference for such processes, however, may encounter “curse of dimensionality” due to high-dimensional covariates. In this article, a technique of sufficient dimension reduction is applied to multiple-type insurance claim data, which uses a copula to model the dependence between different types of claim processes, and incorporates a one-dimensional frailty to fit the dependence of claims “within” the same claim process. A two-step procedure is proposed to estimate model parameters. The first step develops nonparametric estimators of the baseline, the basis of the central subspace and its dimension, and the regression function. Then the second step estimates the copula parameter. Simulations are performed to evaluate and confirm the theoretical results.     

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.     

Testing independence based on Bernstein empirical copula and copula density

In this paper we provide three nonparametric tests of independence between continuous random variables based on the Bernstein copula distribution function and the Bernstein copula density function. The first test is constructed based on a Cramér-von Mises divergence-type functional based on the empirical Bernstein copula process. The two other tests are based on the Bernstein copula density and use Cramér-von Mises and Kullback–Leibler divergence-type functionals, respectively. Furthermore, we study the asymptotic null distribution of each of these test statistics. Finally, we consider a Monte Carlo experiment to investigate the performance of our tests. In particular we examine their size and power which we compare with those of the classical nonparametric tests that are based on the empirical distribution function.     

Asymptotic inference in poverty indices: an empirical processes approach

We derive an empirical poverty index containing most of those proposed in the literature. Then, we study its asymptotic behavior by using empirical processes theory. From the results obtained, we derive a Wald-type test for comparing a vector of theoretical poverty indices to a vector of reference indices. We study the test statistic under the null hypothesis of equality, fixed alternatives, and a sequence of local alternatives. A simulation experiment conducted shows that our test performs well for exponential and Pareto data.     

Copula Density Estimation Using Multiwavelets Based on the Multiresolution Analysis

Recently, wavelet has been used for copula density estimation. A known characteristic of wavelet functions is that they cannot be symmetric, orthogonal, and compact support at the same time while multiwavelets overcome this disadvantage. This article highlights the usefulness of the multiwavelet in order to approximate copula density functions. Possessing three appropriate properties at the same time, high smoothness, and high approximation order properties, multiwavelet can be more precise in copula density approximation. We make this approximation method more accurate by using multiresolution analysis. Finally, we apply our proposed method to approximate the copula density in actuarial data.     

Limit theorems for a general weighted process under random censoring

Necessary and sufficient conditions for weak and strong convergence are derived for the weighted version of a general process under random censoring. To be more explicit, this means that for this process complete analogues are obtained of the Chibisov-O'Reilly theorem, the Lai-Wellner Glivenko-Cantelli theorem, and the James law of the iterated logarithm for the empirical process. The process contains as special cases the so-called basic martingale, the empirical cumulative hazard process, and the product-limit process. As a tool we derive a Kiefer-process-type approximation of our process, which may be of independent interest.