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.     

Statistical properties of a kernel-type estimator of the intensity function of a cyclic Poisson process

We consider a kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process when the period is unknown. We assume that only a single realization of the Poisson process is observed in a bounded window which expands in time. We compute the asymptotic bias, variance, and the mean-squared error of the estimator when the window indefinitely expands.     

Wavelet estimation of conditional density with truncated, censored and dependent data

In this paper we define a new nonlinear wavelet-based estimator of conditional density function for a random left truncation and right censoring model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimators, the MISE expression of the wavelet-based estimators is not affected by the presence of discontinuities in the curves. Also, asymptotic normality of the estimator is established.     

Properties of nonparametric estimators of autocovariance for stationary random fields

Summary We introduce nonparametric estimators of the autocovariance of a stationary random field. One of our estimators has the property that it is itself an autocovatiance. This feature enables the estimator to be used as the basis of simulation studies such as those which are necessary when constructing bootstrap confidence intervals for unknown parameters. Unlike estimators proposed recently by other authors, our own do not require assumptions such as isotropy or monotonicity. Indeed, like nonparametric function estimators considered more widely in the context of curve estimation, our approach demands only smoothness and tail conditions on the underlying curve or surface (here, the autocovariance), and moment and mixing conditions on the random field. We show that by imposing the condition that the estimator be a covariance function we actually reduce the numerical value of integrated squared error.     

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.     

Sharp adaptive estimation of quadratic functionals

Estimation of a quadratic functional of a function observed in the Gaussian white noise model is considered. A data-dependent method for choosing the amount of smoothing is given. The method is based on comparing certain quadratic estimators with each other. It is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the “regular” and “irregular” region. We consider l_p bodies and construct bounds for the risk of the estimator which show that for p=4 the estimator is exactly optimal and for example when p ∈[3,100], then the upper bound is at most 1.055 times larger than the lower bound. We show the connection of the estimator to the theory of optimal recovery. The estimator is a calibration of an estimator which is nearly minimax optimal among quadratic estimators. Writing of this article was financed by Deutsche Forschungsgemeinschaft under project MA1026/6-2, CIES, France, and Jenny and AnttiWihuri Foundation.     

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).     

Statistical inference in a panel data semiparametric regression model with serially correlated errors

We consider a panel data semiparametric partially linear regression model with an unknown vector β of regression coefficients, an unknown nonparametric function g(·) for nonlinear component, and unobservable serially correlated errors. The correlated errors are modeled by a vector autoregressive process which involves a constant intraclass correlation. Applying the pilot estimators of β and g(·), we construct estimators of the autoregressive coefficients, the intraclass correlation and the error variance, and investigate their asymptotic properties. Fitting the error structure results in a new semiparametric two-step estimator of β, which is shown to be asymptotically more efficient than the usual semiparametric least squares estimator in terms of asymptotic covariance matrix. Asymptotic normality of this new estimator is established, and a consistent estimator of its asymptotic covariance matrix is presented. Furthermore, a corresponding estimator of g(·) is also provided. These results can be used to make asymptotically efficient statistical inference. Some simulation studies are conducted to illustrate the finite sample performances of these proposed estimators.     

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^−1/2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.     

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.     

Local linear regression for functional predictor and scalar response

The aim of this work is to introduce a new nonparametric regression technique in the context of functional covariate and scalar response. We propose a local linear regression estimator and study its asymptotic behaviour. Its finite-sample performance is compared with a Nadayara-Watson type kernel regression estimator and with the linear regression estimator via a Monte Carlo study and the analysis of two real data sets. In all the scenarios considered, the local linear regression estimator performs better than the kernel one, in the sense that the mean squared prediction error is lower.     

Optimal prediction for linear regression with infinitely many parameters

The problem of optimal prediction in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that outperforms asymptotically the ordinary least squares predictor. Moreover, if the random errors are Gaussian, the method is asymptotically minimax over ellipsoids in ?₂. The method is based on a regularized least squares estimator with weights of the Pinsker filter. We also consider the case of dynamic linear regression, which is important in the context of transfer function modeling.     

High breakdown estimators for principal components: the projection-pursuit approach revisited

Li and Chen (J. Amer. Statist. Assoc. 80 (1985) 759) proposed a method for principal components using projection-pursuit techniques. In classical principal components one searches for directions with maximal variance, and their approach consists of replacing this variance by a robust scale measure. Li and Chen showed that this estimator is consistent, qualitative robust and inherits the breakdown point of the robust scale estimator. We complete their study by deriving the influence function of the estimators for the eigenvectors, eigenvalues and the associated dispersion matrix. Corresponding Gaussian efficiencies are presented as well. Asymptotic normality of the estimators has been treated in a paper of Cui et al. (Biometrika 90 (2003) 953), complementing the results of this paper. Furthermore, a simple explicit version of the projection-pursuit based estimator is proposed and shown to be fast to compute, orthogonally equivariant, and having the maximal finite-sample breakdown point property. We will illustrate the method with a real data example.     

Tree-based multivariate regression and density estimation with right-censored data

We propose a unified strategy for estimator construction, selection, and performance assessment in the presence of censoring. This approach is entirely driven by the choice of a loss function for the full (uncensored) data structure and can be stated in terms of the following three main steps. (1) First, define the parameter of interest as the minimizer of the expected loss, or risk, for a full data loss function chosen to represent the desired measure of performance. Map the full data loss function into an observed (censored) data loss function having the same expected value and leading to an efficient estimator of this risk. (2) Next, construct candidate estimators based on the loss function for the observed data. (3) Then, apply cross-validation to estimate risk based on the observed data loss function and to select an optimal estimator among the candidates. A number of common estimation procedures follow this approach in the full data situation, but depart from it when faced with the obstacle of evaluating the loss function for censored observations. Here, we argue that one can, and should, also adhere to this estimation road map in censored data situations.Tree-based methods, where the candidate estimators in Step 2 are generated by recursive binary partitioning of a suitably defined covariate space, provide a striking example of the chasm between estimation procedures for full data and censored data (e.g., regression trees as in CART for uncensored data and adaptations to censored data). Common approaches for regression trees bypass the risk estimation problem for censored outcomes by altering the node splitting and tree pruning criteria in manners that are specific to right-censored data. This article describes an application of our unified methodology to tree-based estimation with censored data. The approach encompasses univariate outcome prediction, multivariate outcome prediction, and density estimation, simply by defining a suitable loss function for each of these problems. The proposed method for tree-based estimation with censoring is evaluated using a simulation study and the analysis of CGH copy number and survival data from breast cancer patients.     

Scan clustering: A false discovery approach

We present a method that scans a random field for localized clusters while controlling the fraction of false discoveries. We use a kernel density estimator as the test statistic and adjust for the bias in this estimator by a method we introduce in this paper. We also show how to combine information across multiple bandwidths while maintaining false discovery control.     

Data sharpening methods in multivariate local quadratic regression

This paper is concerned with the conditional bias and variance of local quadratic regression to the multivariate predictor variables. Data sharpening methods of nonparametric regression were first proposed by Choi, Hall, Roussion. Recently, a data sharpening estimator of local linear regression was discussed by Naito and Yoshizaki. In this paper, to improve mainly the fitting precision, we extend their results on the asymptotic bias and variance. Using the data sharpening estimator of multivariate local quadratic regression, we are able to derive higher fitting precision. In particular, our approach is simple to implement, since it has an explicit form, and is convenient when analyzing the asymptotic conditional bias and variance of the estimator at the interior and boundary points of the support of the density function.     

Optimal global rate of convergence in nonparametric regression with left-truncated and right-censored data

In this paper we consider nonparametric regression with left-truncated and right-censored data. An estimator of the regression function is developed when censoring and truncation are independent of covariates and the response. The estimation is based on the product limit estimator of the response variable. Under certain conditions, the L₂ rate of convergence of the estimated regression function is obtained when tensor products of B-splines are used.     

Nonparametric regression estimation with general parametric error covariance

The asymptotic distribution for the local linear estimator in nonparametric regression models is established under a general parametric error covariance with dependent and heterogeneously distributed regressors. A two-step estimation procedure that incorporates the parametric information in the error covariance matrix is proposed. Sufficient conditions for its asymptotic normality are given and its efficiency relative to the local linear estimator is established. We give examples of how our results are useful in some recently studied regression models. A Monte Carlo study confirms the asymptotic theory predictions and compares our estimator with some recently proposed alternative estimation procedures.     

Asymptotic bias and variance for a general class of varying bandwidth density estimators

Summary We consider a general class of varying bandwidth estimators of a probability density function. The class includes the Abramson estimator, transformation kernel density estimator (TKDE), Jones transformation kernel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-Linton-Nielsen estimator (JLN), Taylor series approximations of TKDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinary kernel estimator , it is possible to iterate and compute a sequence of estimates , using each estimate as a pilot estimator in the next step. The first main result is a formula for the bias order. If the bandwidths used in different steps have a common orderh=h(n), the bias of is of orderh ^2km ,k=1, ...,t. Hereh ^m is the bias order of the ideal estimator (defined by using the unknownf as pilot). The second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates. Ifm<, it is possible to make asymptotically equivalent to the ideal estimator.     

Censored multiple regression by the method of average derivatives

This paper proposes a technique [termed censored average derivative estimation (CADE)] for studying estimation of the unknown regression function in nonparametric censored regression models with randomly censored samples. The CADE procedure involves three stages: firstly-transform the censored data into synthetic data or pseudo-responses using the inverse probability censoring weighted (IPCW) technique, secondly estimate the average derivatives of the regression function, and finally approximate the unknown regression function by an estimator of univariate regression using techniques for one-dimensional nonparametric censored regression. The CADE provides an easily implemented methodology for modelling the association between the response and a set of predictor variables when data are randomly censored. It also provides a technique for “dimension reduction” in nonparametric censored regression models. The average derivative estimator is shown to be root-n consistent and asymptotically normal. The estimator of the unknown regression function is a local linear kernel regression estimator and is shown to converge at the optimal one-dimensional nonparametric rate. Monte Carlo experiments show that the proposed estimators work quite well.