A Berry-Esseen type bound in kernel density estimation for strong mixing censored samples

In this paper, we discuss the estimation of a density function based on censored data by the kernel smoothing method when the survival and the censoring times form a stationary α-mixing sequence. A Berry-Esseen type bound is derived for the kernel density estimator at a fixed point x. For practical purposes, a randomly weighted estimator of the density function is also constructed and investigated.     

Dimension estimation in sufficient dimension reduction: A unifying approach

Sufficient Dimension Reduction (SDR) in regression comprises the estimation of the dimension of the smallest (central) dimension reduction subspace and its basis elements. For SDR methods based on a kernel matrix, such as SIR and SAVE, the dimension estimation is equivalent to the estimation of the rank of a random matrix which is the sample based estimate of the kernel. A test for the rank of a random matrix amounts to testing how many of its eigen or singular values are equal to zero. We propose two tests based on the smallest eigen or singular values of the estimated matrix: an asymptotic weighted chi-square test and a Wald-type asymptotic chi-square test. We also provide an asymptotic chi-square test for assessing whether elements of the left singular vectors of the random matrix are zero. These methods together constitute a unified approach for all SDR methods based on a kernel matrix that covers estimation of the central subspace and its dimension, as well as assessment of variable contribution to the lower-dimensional predictor projections with variable selection, a special case. A small power simulation study shows that the proposed and existing tests, specific to each SDR method, perform similarly with respect to power and achievement of the nominal level. Also, the importance of the choice of the number of slices as a tuning parameter is further exhibited.     

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.     

Asymptotic distributions of two “synthetic data” estimators for censored single-index models

The censored single-index model provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored and the link function is unknown. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure time models for survival analysis. This paper proposes two methods for estimation of single-index models with randomly censored samples. We first transform the censored data into synthetic data or pseudo-responses unbiasedly, then obtain estimates of the index coefficients by the rOPG or rMAVE procedures of Xia (2006) [1]. Finally, we estimate the unknown nonparametric link function using techniques for univariate censored nonparametric regression. The estimators for the index coefficients are shown to be root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodologies.     

On the convergence of Newton's method when estimating higher dimensional parameters

In this paper, we consider the estimation of a parameter of interest where the estimator is one of the possibly several solutions of a set of nonlinear empirical equations. Since Newton's method is often used in such a setting to obtain a solution, it is important to know whether the so obtained iteration converges to the locally unique consistent root to the aforementioned parameter of interest. Under some conditions, we show that this is eventually the case when starting the iteration from within a ball about the true parameter whose size does not depend on n. Any preliminary almost surely consistent estimate will eventually lie in such a ball and therefore provides a suitable starting point for large enough n. As examples, we will apply our results in the context of M-estimates, kernel density estimates, as well as minimum distance estimates.     

Asymptotics for Kernel Estimation of Slicing Average Third-Moment Estimation

To estimate central dimension-reduction space in multivariate nonparametric rcgression, Sliced Inverse Regression （SIR）, Sliced Average Variance Estimation （SAVE） and Slicing Average Third-moment Estimation （SAT） have been developed, Since slicing estimation has very different asymptotic behavior for SIR, and SAVE, the relevant study has been madc case by case, when the kernel estimators of SIH and SAVE share similar asymptotic properties. In this paper, we also investigate kernel estimation of SAT. We. prove the asymptotic normality, and show that, compared with tile existing results, the kernel Slnoothing for SIR, SAVE and SAT has very similar asymptotic behavior,     

Boundary kernels for adaptive density estimators on regions with irregular boundaries

In some applications of kernel density estimation the data may have a highly non-uniform distribution and be confined to a compact region. Standard fixed bandwidth density estimates can struggle to cope with the spatially variable smoothing requirements, and will be subject to excessive bias at the boundary of the region. While adaptive kernel estimators can address the first of these issues, the study of boundary kernel methods has been restricted to the fixed bandwidth context. We propose a new linear boundary kernel which reduces the asymptotic order of the bias of an adaptive density estimator at the boundary, and is simple to implement even on an irregular boundary. The properties of this adaptive boundary kernel are examined theoretically. In particular, we demonstrate that the asymptotic performance of the density estimator is maintained when the adaptive bandwidth is defined in terms of a pilot estimate rather than the true underlying density. We examine the performance for finite sample sizes numerically through analysis of simulated and real data sets.     

Empirical likelihood based confidence intervals for copulas

Copula as an effective way of modeling dependence has become more or less a standard tool in risk management, and a wide range of applications of copula models appear in the literature of economics, econometrics, insurance, finance, etc. How to estimate and test a copula plays an important role in practice, and both parametric and nonparametric methods have been studied in the literature. In this paper, we focus on interval estimation and propose an empirical likelihood based confidence interval for a copula. A simulation study and a real data analysis are conducted to compare the finite sample behavior of the proposed empirical likelihood method with the bootstrap method based on either the empirical copula estimator or the kernel smoothing copula estimator.     

Functional semiparametric partially linear model with autoregressive errors

In this paper, we introduce a functional semiparametric model, where a real-valued random variable is explained by the sum of a unknown linear combination of the components of a multivariate random variable and an unknown transformation of a functional random variable. The errors can be autocorrelated. We focus here on the parametric estimation of the coefficients in the linear combination. First, we use a nonparametric kernel method to remove the effect of the functional explanatory variable. Then, we use generalized least squares approach to obtain an estimator of these coefficients. Under some technical assumptions, we prove consistency and asymptotic normality of our estimator. Finally, we present Monte Carlo simulations that illustrate these characteristics.     

Robust regression function estimation

A robust estimator of the regression function is proposed combining kernel methods as introduced for density estimation and robust location estimation techniques. Weak and strong consistency and asymptotic normality are shown under mild conditions on the kernel sequence. The asymptotic variance is a product from a factor depending only on the kernel and a factor similar to the asymptotic variance in robust estimation of location. The estimation is minimax robust in the sense of Huber (1964). Robust estimation of a location parameter. Ann. Math. Statist.33 73–101.     

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).     

A semiparametric density estimator based on elliptical distributions

In the paper we study a semiparametric density estimation method based on the model of an elliptical distribution. The method considered here shows a way to overcome problems arising from the curse of dimensionality. The optimal rate of the uniform strong convergence of the estimator under consideration coincides with the optimal rate for the usual one-dimensional kernel density estimator except in a neighbourhood of the mean. Therefore the optimal rate does not depend on the dimension. Moreover, asymptotic normality of the estimator is proved.     

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.     

Strong convergence in nonparametric regression with truncated dependent data

In this paper we derive rates of uniform strong convergence for the kernel estimator of the regression function in a left-truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The estimation of the covariate’s density is considered as well. Under the assumption that the lifetime observations are bounded, we show that, by an appropriate choice of the bandwidth, both estimators of the covariate’s density and regression function attain the optimal strong convergence rate known from independent complete samples.     

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.     

Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations

In this article, we propose a new estimation methodology to deal with PCA for high-dimension, low-sample-size (HDLSS) data. We first show that HDLSS datasets have different geometric representations depending on whether a ρ-mixing-type dependency appears in variables or not. When the ρ-mixing-type dependency appears in variables, the HDLSS data converge to an n-dimensional surface of unit sphere with increasing dimension. We pay special attention to this phenomenon. We propose a method called the noise-reduction methodology to estimate eigenvalues of a HDLSS dataset. We show that the eigenvalue estimator holds consistency properties along with its limiting distribution in HDLSS context. We consider consistency properties of PC directions. We apply the noise-reduction methodology to estimating PC scores. We also give an application in the discriminant analysis for HDLSS datasets by using the inverse covariance matrix estimator induced by the noise-reduction methodology.     

A bias corrected nonparametric regression estimator

In this article, we propose a new method of bias reduction in nonparametric regression estimation. The proposed new estimator has asymptotic bias order h⁴, where h is a smoothing parameter, in contrast to the usual bias order h² for the local linear regression. In addition, the proposed estimator has the same order of the asymptotic variance as the local linear regression. Our proposed method is closely related to the bias reduction method for kernel density estimation proposed by Chung and Lindsay (2011). However, our method is not a direct extension of their density estimate, but a totally new one based on the bias cancelation result of their proof.     

Asymptotic normality and Berry-Esseen results for conditional density estimator with censored and dependent data

In this paper we derive the asymptotic normality and a Berry-Esseen type bound for the kernel conditional density estimator proposed in Ould-Saïd and Cai (2005) [26] when the censored observations with multivariate covariates form a stationary α-mixing sequence.     

Empirical likelihood analysis of the Buckley-James estimator

The censored linear regression model, also referred to as the accelerated failure time (AFT) model when the logarithm of the survival time is used as the response variable, is widely seen as an alternative to the popular Cox model when the assumption of proportional hazards is questionable. Buckley and James [Linear regression with censored data, Biometrika 66 (1979) 429-436] extended the least squares estimator to the semiparametric censored linear regression model in which the error distribution is completely unspecified. The Buckley-James estimator performs well in many simulation studies and examples. The direct interpretation of the AFT model is also more attractive than the Cox model, as Cox has pointed out, in practical situations. However, the application of the Buckley-James estimation was limited in practice mainly due to its illusive variance. In this paper, we use the empirical likelihood method to derive a new test and confidence interval based on the Buckley-James estimator of the regression coefficient. A standard chi-square distribution is used to calculate the P-value and the confidence interval. The proposed empirical likelihood method does not involve variance estimation. It also shows much better small sample performance than some existing methods in our simulation studies.     

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.