arXiv subjects

Holger Dette

Publications and source records attributed to Holger Dette.

At least 181 records · Page 10Linked to original sources

arXiv2014

Detecting Gradual Changes in Locally Stationary Processes

In a wide range of applications, the stochastic properties of the observed time series change over time. The changes often occur gradually rather than abruptly: the prop- erties are (approximately) constant for some time and then slowly start to change. In such situations, it is frequently of interest to locate the time point where the properties start to vary. In contrast to the analysis of abrupt changes, methods for detecting smooth or gradual change points are less developed and often require strong paramet- ric assumptions. In this paper, we develop a fully nonparametric method to estimate a smooth change point in a locally stationary framework. We set up a general procedure which allows to deal with a wide variety of stochastic properties including the mean, (auto)covariances and higher-order moments. The theoretical part of the paper estab- lishes the convergence rate of the new estimator. In addition, we examine its finite sample performance by means of a simulation study and illustrate the methodology by applications to temperature and financial return data.

stat.ME↗

arXiv2013

A test for stationarity based on empirical processes

In this paper we investigate the problem of testing the assumption of stationarity in locally stationary processes. The test is based on an estimate of a Kolmogorov-Smirnov type distance between the true time varying spectral density and its best approximation through a stationary spectral density. Convergence of a time varying empirical spectral process indexed by a class of certain functions is proved, and furthermore the consistency of a bootstrap procedure is shown which is used to approximate the limiting distribution of the test statistic. Compared to other methods proposed in the literature for the problem of testing for stationarity the new approach has at least two advantages: On one hand, the test can detect local alternatives converging to the null hypothesis at any rate $g_T\to0$ such that $g_TT^{1/2}\to \infty$, where $T$ denotes the sample size. On the other hand, the estimator is based on only one regularization parameter while most alternative procedures require two. Finite sample properties of the method are investigated by means of a simulation study, and a comparison with several other tests is provided which have been proposed in the literature.

math.ST↗

arXiv2013

Optimal designs for multi-response generalized linear models with applications in thermal spraying

We consider the problem of designing experiments for investigating particle in-flight properties in thermal spraying. Observations are available on an extensive design for an initial day and thereafter in limited number for any particular day. Generalized linear models including additional day effects are used for analyzing the process, where the models vary with respect to different responses. We construct robust D-optimal designs to collect additional data on any current day, which are efficient for the estimation of the parameters in all models under consideration. These designs improve a reference fractional factorial design substantially. We also investigate designs, which maximize the power of the test for an additional day effect. The results are used to design additional experiments of the thermal spraying process and a comparison of the statistical analysis based on a reference design as well as on a selected D-optimal design is performed.

stat.AP↗

arXiv2013

Multiplier bootstrap of tail copulas with applications

For the problem of estimating lower tail and upper tail copulas, we propose two bootstrap procedures for approximating the distribution of the corresponding empirical tail copulas. The first method uses a multiplier bootstrap of the empirical tail copula process and requires estimation of the partial derivatives of the tail copula. The second method avoids this estimation problem and uses multipliers in the two-dimensional empirical distribution function and in the estimates of the marginal distributions. For both multiplier bootstrap procedures, we prove consistency. For these investigations, we demonstrate that the common assumption of the existence of continuous partial derivatives in the the literature on tail copula estimation is so restrictive, such that the tail copula corresponding to tail independence is the only tail copula with this property. Moreover, we are able to solve this problem and prove weak convergence of the empirical tail copula process under nonrestrictive smoothness assumptions that are satisfied for many commonly used models. These results are applied in several statistical problems, including minimum distance estimation and goodness-of-fit testing.

math.ST↗

arXiv2013

Smooth backfitting in additive inverse regression

We consider the problem of estimating an additive regression function in an inverse regres- sion model with a convolution type operator. A smooth backfitting procedure is developed and asymptotic normality of the resulting estimator is established. Compared to other meth- ods for the estimation in additive models the new approach neither requires observations on a regular grid nor the estimation of the joint density of the predictor. It is also demonstrated by means of a simulation study that the backfitting estimator outperforms the marginal in- tegration method at least by a factor two with respect to the integrated mean squared error criterion.

stat.ME↗

arXiv2013

Optimal designs for nonlinear regression models with respect to non-informative priors

In nonlinear regression models the Fisher information depends on the parameters of the model. Consequently, optimal designs maximizing some functional of the information matrix cannot be implemented directly but require some preliminary knowledge about the unknown parameters. Bayesian optimality criteria provide an attractive solution to this problem. These criteria depend sensitively on a reasonable specification of a prior distribution for the model parameters which might not be available in all applications. In this paper we investigate Bayesian optimality criteria with non-informative prior dis- tributions. In particular, we study the Jeffreys and the Berger-Bernardo prior for which the corresponding optimality criteria are not necessarily concave. Several examples are investigated where optimal designs with respect to the new criteria are calculated and compared to Bayesian optimal designs based on a uniform and a functional uniform prior.

stat.ME↗

arXiv2013

Misspecification in copula-based regression

In a recent paper Noh et al. (2013) 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 paper investigates the effect which occurs under misspecification of the parametric model. We demonstrate 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 more exible copula models such as vine copulae (with the common parametric families) are used in the estimation procedure.

stat.ME↗

arXiv2013

Robust T-optimal discriminating designs

This paper considers the problem of constructing optimal discriminating experimental designs for competing regression models on the basis of the T-optimality criterion introduced by Atkinson and Fedorov [Biometrika 62 (1975) 57-70]. T-optimal designs depend on unknown model parameters and it is demonstrated that these designs are sensitive with respect to misspecification. As a solution to this problem we propose a Bayesian and standardized maximin approach to construct robust and efficient discriminating designs on the basis of the T-optimality criterion. It is shown that the corresponding Bayesian and standardized maximin optimality criteria are closely related to linear optimality criteria. For the problem of discriminating between two polynomial regression models which differ in the degree by two the robust T-optimal discriminating designs can be found explicitly. The results are illustrated in several examples.

math.ST↗

arXiv2013

Detection of multiple structural breaks in multivariate time series

We propose a new nonparametric procedure for the detection and estimation of multiple structural breaks in the autocovariance function of a multivariate (second- order) piecewise stationary process, which also identifies the components of the series where the breaks occur. The new method is based on a comparison of the estimated spectral distribution on different segments of the observed time series and consists of three steps: it starts with a consistent test, which allows to prove the existence of structural breaks at a controlled type I error. Secondly, it estimates sets containing possible break points and finally these sets are reduced to identify the relevant structural breaks and corresponding components which are responsible for the changes in the autocovariance structure. In contrast to all other methods which have been proposed in the literature, our approach does not make any parametric assumptions, is not especially designed for detecting one single change point and addresses the problem of multiple structural breaks in the autocovariance function directly with no use of the binary segmentation algorithm. We prove that the new procedure detects all components and the corresponding locations where structural breaks occur with probability converging to one as the sample size increases and provide data-driven rules for the selection of all regularization parameters. The results are illustrated by analyzing financial returns, and in a simulation study it is demonstrated that the new procedure outperforms the currently available nonparametric methods for detecting breaks in the dependency structure of multivariate time series.

math.ST↗

arXiv2013

Complete classes of designs for nonlinear regression models and principal representations of moment spaces

In a recent paper Yang and Stufken [Ann. Statist. 40 (2012a) 1665-1685] gave sufficient conditions for complete classes of designs for nonlinear regression models. In this note we demonstrate that there is an alternative way to validate this result. Our main argument utilizes the fact that boundary points of moment spaces generated by Chebyshev systems possess unique representations.

math.ST↗

arXiv2013

Censored quantile regression processes under dependence and penalization

We consider quantile regression processes from censored data under dependent data structures and derive a uniform Bahadur representation for those processes. We also consider cases where the dimension of the parameter in the quantile regression model is large. It is demonstrated that traditional penalized estimators such as the adaptive lasso yield sub-optimal rates if the coefficients of the quantile regression cross zero. New penalization techniques are introduced which are able to deal with specific problems of censored data and yield estimates with an optimal rate. In contrast to most of the literature, the asymptotic analysis does not require the assumption of independent observations, but is based on rather weak assumptions, which are satisfied for many kinds of dependent data.

math.ST↗

arXiv2013

Optimal discriminating designs for several competing regression models

The problem of constructing optimal discriminating designs for a class of regression models is considered. We investigate a version of the $T_p$-optimality criterion as introduced by Atkinson and Fedorov [Biometrika 62 (1975a) 289-303]. The numerical construction of optimal designs is very hard and challenging, if the number of pairwise comparisons is larger than 2. It is demonstrated that optimal designs with respect to this type of criteria can be obtained by solving (nonlinear) vector-valued approximation problems. We use a characterization of the best approximations to develop an efficient algorithm for the determination of the optimal discriminating designs. The new procedure is compared with the currently available methods in several numerical examples, and we demonstrate that the new method can find optimal discriminating designs in situations where the currently available procedures fail.

math.ST↗

arXiv2013

Measuring stationarity in long-memory processes

In this paper we consider the problem of measuring stationarity in locally stationary long-memory processes. We introduce an $L_2$-distance between the spectral density of the locally stationary process and its best approximation under the assumption of stationarity. The distance is estimated by a numerical approximation of the integrated spectral periodogram and asymptotic normality of the resulting estimate is established. The results can be used to construct a simple test for the hypothesis of stationarity in locally stationary long-range dependent processes. We also propose a bootstrap procedure to improve the approximation of the nominal level and prove its consistency. Throughout the paper, we will work with Riemann sums of a squared periodogram instead of integrals (as it is usually done in the literature) and as a by-product of independent interest it is demonstrated that the two approaches behave differently in the limit.

math.ST↗

arXiv2013

Optimal design for linear models with correlated observations

In the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated. A necessary condition for the optimality of a given design is provided, which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data. If the regression functions are eigenfunctions of an integral operator defined by the covariance kernel, it is shown that the corresponding measure defines a universally optimal design. For several models universally optimal designs can be identified explicitly. In particular, it is proved that the uniform distribution is universally optimal for a class of trigonometric regression models with a broad class of covariance kernels and that the arcsine distribution is universally optimal for the polynomial regression model with correlation structure defined by the logarithmic potential. To the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations, which are not restricted to the location scale model.

math.ST↗

arXiv2012

Zeros and ratio asymptotics for matrix orthogonal polynomials

Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients $A_n$ and $B_n$ having limits $A$ and $B$ respectively (the matrix Nevai class) were obtained by Durán. In the present paper we obtain an alternative description of the limiting ratio. We generalize it to recurrence coefficients which are asymptotically periodic with higher periodicity, and/or which are slowly varying in function of a parameter. Under such assumptions, we also find the limiting zero distribution of the matrix orthogonal polynomials, generalizing results by Durán-López-Saff and Dette-Reuther to the non-Hermitian case. Our proofs are based on "normal family" arguments and on the solution to a quadratic eigenvalue problem. As an application of our results we obtain new explicit formulas for the spectral measures of the matrix Chebyshev polynomials of the first and second kind, and we derive the asymptotic eigenvalue distribution for a class of random band matrices generalizing the tridiagonal matrices introduced by Dumitriu-Edelman.

math.CA↗

arXiv2012

Model checks for the volatility under microstructure noise

We consider the problem of testing the parametric form of the volatility for high frequency data. It is demonstrated that in the presence of microstructure noise commonly used tests do not keep the preassigned level and are inconsistent. The concept of preaveraging is used to construct new tests, which do not suffer from these drawbacks. These tests are based on a Kolmogorov-Smirnov or Cramer-von-Mises functional of an integrated stochastic process, for which weak convergence to a (conditional) Gaussian process is established. The finite sample properties of a bootstrap version of the test are illustrated by means of a simulation study.

math.ST↗

arXiv2012

Distributions on unbounded moment spaces and random moment sequences

In this paper we define distributions on moment spaces corresponding to measures on the real line with an unbounded support. We identify these distributions as limiting distributions of random moment vectors defined on compact moment spaces and as distributions corresponding to random spectral measures associated with the Jacobi, Laguerre and Hermite ensemble from random matrix theory. For random vectors on the unbounded moment spaces we prove a central limit theorem where the centering vectors correspond to the moments of the Marchenko-Pastur distribution and Wigner's semi-circle law.

math.PR↗

arXiv2012

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 $L^2$-distances - the new test is able to detect local alternatives which converge to the null hypothesis with any rate $a_n \to 0$ such that $a_n \sqrt{n} \to \infty$ (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.

stat.ME↗