Shrinkage estimation in the frequency domain of multivariate time series

In this paper on developing shrinkage for spectral analysis of multivariate time series of high dimensionality, we propose a new nonparametric estimator of the spectral matrix with two appealing properties. First, compared to the traditional smoothed periodogram our shrinkage estimator has a smaller L₂ risk. Second, the proposed shrinkage estimator is numerically more stable due to a smaller condition number. We use the concept of “Kolmogorov” asymptotics where simultaneously the sample size and the dimensionality tend to infinity, to show that the smoothed periodogram is not consistent and to derive the asymptotic properties of our regularized estimator. This estimator is shown to have asymptotically minimal risk among all linear combinations of the identity and the averaged periodogram matrix. Compared to existing work on shrinkage in the time domain, our results show that in the frequency domain it is necessary to take the size of the smoothing span as “effective sample size” into account. Furthermore, we perform extensive Monte Carlo studies showing the overwhelming gain in terms of lower L₂ risk of our shrinkage estimator, even in situations of oversmoothing the periodogram by using a large smoothing span.     

A multivariate version of Hoeffding’s Phi-Square

A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.     

Hybrid bootstrap aided unit root testing

In this paper, we propose a hybrid bootstrap procedure for augmented Dickey-Fuller (ADF) tests for the presence of a unit root. This hybrid proposal combines a time domain parametric autoregressive fit to the data and a nonparametric correction applied in the frequency domain to capture features that are possibly not represented by the parametric model. It is known that considerable size and power problems can occur in small samples for unit root testing in the presence of an MA parameter using critical values of the asymptotic Dickey-Fuller distribution. The benefit of the sieve bootstrap in this situation has been investigated by Chang and Park (J Time Ser Anal 24:379–400, 2003). They showed asymptotic validity as well as substantial improvements for small sample sizes, but the actual sizes of their bootstrap tests were still quite far away from the nominal size. The finite sample performances of our procedure are extensively investigated through Monte Carlo simulations and compared to the sieve bootstrap approach. Regarding the size of the tests, our results show that the hybrid bootstrap remarkably outperforms the sieve bootstrap.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

Testing model assumptions in functional regression models

In the functional regression model where the responses are curves, new tests for the functional form of the regression and the variance function are proposed, which are based on a stochastic process estimating L²-distances. Our approach avoids the explicit estimation of the functional regression and it is shown that normalized versions of the proposed test statistics converge weakly. The finite sample properties of the tests are illustrated by means of a small simulation study. It is also demonstrated that for small samples, bootstrap versions of the tests improve the quality of the approximation of the nominal level.     

An asymptotics look at the generalized inference

This paper provides an asymptotics look at the generalized inference through showing connections between the generalized inference and two widely used asymptotic methods, the bootstrap and plug-in method. A generalized bootstrap method and a generalized plug-in method are introduced. The generalized bootstrap method can not only be used to prove asymptotic frequentist properties of existing generalized confidence regions through viewing fiducial generalized pivotal quantities as generalized bootstrap variables, but also yield new confidence regions for the situations where the generalized inference is unavailable. Some examples are presented to illustrate the method. In addition, the generalized F-test (Weerahandi, 1995 [26]) can be derived by the generalized plug-in method, then its asymptotic validity is obtained.     

Modified Whittle estimation of multilateral models on a lattice

In the estimation of parametric models for stationary spatial or spatio-temporal data on a d-dimensional lattice, for d?2, the achievement of asymptotic efficiency under Gaussianity, and asymptotic normality more generally, with standard convergence rate, faces two obstacles. One is the “edge effect”, which worsens with increasing d. The other is the possible difficulty of computing a continuous-frequency form of Whittle estimate or a time domain Gaussian maximum likelihood estimate, due mainly to the Jacobian term. This is especially a problem in “multilateral” models, which are naturally expressed in terms of lagged values in both directions for one or more of the d dimensions. An extension of the discrete-frequency Whittle estimate from the time series literature deals conveniently with the computational problem, but when subjected to a standard device for avoiding the edge effect has disastrous asymptotic performance, along with finite sample numerical drawbacks, the objective function lacking a minimum-distance interpretation and losing any global convexity properties. We overcome these problems by first optimizing a standard, guaranteed non-negative, discrete-frequency, Whittle function, without edge-effect correction, providing an estimate with a slow convergence rate, then improving this by a sequence of computationally convenient approximate Newton iterations using a modified, almost-unbiased periodogram, the desired asymptotic properties being achieved after finitely many steps. The asymptotic regime allows increase in both directions of all d dimensions, with the central limit theorem established after re-ordering as a triangular array. However our work offers something new for “unilateral” models also. When the data are non-Gaussian, asymptotic variances of all parameter estimates may be affected, and we propose consistent, non-negative definite estimates of the asymptotic variance matrix.     

Kernel-based goodness-of-fit tests for copulas with fixed smoothing parameters

We study a test statistic based on the integrated squared difference between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. We show for fixed smoothing parameters that the test is consistent and that the asymptotic properties are driven by a U-statistic of order 4 with degeneracy of order 1. For practical implementation we suggest to compute the critical values through a semiparametric bootstrap. Monte Carlo results show that the bootstrap procedure performs well in small samples. In particular, size and power are less sensitive to smoothing parameter choice than they are under the asymptotic approximation obtained for a vanishing bandwidth.     

The inverted complex Wishart distribution and its application to spectral estimation

The inverted complex Wishart distribution and its use for the construction of spectral estimates are studied. The density, some marginals of the distribution, and the first- and second-order moments are given. For a vector-valued time series, estimation of the spectral density at a collection of frequencies and estimation of the increments of the spectral distribution function in each of a set of frequency bands are considered. A formal procedure applies Bayes theorem, where the complex Wishart is used to represent the distribution of an average of adjacent periodogram values. A conjugate prior distribution for each parameter is an inverted complex Wishart distribution. Use of the procedure for estimation of a 2 × 2 spectral density matrix is discussed.     

A new test for sphericity of the covariance matrix for high dimensional data

In this paper we propose a new test procedure for sphericity of the covariance matrix when the dimensionality, p, exceeds that of the sample size, N=n+1. Under the assumptions that (A) as p→∞ for i=1,…,16 and (B) p/n→c<∞ known as the concentration, a new statistic is developed utilizing the ratio of the fourth and second arithmetic means of the eigenvalues of the sample covariance matrix. The newly defined test has many desirable general asymptotic properties, such as normality and consistency when (n,p)→∞. Our simulation results show that the new test is comparable to, and in some cases more powerful than, the tests for sphericity in the current literature.     

Bootstrap,wild bootstrap,and asymptotic normality

Summary We show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend onn. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap.This work has been supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 123 Stochastische Mathematische Modelle     

Bootstrap approximation of tail dependence function

For estimating a rare event via the multivariate extreme value theory, the so-called tail dependence function has to be investigated (see [L. de Haan, J. de Ronde, Sea and wind: Multivariate extremes at work, Extremes 1 (1998) 7-45]). A simple, but effective estimator for the tail dependence function is the tail empirical distribution function, see [X. Huang, Statistics of Bivariate Extreme Values, Ph.D. Thesis, Tinbergen Institute Research Series, 1992] or [R. Schmidt, U. Stadtmüller, Nonparametric estimation of tail dependence, Scand. J. Stat. 33 (2006) 307-335]. In this paper, we first derive a bootstrap approximation for a tail dependence function with an approximation rate via the construction approach developed by [K. Chen, S.H. Lo, On a mapping approach to investigating the bootstrap accuracy, Probab. Theory Relat. Fields 107 (1997) 197-217], and then apply it to construct a confidence band for the tail dependence function. A simulation study is conducted to assess the accuracy of the bootstrap approach.     

Multivariate logistic regression with incomplete covariate and auxiliary information

In this article, we propose and explore a multivariate logistic regression model for analyzing multiple binary outcomes with incomplete covariate data where auxiliary information is available. The auxiliary data are extraneous to the regression model of interest but predictive of the covariate with missing data. Horton and Laird [N.J. Horton, N.M. Laird, Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information, Biometrics 57 (2001) 34–42] describe how the auxiliary information can be incorporated into a regression model for a single binary outcome with missing covariates, and hence the efficiency of the regression estimators can be improved. We consider extending the method of [9] to the case of a multivariate logistic regression model for multiple correlated outcomes, and with missing covariates and completely observed auxiliary information. We demonstrate that in the case of moderate to strong associations among the multiple outcomes, one can achieve considerable gains in efficiency from estimators in a multivariate model as compared to the marginal estimators of the same parameters.     

Low dimensional semiparametric estimation in a censored regression model

A new estimation procedure for a partial linear additive model with censored responses is proposed. To this aim, ideas of Lewbel and Linton [A. Lewbel, O. Linton, Nonparametric censored and truncated regression, Econometrica 70 (2002) 765-779] on censored model regression are combined with those of Kim et al. [W. Kim, O. Linton, N.W. Hengartner, A computationally efficient estimator for additive nonparametric regression with bootstrap confidence intervals, Journal of Computational and Graphical Statistics, 8 (1999) 278-297] on marginal integration and those on average derivatives. This allows for dimension reduction, interpretability and — depending on the context — for weights yielding computationally attractive estimates. Asymptotic behavior is provided for all proposed estimators.     

Second-order accuracy of depth-based bootstrap confidence regions

We consider the problem of setting bootstrap confidence regions for multivariate parameters based on data depth functions. We prove, under mild regularity conditions, that depth-based bootstrap confidence regions are second-order accurate in the sense that their coverage error is of order n⁻¹, given a random sample of size n. The results hold in general for depth functions of types A and D, which cover as special cases the Tukey depth, the majority depth, and the simplicial depth. A simulation study is also provided to investigate empirically the bootstrap confidence regions constructed using these three depth functions.     

Cross-validated bagged learning

Many applications aim to learn a high dimensional parameter of a data generating distribution based on a sample of independent and identically distributed observations. For example, the goal might be to estimate the conditional mean of an outcome given a list of input variables. In this prediction context, bootstrap aggregating (bagging) has been introduced as a method to reduce the variance of a given estimator at little cost to bias. Bagging involves applying an estimator to multiple bootstrap samples and averaging the result across bootstrap samples. In order to address the curse of dimensionality, a common practice has been to apply bagging to estimators which themselves use cross-validation, thereby using cross-validation within a bootstrap sample to select fine-tuning parameters trading off bias and variance of the bootstrap sample-specific candidate estimators. In this article we point out that in order to achieve the correct bias variance trade-off for the parameter of interest, one should apply the cross-validation selector externally to candidate bagged estimators indexed by these fine-tuning parameters. We use three simulations to compare the new cross-validated bagging method with bagging of cross-validated estimators and bagging of non-cross-validated estimators.     

Robustness of Stein-type estimators under a non-scalar error covariance structure

The Stein-rule (SR) and positive-part Stein-rule (PSR) estimators are two popular shrinkage techniques used in linear regression, yet very little is known about the robustness of these estimators to the disturbances’ deviation from the white noise assumption. Recent studies have shown that the OLS estimator is quite robust, but whether this is so for the SR and PSR estimators is less clear as these estimators also depend on the F statistic which is highly susceptible to covariance misspecification. This study attempts to evaluate the effects of misspecifying the disturbances as white noise on the SR and PSR estimators by a sensitivity analysis. Sensitivity statistics of the SR and PSR estimators are derived and their properties are analyzed. We find that the sensitivity statistics of these estimators exhibit very similar properties and both estimators are extremely robust to MA(1) disturbances and reasonably robust to AR(1) disturbances except for the cases of severe autocorrelation. The results are useful in light of the rising interest of the SR and PSR techniques in the applied literature.     

On nondifferentiable functions and the bootstrap

Summary We investigate a class of statistical problems, where usual bootstrap methods fail, and discuss two alternative solutions. In particular, a stochastic procedure for constructing confidence sets is proposed. Special applications are the eigenvalues of a covariance matrix and minimum distance functionals.Work supported by the Miller Institute for Basic Research in Science, Berkeley     

Concordance measures for multivariate non-continuous random vectors

A notion of multivariate concordance suitable for non-continuous random variables is defined and many of its properties are established. This allows the definition of multivariate, non-continuous versions of Kendall’s tau, Spearman’s rho and Spearman’s footrule, which are concordance measures. Since the maximum values of these association measures are not +1 in general, a special attention is given to the computation of upper bounds. The latter turn out to be multivariate generalizations of earlier findings made by Nešlehová (2007) [9] and Denuit and Lambert (2005) [2]. They are easy to compute and can be estimated from a data set of (possibly) discontinuous random vectors. Corrected versions are considered as well.     

Statistical inference of partially linear regression models with heteroscedastic errors

The authors study a heteroscedastic partially linear regression model and develop an inferential procedure for it. This includes a test of heteroscedasticity, a two-step estimator of the heteroscedastic variance function, semiparametric generalized least-squares estimators of the parametric and nonparametric components of the model, and a bootstrap goodness of fit test to see whether the nonparametric component can be parametrized.