We propose a general semiparametric variance function model in a fixed design regression setting. In this model, the regression function is assumed to be smooth and is modelled nonparametrically, whereas the relation between the variance and the mean regression function is assumed to follow a generalized linear model. Almost all variance function models that were considered in the literature emerge as special cases. Least-squares-types estimates for the parameters of this…
This paper concerns estimating a mixing density function $g$ and its derivatives based on iid observations from $f(x) = \int f(x \mid \theta)g(\theta)d\theta$, where $f(x \mid \theta)$ is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive kernel estimators, upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence. If $f(x \mid \theta_0) \geq \varepsilon^{x + 1} \forall x$…
The traditional kernel density estimator of an unknown density is by construction completely nonparametric in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example, the normal, while not losing much in precision when the true density is…
Consider a statistical procedure (Method A) which is based on $n$ observations and a less effective procedure (Method B) which requires a larger number $k_n$ of observations to give equal performance under a certain criterion. To compare two different procedures, Hodges and Lehmann suggested that the difference $k_n - n$, called the deficiency of Method B with respect to Method A, is the most natural quantity to examine. In this article, the performance of two kernel…
Most hazard regression models in survival analysis specify a given functional form to describe the influence of the covariates on the hazard rate. For instance, Cox's model assumes that the covariates act multiplicatively on the hazard rate, and Aalen's additive risk model stipulates that the covariates have a linear additive effect on the hazard rate. In this paper we study a fully nonparametric model which makes no assumption on the association between the hazard rate and…
The common unknown probability law $P$ of a random sample $Y_1,\ldots, Y_n$ is assigned a Dirichlet process prior with index $\alpha$. It is shown that the posterior joint density of several moments of $P$ converges, as $\alpha(\mathbb{R})\rightarrow 0$, to a multivariate B-spline, which is, therefore, the Bayesian bootstrap joint density of the moments. The result provides the basis for possible default nonparametric Bayesian inference on unknown moments.
We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based density estimators. We show that, unlike the analogous situation for kernel density estimators, this MISE formula is relatively unaffected by assumptions of continuity. In particular, it is available for densities which are smooth in only a piecewise sense. Another difference is that in the wavelet case the classical MISE formula is valid only for sufficiently small…
A data-based procedure is introduced for local bandwidth selection for kernel estimation of a regression function at a point. The estimated bandwidth is shown to be consistent and asymptotically normal as an estimator of the (asymptotic) optimal value for minimum mean square estimation. Simulation studies indicate satisfactory behavior of the new bandwidth estimator in finite samples. The findings are improvements over a global bandwidth estimator. The same methodology works…
An important gap, between the classical mathematical theory and the practice and implementation of nonparametric curve estimation, is due to the fact that the usual norms on function spaces measure something different from what the eye can see visually in a graphical presentation. Mathematical error criteria that more closely follow “visual impression” are developed and analyzed from both graphical and mathematical viewpoints. Examples from wavelet regression and kernel…
A data-based procedure is introduced for local bandwidth selection for kernel estimation of a regression function at a point. The estimated bandwidth is shown to be consistent and asymptotically normal as an estimator of the (asymptotic) optimal value for minimum mean square estimation. Simulation studies indicate satisfactory behavior of the new bandwidth estimator in finite samples. The findings are improvements over a global bandwidth estimator. The same methodology works…
C. K. Chu · K. F. Cheng·刊期 1995-06 · 卷 82（2）·Biometrika已编入正刊
利用误分类二元响应的非参数回归估计AI 解读
For random design nonparametric regression, in the case that the responses are binary and subject to misclassification, the performance of the kernel estimator is investigated. The kernel estimator is generally biased for the local proportion. To adjust for the bias, the double sampling scheme of Tenenbein (1970, 1971) is considered. A plugged-in kernel estimator and an imputed kernel estimator, which adjust for the effect of misclassification on the kernel estimator, are…
This paper presents a number of consistency results for nonparametric kernel estimators of density and regression functions and their derivatives. These results are particularly useful in semiparametric estimation and testing problems that rely on preliminary nonparametric estimators, as in Andrews (1994, Econometrica 62, 43–72). The results allow for near-epoch dependent, nonidentically distributed random variables, data-dependent bandwidth sequences, preliminary estimation…
Given data from a sample of noisy curves, we consider a nonlinear parametric regression model with unknown model function. An iterative algorithm for estimating individual parameters as well as the model function is introduced under the assumption of a certain shape invariance: the individual regression curves are obtained from a common shape function by linear transformations of the axes. Our algorithm is based on least-squares methods for parameter estimation and on…
A conservative finite-sample simultaneous confidence envelope for a density can be found by solving a finite set of finite-dimensional linear programming problems if the density is known to be monotonic or to have at most $k$ modes relative to a positive weight function. The dimension of the problems is at most $(n/\log n)^{1/3}$, where $n$ is the number of observations. The linear programs find densities attaining the largest and smallest values at a point among cumulative…
Let $Y_1,\ldots, Y_n$ be independent identically distributed with density $p_0$ and let $\mathscr{F}$ be a space of densities. We show that the supremum of the likelihood ratios $\prod^n_{i=1} p(Y_i)/p_0(Y_i)$, where the supremum is over $p \in \mathscr{F}$ with $\|p^{1/2} - p^{1/2}_0\|_2 \geq \varepsilon$, is exponentially small with probability exponentially close to 1. The exponent is proportional to $n\varepsilon^2$. The only condition required for this to hold is that $\varepsilon$…
O. LINTON · J. P. NIELSEN·刊期 1995-03-01 · 卷 82（1）·Biometrika已编入正刊
基于边际积分的结构化非参数回归核估计AI 解读
We define a simple kernel procedure based on marginal integration that estimates the relevant univariate quantity in both additive and multiplicative nonparametric regression.
A. T. WALDEN · E. J. McCOY · D. B. PERCIVAL·刊期 1995-03-01 · 卷 82（1）·Biometrika已编入正刊
多窗谱估计量的有效带宽AI 解读
The bandwidth of a spectral estimator is a measure of the minimum separation in frequency between approximately uncorrelated spectral estimates. We determine an effective bandwidth measure for multitaper spectral estimators, a relatively new and very powerful class of spectral estimators proving to be very valuable whenever the spectrum of interest is detailed and/ or varies rapidly with a large dynamic range. The multitaper spectral estimator is the average of several…
We investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi-likelihood contexts. In the ordinary regression case, local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity. One noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators. These…
Linear splines and their tensor products are used to estimate the conditional log-hazard function based on possibly censored, positive response data and one or more covariates. An automatic procedure involving the maximum likelihood method, stepwise addition, stepwise deletion, and the Bayes Information Criterion is used to select the final model. The possible models contain proportional hazards models as a subclass, which makes it possible to diagnose departures from…
A. T. Walden · E. J. McCoy · D. B. Percival·刊期 1995-03 · 卷 82（1）·Biometrika已编入正刊
多窗谱估计量的有效带宽AI 解读
The bandwidth of a spectral estimator is a measure of the minimum separation in frequency between approximately uncorrelated spectral estimates. We determine an effective bandwidth measure for multitaper spectral estimators, a relatively new and very powerful class of spectral estimators proving to be very valuable whenever the spectrum of interest is detailed and/ or varies rapidly with a large dynamic range. The multitaper spectral estimator is the average of several…
We consider the problem of estimating a pdf $f$ from samples $X_1, X_2, \ldots, X_n$ of a random variable with pdf $\mathscr{K}f$, where $\mathscr{K}$ is a compact integral operator. We employ a maximum smoothed likelihood formalism inspired by a nonlinearly smoothed version of the EMS algorithm of Silverman, Jones, Wilson and Nychka. We show that this nonlinearly smoothed algorithm is itself an EM algorithm, which helps explain the strong convergence properties of the…
R. J. Carroll · R. K. Knickerbocker · C. Y. Wang·刊期 1995-02-01 · 卷 23（1）·The Annals of Statistics已编入正刊
协变量含测量误差的半参数回归模型中的降维AI 解读
We consider a semiparametric estimation method for general regression models when some of the predictors are measured with error. The technique relies on a kernel regression of the "true" covariate on all the observed covariates and surrogates. This requires a nonparametric regression in as many dimensions as there are covariates and surrogates. The usual theory copes with such higher-dimensional problems by using higher-order kernels, but this is unrealistic for most…
Variable window width kernel density estimators, with the width varying proportionally to the square root of the density, have been thought to have superior asymptotic properties. The rate of convergence has been claimed to be as good as those typical for higher-order kernels, which makes the variable width estimators more attractive because no adjustment is needed to handle the negativity usually entailed by the latter. However, in a recent paper, Terrell and Scott show…
Elias Masry · Dag Tjøstheim·刊期 1995-02 · 卷 11（2）·Econometric Theory已编入正刊
非线性 ARCH 时间序列的非参数估计与识别：强收敛和渐近正态性AI 解读
We consider the estimation and identification of the functional structures of nonlinear econometric systems of the ARCH type. We employ nonparametric kernel estimates for the nonlinear functions characterizing the systems, and we establish strong consistency along with sharp rates of convergence under mild regularity conditions. We also prove the asymptotic normality of the estimates.
This paper concerns estimating a mixing density function g and its derivatives based on iid observations from f(x)=∫f(x∣θ)g(θ)dθ, where f(x∣θ) is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive kernel estimators, upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence. If f(x∣θ0)≥εx+1∀x…
Consider a statistical procedure (Method A) which is based on n observations and a less effective procedure (Method B) which requires a larger number kn of observations to give equal performance under a certain criterion. To compare two different procedures, Hodges and Lehmann suggested that the difference kn−n, called the deficiency of Method B with respect to Method A, is the most natural quantity to examine. In this article, the performance of two kernel…
The common unknown probability law P of a random sample Y1,…,Yn is assigned a Dirichlet process prior with index α. It is shown that the posterior joint density of several moments of P converges, as α(R)→0, to a multivariate B-spline, which is, therefore, the Bayesian bootstrap joint density of the moments. The result provides the basis for possible default nonparametric Bayesian inference on unknown moments.
A conservative finite-sample simultaneous confidence envelope for a density can be found by solving a finite set of finite-dimensional linear programming problems if the density is known to be monotonic or to have at most k modes relative to a positive weight function. The dimension of the problems is at most (n/logn)1/3, where n is the number of observations. The linear programs find densities attaining the largest and smallest values at a point among cumulative…
Let Y1,…,Yn be independent identically distributed with density p0 and let F be a space of densities. We show that the supremum of the likelihood ratios ∏i=1np(Yi)/p0(Yi), where the supremum is over p∈F with ∥p1/2−p01/2∥2≥ε, is exponentially small with probability exponentially close to 1. The exponent is proportional to nε2. The only condition required for this to hold is that ε…
We consider the problem of estimating a pdf f from samples X1,X2,…,Xn of a random variable with pdf Kf, where K is a compact integral operator. We employ a maximum smoothed likelihood formalism inspired by a nonlinearly smoothed version of the EMS algorithm of Silverman, Jones, Wilson and Nychka. We show that this nonlinearly smoothed algorithm is itself an EM algorithm, which helps explain the strong convergence properties of the…