866 research outputs found
Weak convergence of the weighted empirical beta copula process
The empirical copula has proved to be useful in the construction and
understanding of many statistical procedures related to dependence within
random vectors. The empirical beta copula is a smoothed version of the
empirical copula that enjoys better finite-sample properties. At the core lie
fundamental results on the weak convergence of the empirical copula and
empirical beta copula processes. Their scope of application can be increased by
considering weighted versions of these processes. In this paper we show weak
convergence for the weighted empirical beta copula process. The weak
convergence result for the weighted empirical beta copula process is stronger
than the one for the empirical copula and its use is more straightforward. The
simplicity of its application is illustrated for weighted Cram\'er--von Mises
tests for independence and for the estimation of the Pickands dependence
function of an extreme-value copula.Comment: 19 pages, 2 figure
Subsampling (weighted smooth) empirical copula processes
A key tool to carry out inference on the unknown copula when modeling a
continuous multivariate distribution is a nonparametric estimator known as the
empirical copula. One popular way of approximating its sampling distribution
consists of using the multiplier bootstrap. The latter is however characterized
by a high implementation cost. Given the rank-based nature of the empirical
copula, the classical empirical bootstrap of Efron does not appear to be a
natural alternative, as it relies on resamples which contain ties. The aim of
this work is to investigate the use of subsampling in the aforementioned
framework. The latter consists of basing the inference on statistic values
computed from subsamples of the initial data. One of its advantages in the
rank-based context under consideration is that the formed subsamples do not
contain ties. Another advantage is its asymptotic validity under minimalistic
conditions. In this work, we show the asymptotic validity of subsampling for
several (weighted, smooth) empirical copula processes both in the case of
serially independent observations and time series. In the former case,
subsampling is observed to be substantially better than the empirical bootstrap
and equivalent, overall, to the multiplier bootstrap in terms of finite-sample
performance.Comment: 34 pages, 5 figures, 4 + 8 table
Quantile spectral processes: Asymptotic analysis and inference
Quantile- and copula-related spectral concepts recently have been considered
by various authors. Those spectra, in their most general form, provide a full
characterization of the copulas associated with the pairs in a
process , and account for important dynamic features,
such as changes in the conditional shape (skewness, kurtosis),
time-irreversibility, or dependence in the extremes that their traditional
counterparts cannot capture. Despite various proposals for estimation
strategies, only quite incomplete asymptotic distributional results are
available so far for the proposed estimators, which constitutes an important
obstacle for their practical application. In this paper, we provide a detailed
asymptotic analysis of a class of smoothed rank-based cross-periodograms
associated with the copula spectral density kernels introduced in Dette et al.
[Bernoulli 21 (2015) 781-831]. We show that, for a very general class of
(possibly nonlinear) processes, properly scaled and centered smoothed versions
of those cross-periodograms, indexed by couples of quantile levels, converge
weakly, as stochastic processes, to Gaussian processes. A first application of
those results is the construction of asymptotic confidence intervals for copula
spectral density kernels. The same convergence results also provide asymptotic
distributions (under serially dependent observations) for a new class of
rank-based spectral methods involving the Fourier transforms of rank-based
serial statistics such as the Spearman, Blomqvist or Gini autocovariance
coefficients.Comment: Published at http://dx.doi.org/10.3150/15-BEJ711 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Statistics for Copula-based Measures of Multivariate Association - Theory and Applications to Financial Data
Concepts of association or dependence play a central role when considering multiple random sources in statistical models as they describe the relationship between two or more random variables. In particular, the concept of copulas has proven to be useful in many fields of application and research. Copulas split the multivariate distribution function of a random vector into the univariate marginal distribution functions and the dependence structure represented by the copula. This dissertation addresses the modeling, the estimation and the statistical inference of multivariate versions of copula-based measures of association such as Spearman's rho. Special focus is put on the analysis of the statistical properties of related nonparametric estimators as well as the derivation of statistical hypothesis tests. The latter may be used to verify specific modeling assumptions such as, for example, equal pairwise rank correlation. Further, statistical tests are developed to identify significant changes of association over time. The theoretical results are illustrated with applications to financial data
Estimation of Copula-Based Semiparametric Time Series Models
This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric invariant (or marginal) distributions and parametric copula functions that capture the temporal dependence of the processes; the implied transition distributions are all semiparametric. Models in this class are easy to simulate, and can be expressed as semiparametric regression transformation models. One advantage of this copula approach is to separate out the temporal dependence(such as tail dependence) from the marginal behavior (such as fat tailedness) of a time series. We present conditions under which processes generated by models in this class are -mixing; naturally, these conditions depend only on the copula specification. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. Estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles are easily obtained from estimators of the marginal distribution and the copula parameter; their consistency and asymptotic normality can be obtained using the Delta method. In addition, the semiparametric conditional quantile estimators are automatically monotonic across quantiles.Copula; Nonlinear Markov models; Semiparametric estimation;Conditional quantile
Penalized Sieve Estimation and Inference of Semi-Nonparametric Dynamic Models: A Selective Review
In this selective review, we first provide some empirical examples that motivate the usefulness of semi-nonparametric techniques in modelling economic and financial time series. We describe popular classes of semi-nonparametric dynamic models and some temporal dependence properties. We then present penalized sieve extremum (PSE) estimation as a general method for semi-nonparametric models with cross-sectional, panel, time series, or spatial data. The method is especially powerful in estimating difficult ill-posed inverse problems such as semi-nonparametric mixtures or conditional moment restrictions. We review recent advances on inference and large sample properties of the PSE estimators, which include (1) consistency and convergence rates of the PSE estimator of the nonparametric part; (2) limiting distributions of plug-in PSE estimators of functionals that are either smooth (i.e., root-n estimable) or non-smooth (i.e., slower than root-n estimable); (3) simple criterion-based inference for plug-in PSE estimation of smooth or non-smooth functionals; and (4) root-n asymptotic normality of semiparametric two-step estimators and their consistent variance estimators. Examples from dynamic asset pricing, nonlinear spatial VAR, semiparametric GARCH, and copula-based multivariate financial models are used to illustrate the general results
Conditional empirical copula processes and generalized dependence measures
We study the weak convergence of conditional empirical copula processes, when
the conditioning event has a nonzero probability. The validity of several
bootstrap schemes is stated, including the exchangeable bootstrap. We define
general - possibly conditional - multivariate dependence measures and their
estimators. By applying our theoretical results, we prove the asymptotic
normality of some estimators of such dependence measures.Comment: 29 pages, 4 figure
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