Stanislav Volgushev

Empirical and sequential empirical copula processes under serial dependence

Empirical and sequential empirical copula processes play a central role for statistical inference on copulas. However, as pointed out by Johan Segers [J. Segers, Asymptotics of empirical copula processes under non-restrictive smoothness assumptions, Bernoulli 18 (3) (2012) 764–782] the usual assumptions under which these processes have been studied so far are too restrictive. In this paper, we provide a unified approach to the analysis of empirical and sequential empirical copula processes that circumvents those restrictive assumptions in a very general setting. In particular, our methods allow for an easy analysis of copula processes and appropriate bootstrap approximations in the setting of sequentially dependent data. One particularly useful finding is that certain sequential empirical copula processes converge without any smoothness assumptions on the copula.

New estimators of the Pickands dependence function and a test for extreme-value dependence

We propose a new class of estimators for Pickands dependence function which is based on the best L 2 -approximation of the logarithm of the copula by logarithms of extremevalue copulas. An explicit integral representation of the best approximation is derived and it is shown that this approximation satises the boundary conditions of a Pickands dependence function. The estimators ^ A(t) are obtained by replacing the unknown copula by its empirical counterpart and weak convergence of the process p nf ^ A(t) A(t)g t2[0;1] is shown. A comparison with the commonly used estimators is performed from a theoretical point of view and by means of a simulation study. Our asymptotic and numerical results indicate that some of the new estimators outperform the rank-based versions of Pickands estimator and an estimator which was recently proposed by Genest and Segers (2009). As a by-product of our results we obtain a simple test for the hypothesis of an extreme-value copula, which is consistent against all alternatives with continuous partial derivatives of rst order satisfying C(u;v) uv.

Of Copulas, Quantiles, Ranks and Spectra - An L1-Approach to Spectral Analysis

In this paper we present an alternative method for the spectral analysis of a strictly stationary time series {Yt}t2Z. We define a “new” spectrum as the Fourier transform of the differences between copulas of the pairs (Yt, Yt−k) and the independence copula. This object is called copula spectral density kernel and allows to separate marginal and serial aspects of a time series. We show that it is intrinsically related to the concept of quantile regression. Like in quantile regression, which provides more information about the conditional distribution than the classical location-scale model, the copula spectral density kernel is more informative than the spectral density obtained from the autocovariances. In particular the approach provides a complete description of the distributions of all pairs (Yt, Yt−k). Moreover, it inherits the robustness properties of classical quantile regression, because it does not require any distributional assumptions such as the existence of finite moments. In order to estimate the copula spectral density kernel we introduce rank-based Laplace periodograms which are calculated as bilinear forms of weighted L1-projections of the ranks of the observed time series onto a harmonic regression model. We establish the asymptotic distribution of those periodograms, and the consistency of adequately smoothed versions. The finite-sample properties of the new methodology, and its potential for applications are briefly investigated by simulations and a short empirical example.

A test for Archimedeanity in bivariate copula models

We propose a new test for the hypothesis that a bivariate copula is an Archimedean copula which can be used as a preliminary step before further dependence modeling. The corresponding test statistic is based on a combination of two measures resulting from the characterization of Archimedean copulas by the property of associativity and by a strict upper bound on the diagonal by the Frechet-Hoeffding upper bound. We prove weak convergence of this statistic and show that the critical values of the corresponding test can be determined by the multiplier bootstrap method. The test is shown to be consistent against all departures from Archimedeanity. A simulation study is presented which illustrates the finite-sample properties of the new test.

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 (Xt,Xt-k) in a process (Xt )t Z, 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.

When uniform weak convergence fails: empirical processes for dependence functions via epi- and hypographs

For copulas whose partial derivatives are not continuous everywhere on the interior of the unit cube, the empirical copula process does not converge weakly with respect to the supremum distance. This makes it hard to verify asymptotic properties of inference procedures for such copulas. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epi- and hypoconvergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies Lp convergence. Even in cases where uniform convergence fails, weak convergence with respect to the new metric is established for empirical copula and tail dependence processes. No additional assumptions are needed for tail dependence functions, and for copulas, the assumptions reduce to existence and continuity of the partial derivatives almost everywhere on the unit cube. The results are applied to obtain asymptotic properties of minimum distance estimators, goodness-of-fit tests and resampling procedures.

Onset Dynamics of Action Potentials in Rat Neocortical Neurons and Identified Snail Neurons: Quantification of the Difference

The generation of action potentials (APs) is a key process in the operation of nerve cells and the communication between neurons. Action potentials in mammalian central neurons are characterized by an exceptionally fast onset dynamics, which differs from the typically slow and gradual onset dynamics seen in identified snail neurons. Here we describe a novel method of analysis which provides a quantitative measure of the onset dynamics of action potentials. This method captures the difference between the fast, step-like onset of APs in rat neocortical neurons and the gradual, exponential-like AP onset in identified snail neurons. The quantitative measure of the AP onset dynamics, provided by the method, allows us to perform quantitative analyses of factors influencing the dynamics.

Some Comments on Copula-Based Regression

In a recent article, Noh, El Ghouch, and Bouezmarni proposed a new semiparametric estimate of a regression function with a multivariate predictor, which is based on a specification of the dependence structure between the predictor and the response by means of a parametric copula. This comment investigates the effect which occurs under misspecification of the parametric model. We demonstrate by means of several examples that even for a one or two-dimensional predictor the error caused by a “wrong” specification of the parametric family is rather severe, if the regression is not monotone in one of the components of the predictor. Moreover, we also show that these problems occur for all of the commonly used copula families and we illustrate in several examples that the copula-based regression may lead to invalid results even when flexible copula models such as vine copulas (with the common parametric families) are used in the estimation procedure.

Energy-efficient encoding by shifting spikes in neocortical neurons

The speed of computations in neocortical networks critically depends on the ability of populations of spiking neurons to rapidly detect subtle changes in the input and translate them into firing rate changes. However, high sensitivity to perturbations may lead to explosion of noise and increased energy consumption. Can neuronal networks reconcile the requirements for high sensitivity, operation in a low‐noise regime, and constrained energy consumption? Using intracellular recordings in slices from the rat visual cortex, we show that layer 2/3 pyramidal neurons are highly sensitive to minor input perturbations. They can change their population firing rate in response to small artificial excitatory postsynaptic currents (aEPSCs) immersed in fluctuating noise very quickly, within 2–2.5 ms. These quick responses were mediated by the generation of new, additional action potentials (APs), but also by shifting spikes into the response peak. In that latter case, the spike count increase during the peak and the decrease after the peak cancelled each other, thus producing quick responses without increases in total spike count and associated energy costs. The contribution of spikes from one or the other source depended on the aEPSCs timing relative to the waves of depolarization produced by ongoing activity. Neurons responded by shifting spikes to aEPSCs arriving at the beginning of a depolarization wave, but generated additional spikes in response to aEPSCs arriving towards the end of a wave. We conclude that neuronal networks can combine high sensitivity to perturbations and operation in a low‐noise regime. Moreover, certain patterns of ongoing activity favor this combination and energy‐efficient computations.

Significance testing in quantile regression

We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that in contrast to the nonparametric approach based on estimation of L2-distances the new test is able to detect local alternatives which converge to the null hypothesis with any rate an → 0 such that an √ n → ∞ (here n denotes the sample size). We also present a small simulation study illustrating the finite sample properties of a bootstrap version of the the corresponding Kolmogorov-Smirnov test. AMS Classification: 62G10, 62G08, 62G30