Nonparametric variance function estimation with missing data

In this paper, a fixed design regression model where the errors follow a strictly stationary process is considered. In this model the conditional mean function and the conditional variance function are unknown curves. Correlated errors when observations are missing in the response variable are assumed. Four nonparametric estimators of the conditional variance function based on local polynomial fitting are proposed. Expressions of the asymptotic bias and variance of these estimators are obtained. A simulation study illustrates the behavior of the proposed estimators.     

Robust estimation in a nonlinear cointegration model

This paper considers the nonparametric M-estimator in a nonlinear cointegration type model. The local time density argument, which was developed by Phillips and Park (1998) [6] and Wang and Phillips (2009) [9], is applied to establish the asymptotic theory for the nonparametric M-estimator. The weak consistency and the asymptotic distribution of the proposed estimator are established under mild conditions. Meanwhile, the asymptotic distribution of the local least squares estimator and the local least absolute distance estimator can be obtained as applications of our main results. Furthermore, an iterated procedure for obtaining the nonparametric M-estimator and a cross-validation bandwidth selection method are discussed, and some numerical examples are provided to show that the proposed methods perform well in the finite sample case.     

One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic

We consider one-step estimation of parameters that represent the strength of spatial dependence in a geostatistical or lattice spatial model. While the maximum likelihood estimators (MLE) of spatial dependence parameters are known to have various desirable properties, they do not have closed-form expressions. Therefore, we consider a one-step alternative to maximum likelihood estimation based on solving an approximate (i.e., one-step) profile likelihood estimating equation. The resulting approximate profile likelihood estimator (APLE) has a closed-form representation, making it a suitable alternative to the widely used Moran’s I statistic. Since the finite-sample and asymptotic properties of one-step estimators of covariance-function parameters have not been studied rigorously, we explore these properties for the APLE of the spatial dependence parameter in the simultaneous autoregressive (SAR) model. Motivated by the APLE statistic’s closed from, we develop exploratory spatial data analysis tools that capture regions of local clustering or the extent to which the strength of spatial dependence varies across space. We illustrate these exploratory tools using both simulated data and observed crime rates in Columbus, OH.     

Time-varying coefficient estimation in differential equation models with noisy time-varying covariates

We study the problem of estimating time-varying coefficients in ordinary differential equations. Current theory only applies to the case when the associated state variables are observed without measurement errors as presented in Chen and Wu (2008) [4] and [5]. The difficulty arises from the quadratic functional of observations that one needs to deal with instead of the linear functional that appears when state variables contain no measurement errors. We derive the asymptotic bias and variance for the previously proposed two-step estimators using quadratic regression functional theory.     

Superefficient estimation of the marginals by exploiting knowledge on the copula

We consider the problem of estimating the marginals in the case where there is knowledge on the copula. If the copula is smooth, it is known that it is possible to improve on the empirical distribution functions: optimal estimators still have a rate of convergence n^−1/2, but a smaller asymptotic variance. In this paper we show that for non-smooth copulas it is sometimes possible to construct superefficient estimators of the marginals: we construct both a copula and, exploiting the information our copula provides, estimators of the marginals with the rate of convergence logn/n.     

Consistent estimation of the intensity function of a cyclic Poisson process

We construct and investigate a consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process when the period is unknown. We do not assume any particular parametric form for the intensity function, nor do we even assume that it is continuous. Moreover, we consider the situation when only a single realization of the Poisson process is available, and only in a bounded window. We prove, in particular, that the proposed estimator is consistent when the size of the window indefinitely expands. We also obtain complete convergence of the estimator.     

Strong consistency of nearest neighbor regression function estimators

For a well-known class of nonparametric regression function estimators of nearest neighbor type the uniform measure of deviation from the estimators to the true regression function is studied. Under weak regularity conditions it is shown that the estimators are uniformly consistent with probability one and the corresponding rate of convergence is near-optimal.     

Frontier estimation via kernel regression on high power-transformed data

We present a new method for estimating the frontier of a multidimensional sample. The estimator is based on a kernel regression on the power-transformed data. We assume that the exponent of the transformation goes to infinity while the bandwidth of the kernel goes to zero. We give conditions on these two parameters to obtain complete convergence and asymptotic normality. The good performance of the estimator is illustrated on some finite sample situations.     

Functional nonparametric estimation of conditional extreme quantiles

We address the estimation of quantiles from heavy-tailed distributions when functional covariate information is available and in the case where the order of the quantile converges to one as the sample size increases. Such “extreme” quantiles can be located in the range of the data or near and even beyond the boundary of the sample, depending on the convergence rate of their order to one. Nonparametric estimators of these functional extreme quantiles are introduced, their asymptotic distributions are established and their finite sample behavior is investigated.     

Nonlinear least-squares estimation

The paper uses empirical process techniques to study the asymptotics of the least-squares estimator (LSE) for the fitting of a nonlinear regression function. By combining and extending ideas of Wu and Van de Geer, it establishes new consistency and central limit theorems that hold under only second moment assumptions on the errors. An application to a delicate example of Wu's illustrates the use of the new theorems, leading to a normal approximation to the LSE with unusual logarithmic rescalings.     

On an asymptotically more efficient estimation of the single-index model

In this note, we revisit the single-index model with heteroscedastic error, and recommend an estimating equation method in terms of transferring restricted least squares to unrestricted least squares: the estimator of the index parameter is asymptotically more efficient than existing estimators in the literature in the sense that it is of a smaller limiting variance.     

Efficient estimation in conditional single-index regression

Semiparametric single-index regression involves an unknown finite-dimensional parameter and an unknown (link) function. We consider estimation of the parameter via the pseudo-maximum likelihood method. For this purpose we estimate the conditional density of the response given a candidate index and maximize the obtained likelihood. We show that this technique of adaptation yields an asymptotically efficient estimator: it has minimal variance among all estimators.     

A moving window approach for nonparametric estimation of the conditional tail index

We present a nonparametric family of estimators for the tail index of a Pareto-type distribution when covariate information is available. Our estimators are based on a weighted sum of the log-spacings between some selected observations. This selection is achieved through a moving window approach on the covariate domain and a random threshold on the variable of interest. Asymptotic normality is proved under mild regularity conditions and illustrated for some weight functions. Finite sample performances are presented on a real data study.     

Efficient nonparametric estimation of distribution density in the basis of algebraic polynomials

A nonparametric estimatef ^* of an unknown distribution densityf W is called locally minimax iff it is minimax for all not too small neighborhoodsW _g ,g W, simultaneously, whereW is some dense subset ofW. Radaviius and Rudzkis proved the existence of such an estimate under some general conditions. However, the construction of the estimate is rather complicated. In this paper, a new estimate is proposed. This estimate is locally minimax under some additional assumptions which usually hold for orthobases of algebraic polynomial and is almost as simple as the linear projective estimate. Thus, it takes a form convenient for the construction of an adaptive estimator, which does not usea-priori information about the smoothness of the density. The adaptive estimation problem is briefly discussed and an unknown density fitting by Jacobi polynomials is investigated more explicitly.     

A note on nonparametric density estimation for dependent variables using a delta sequence

Summary A general method based on “delta sequences” due to Walter and Blum [12] is extended to sequences of strictly stationary mixing random variables having the same marginal distribution admitting a Lebesgue probability density function. It is proved that, under certain conditions, the rate of mean square convergence obtained in the i.i.d. case by Walter and Blum, continues to hold. University of Petroleum and Minerals     

Smooth estimation of survival and density functions for a stationary associated process using Poisson weights

Let {X_n,n≥1} be a sequence of stationary non-negative associated random variables with common marginal density f(x). Here we use the empirical survival function as studied in Bagai and Prakasa Rao (1991) and apply the smoothing technique proposed by Gawronski (1980) (see also Chaubey and Sen, 1996) in proposing a smooth estimator of the density function f and that of the corresponding survival function. Some asymptotic properties of the resulting estimators, similar to those obtained in Chaubey and Sen (1996) for the i.i.d. case, are derived. A simulation study has been carried out to compare the new estimator to the kernel estimator of a density function given in Bagai and Prakasa Rao (1996) and the estimator in Buch-Larsen et al. (2005).     

Nonparametric regression function estimation with surrogate data and validation sampling

This paper develops estimation approaches for nonparametric regression analysis with surrogate data and validation sampling when response variables are measured with errors. Without assuming any error model structure between the true responses and the surrogate variables, a regression calibration kernel regression estimate is defined with the help of validation data. The proposed estimator is proved to be asymptotically normal and the convergence rate is also derived. A simulation study is conducted to compare the proposed estimators with the standard Nadaraya-Watson estimators with the true observations in the validation data set and the complete observations, respectively. The Nadaraya-Watson estimator with the complete observations can serve as a gold standard, even though it is practically unachievable because of the measurement errors.     

Likelihood-based kernel estimation in semiparametric errors-in-covariables models with validation data

We present methods to handle error-in-variables models. Kernel-based likelihood score estimating equation methods are developed for estimating conditional density parameters. In particular, a semiparametric likelihood method is proposed for sufficiently using the information in the data. The asymptotic distribution theory is derived. Small sample simulations and a real data set are used to illustrate the proposed estimation methods.     

Survival function estimation when lifetime and censoring time are dependent

In this paper we consider a model for dependent censoring and derive a consistent asymptotically normal estimator for the underlying survival distribution from a sample of censored data. The methodology is illustrated with an application to the analysis of cancer data. Some simulations to evaluate the performance of our estimator are also presented. The results indicate that our estimator performs reasonably well in comparison to the other dependent censoring survival curve estimators.     

Some inequalities for strong mixing random variables with applications to density estimation

In this paper, we establish an inequality of the characteristic functions for strongly mixing random vectors, by which, an upper bound is provided for the supremum of the absolute value of the difference of two multivariate probability density functions based on strongly mixing random vectors. As its application, we consider the consistency and asymptotic normality of a kernel estimate of a density function under strong mixing. Our results generalize some known results in the literature.