arXiv subjects
Publications and source records attributed to Natalie Neumeyer.
A new approach for nonparametric estimation of the quantile density function (sparsity function) and its derivatives is suggested which is based on local polynomial estimation. The estimator has more advantageous properties at the boundaries than classical quantile density estimators. Asymptotic normality is shown and the bias, asymptotic variance as well as boundary properties are compared with other estimators.
Consider a nonparametric regression model with one-dimensional covariates and a continuous regression function. Assume that the regression function from the left of the covariate support starts equal to zero and then changes at some unknown point. Our aim is to estimate this gradual change point. We define and compare various consistent estimators based on a new general optimization method in the case where the aim is to estimate the largest minimization point of some objective function. We discuss rates of convergence and estimating the regression function based on the gradual change structure. Bootstrap bias approximation is discussed. Further applications in a two sample case are considered, where two continuous regression functions first equal and then change at some point of interest.
We consider the error distribution in functional linear models with scalar response and functional covariate. Different asymptotic expansions of the empirical distribution function and the empirical characteristic function based on estimated residuals under different model assumptions are discussed. The results are applied for simple and composite goodness-of-fit testing for the error distribution, in particular testing for normal distribution.
In a multivariate nonparametric regression setting we construct explicit asymptotic uniform confidence bands for centered purely random forests. Since the most popular example in this class of random forests, namely the uniformly centered purely random forests, is well known to suffer from suboptimal rates, we propose a new type of purely random forests, called the Ehrenfest centered purely random forests, which achieve minimax optimal rates. Our main confidence band theorem applies to both random forests. The proof is based on an interpretation of random forests as generalized U-Statistics together with a Gaussian approximation of the supremum of empirical processes. Our theoretical findings are illustrated in simulation examples.
Asymptotic uniform confidence bands are constructed for a multivariate nonparametric regression model with heteroscedastic noise, employing histogram estimators under flexible partition conditions. The construction is especially applicable to unsmooth regression functions of H\"older regularity less than one. While the radius of the confidence bands could be approximated via the Gumbel distribution, our construction does not depend on an extreme value distribution, but instead can be explicitly calculated for the chosen partition.
We consider linear models with scalar responses and covariates from a separable Hilbert space. The aim is to detect change points in the error distribution, based on sequential residual empirical distribution functions. Expansions for those estimated functions are more challenging in models with infinite-dimensional covariates than in regression models with scalar or vector-valued covariates due to a slower rate of convergence of the parameter estimators. Yet the suggested change point test is asymptotically distribution-free and consistent for one-change point alternatives. In the latter case we also show consistency of a change point estimator.
We consider the copula mapping, which maps a joint cumulative distribution function to the corresponding copula. Its Hadamard differentiablity was shown in van der Vaart and Wellner (1996), Fermanian et al. (2004) and (under less strict assumptions) in Bücher and Volgushev (2013). This differentiability result has proved to be a powerful tool to show weak convergence of empirical copula processes in various settings using the functional delta method. We state a generalization of the Hadamard differentiability results that simplifies the derivations of asymptotic expansions and weak convergence of empirical copula processes in the presence of covariates. The usefulness of this result is illustrated on several applications which include a multidimensional functional linear model, where the copula of the error vector describes the dependency between the components of the vector of observations, given the functional covariate.
In transformation regression models the response is transformed before fitting a regression model to covariates and transformed response. We assume such a model where the errors are independent from the covariates and the regression function is modeled nonparametrically. We suggest a test for goodness-of-fit of a parametric transformation class based on a distance between a nonparametric transformation estimator and the parametric class. We present asymptotic theory under the null hypothesis of validity of the semi-parametric model and under local alternatives. A bootstrap algorithm is suggested in order to apply the test. We also consider relevant hypotheses to distinguish between large and small distances of the parametric transformation class to the `true' transformation.
We consider a nonparametric heteroscedastic time series regression model and suggest testing procedures to detect changes in the conditional variance function. The tests are based on a sequential marked empirical process and thus combine classical CUSUM tests with marked empirical process approaches known from goodness-of-fit testing. The tests are consistent against general alternatives of a change in the conditional variance function, a feature that classical CUSUM tests are lacking. We derive a simple limiting distribution and in the case of univariate covariates even obtain asymptotically distribution-free tests. We demonstrate the good performance of the tests in a simulation study and consider exchange rate data as a real data application.
This paper deals with a situation when one is interested in the dependence structure of a multidimensional response variable in the presence of a multivariate covariate. It is assumed that the covariate affects only the marginal distributions through regression models while the dependence structure, which is described by a copula, is unaffected. A parametric estimation of the copula function is considered with focus on the maximum pseudo-likelihood method. It is proved that under some appropriate regularity assumptions the estimator calculated from the residuals is asymptotically equivalent to the estimator based on the unobserved errors. In such case one can ignore the fact that the response is first adjusted for the effect of the covariate. A Monte Carlo simulation study explores (among others) situations where the regularity assumptions are not satisfied and the claimed result does not hold. It shows that in such situations the maximum pseudo-likelihood estimator may behave poorly and the moment estimation of the copula parameter is of interest. Our results complement the results available for nonparametric estimation of the copula function.
In the context of nonparametric regression models with one-sided errors, we consider parametric transformations of the response variable in order to obtain independence between the errors and the covariates. We focus in this paper on stritcly increasing and continuous transformations. In view of estimating the tranformation parameter, we use a minimum distance approach and show the uniform consistency of the estimator under mild conditions. The boundary curve, i.e. the regression function, is estimated applying a smoothed version of a local constant approximation for which we also prove the uniform consistency. We deal with both cases of random covariates and deterministic (fixed) design points. To highlight the applicability of the procedures and to demonstrate their performance, the small sample behavior is investigated in a simulation study using the so-called Yeo-Johnson transformations.
We consider semiparametric transformation models, where after pre-estimation of a parametric transformation of the response the data are modeled by means of nonparametric regression. We suggest subsequent procedures for testing lack-of-fit of the regression function and for significance of covariables, which - in contrast to procedures from the literature - are asymptotically not influenced by the pre-estimation of the transformation. The test statistics are asymptotically pivotal and have the same asymptotic distribution as in regression models without transformation. We show validity of a multiplier bootstrap procedure which is easier to implement and much less computationally demanding than bootstrap procedures based on the transformation model. In a simulation study we demonstrate the superior performance of the procedure in comparison with the competitors from the literature.
A weakly dependent time series regression model with multivariate covariates and univariate observations is considered, for which we develop a procedure to detect whether the nonparametric conditional mean function is stable in time against change point alternatives. Our proposal is based on a modified CUSUM type test procedure, which uses a sequential marked empirical process of residuals. We show weak convergence of the considered process to a centered Gaussian process under the null hypothesis of no change in the mean function and a stationarity assumption. This requires some sophisticated arguments for sequential empirical processes of weakly dependent variables. As a consequence we obtain convergence of Kolmogorov-Smirnov and Cramér-von Mises type test statistics. The proposed procedure acquires a very simple limiting distribution and nice consistency properties, features from which related tests are lacking. We moreover suggest a bootstrap version of the procedure and discuss its applicability in the case of unstable variances.
This paper is concerned with modeling the dependence structure of two (or more) time-series in the presence of a (possible multivariate) covariate which may include past values of the time series. We assume that the covariate influences only the conditional mean and the conditional variance of each of the time series but the distribution of the standardized innovations is not influenced by the covariate and is stable in time. The joint distribution of the time series is then determined by the conditional means, the conditional variances and the marginal distributions of the innovations, which we estimate nonparametrically, and the copula of the innovations, which represents the dependency structure. We consider a nonparametric as well as a semiparametric estimator based on the estimated residuals. We show that under suitable assumptions these copula estimators are asymptotically equivalent to estimators that would be based on the unobserved innovations. The theoretical results are illustrated by simulations and a real data example.
In this paper we consider a location model of the form $Y = m(X) + \varepsilon$, where $m(\cdot)$ is the unknown regression function, the error $\varepsilon$ is independent of the $p$-dimensional covariate $X$ and $E(\varepsilon)=0$. Given i.i.d. data $(X_1,Y_1),\ldots,(X_n,Y_n)$ and given an estimator $\hat m(\cdot)$ of the function $m(\cdot)$ (which can be parametric or nonparametric of nature), we estimate the distribution of the error term $\varepsilon$ by the empirical distribution of the residuals $Y_i-\hat m(X_i)$, $i=1,\ldots,n$. To approximate the distribution of this estimator, Koul and Lahiri (1994) and Neumeyer (2008, 2009) proposed bootstrap procedures, based on smoothing the residuals either before or after drawing bootstrap samples. So far it has been an open question whether a classical non-smooth residual bootstrap is asymptotically valid in this context. In this paper we solve this open problem, and show that the non-smooth residual bootstrap is consistent. We illustrate this theoretical result by means of simulations, that show the accuracy of this bootstrap procedure for various models, testing procedures and sample sizes.
Consider a nonparametric regression model with one-sided errors and regression function in a general Hölder class. We estimate the regression function via minimization of the local integral of a polynomial approximation. We show uniform rates of convergence for the simple regression estimator as well as for a smooth version. These rates carry over to mean regression models with a symmetric and bounded error distribution. In such a setting, one obtains faster rates for irregular error distributions concentrating sufficient mass near the endpoints than for the usual regular distributions. The results are applied to prove asymptotic $\sqrt{n}$-equivalence of a residual-based (sequential) empirical distribution function to the (sequential) empirical distribution function of unobserved errors in the case of irregular error distributions. This result is remarkably different from corresponding results in mean regression with regular errors. It can readily be applied to develop goodness-of-fit tests for the error distribution. We present some examples and investigate the small sample performance in a simulation study. We further discuss asymptotically distribution-free hypotheses tests for independence of the error distribution from the points of measurement and for monotonicity of the boundary function as well.
In this paper an autoregressive time series model with conditional heteroscedasticity is considered, where both conditional mean and conditional variance function are modeled nonparametrically. A test for the model assumption of independence of innovations from past time series values is suggested. The test is based on an weighted $L^2$-distance of empirical characteristic functions. The asymptotic distribution under the null hypothesis of independence is derived and consistency against fixed alternatives is shown. A smooth autoregressive residual bootstrap procedure is suggested and its performance is shown in a simulation study.
In this paper the nonparametric quantile regression model is considered in a location-scale context. The asymptotic properties of the empirical independence process based on covariates and estimated residuals are investigated. In particular an asymptotic expansion and weak convergence to a Gaussian process are proved. The results can, on the one hand, be applied to test for validity of the location-scale model. On the other hand, they allow to derive various specification tests in conditional quantile location-scale models. In detail a test for monotonicity of the conditional quantile curve is investigated. For the test for validity of the location-scale model as well as for the monotonicity test smooth residual bootstrap versions of Kolmogorov-Smirnov and Cramer-von Mises type test statistics are suggested. We give rigorous proofs for bootstrap versions of the weak convergence results. The performance of the tests is demonstrated in a simulation study.