Nonparametric estimation of the hazard function by using a model selection method: estimation of cancer deaths in Hiroshima atomic bomb survivors

Summary.  Controversy has intensified regarding the death-rate from cancer that is induced by a dose of radiation. In the models that are usually considered the hazard function is an increasing function of the dose of radiation. Such models can mask local variations. We consider the models of excess relative risk and of absolute risk and propose a nonparametric estimation of the effect of the dose by using a model selection procedure. This estimation deals with stratified data. We approximate the function of the dose by a collection of splines and select the best one according to the Akaike information criterion. In the same way between the models of excess relative risk or excess absolute risk, we choose the model that best fits the data. We propose a bootstrap method for calculating a pointwise confidence interval of the dose function. We apply our method for estimating the solid cancer and leukaemia death hazard functions to Hiroshima.     

Functional data: local linear estimation of the conditional density and its application

In this paper, we introduce a new nonparametric estimation procedure of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space. Under some general conditions, we establish both the pointwise and the uniform almost-complete consistencies with convergence rates of the conditional density estimator related to this estimation procedure. Moreover, we give some particular cases of our results which can also be considered as novel in the finite-dimensional setting. Notice also that the results of this paper are used to derive some asymptotic properties of the local linear estimator of the conditional mode.     

Adaptive nonparametric estimation in the presence of dependence

We consider nonparametric estimation problems in the presence of dependent data, notably nonparametric regression with random design and nonparametric density estimation. The proposed estimation procedure is based on a dimension reduction. The minimax optimal rate of convergence of the estimator is derived assuming a sufficiently weak dependence characterised by fast decreasing mixing coefficients. We illustrate these results by considering classical smoothness assumptions. However, the proposed estimator requires an optimal choice of a dimension parameter depending on certain characteristics of the function of interest, which are not known in practice. The main issue addressed in our work is an adaptive choice of this dimension parameter combining model selection and Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski [(2011), ‘Bandwidth Selection in Kernel Density Estimation: Oracle Inequalities and Adaptive Minimax Optimality’, The Annals of Statistics, 39, 1608–1632]. We show that this data-driven estimator can attain the lower risk bound up to a constant provided a fast decay of the mixing coefficients.     

Asymptotically pointwise optimal and asymptotically optimal stopping times in the Bayesian inference

The paper deals with the problem of determining asymptotically pointwise optimal and asymptotically optimal stopping times in the Bayesian inference. The sufficient conditions are given for a family of stopping times to be asymptotically pointwise optimal and asymptotically optimal with respect to a continuous time process. As an example a sequential estimation of the intensity of the Poisson process is considered. Under a gamma prior distribution, an asymptotically pointwise optimal and asymptotically optimal rule is given using a LINEX loss function and the cost c per unit time.     

ON ESTIMATION OF THE POPULATION SPECTRAL DISTRIBUTION FROM A HIGH‐DIMENSIONAL SAMPLE COVARIANCE MATRIX

Sample covariance matrices play a central role in numerous popular statistical methodologies, for example principal components analysis, Kalman filtering and independent component analysis. However, modern random matrix theory indicates that, when the dimension of a random vector is not negligible with respect to the sample size, the sample covariance matrix demonstrates significant deviations from the underlying population covariance matrix. There is an urgent need to develop new estimation tools in such cases with high‐dimensional data to recover the characteristics of the population covariance matrix from the observed sample covariance matrix. We propose a novel solution to this problem based on the method of moments. When the parametric dimension of the population spectrum is finite and known, we prove that the proposed estimator is strongly consistent and asymptotically Gaussian. Otherwise, we combine the first estimation method with a cross‐validation procedure to select the unknown model dimension. Simulation experiments demonstrate the consistency of the proposed procedure. We also indicate possible extensions of the proposed estimator to the case where the population spectrum has a density.     

Semiparametric jump-preserving estimation for single-index models

Estimation of the single-index model with a discontinuous unknown link function is considered in this paper. Existed refined minimum average variance estimation (rMAVE) method can estimate the single-index parameter and unknown link function simultaneously by minimising the average pointwise conditional variance, where the conditional variance can be estimated using the local linear fit method with centred kernel function. When there are jumps in the link function, big biases around jumps can appear. For this reason, we embed the jump-preserving technique in the rMAVE method, then propose an adaptive jump-preserving estimation procedure for the single-index model. Concretely speaking, the conditional variance is obtained by the one among local linear fits with centred, left-sided and right-sided kernel functions who has minimum weighted residual mean squares. The resulting estimators can preserve the jumps well and also give smooth estimates of the continuity parts. Asymptotic properties are established under some mild conditions. Simulations and real data analysis show the proposed method works well.     

Statistical inference on asymptotic properties of two estimators for the partially linear single-index models

The outer product of gradients (OPG) estimation procedure based on least squares (LS) approach has been presented by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] to estimate the single-index parameter in partially linear single-index models (PLSIM). However, its asymptotic property has not been established yet and the efficiency of LS-based method can be significantly affected by outliers and heavy-tailed distributions. In this paper, we firstly derive the asymptotic property of OPG estimator developed by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] in theory, and a novel robust estimation procedure combining the ideas of OPG and local rank (LR) inference is further developed for PLSIM along with its theoretical property. Then, we theoretically derive the asymptotic relative efficiency (ARE) of the proposed LR-based procedure with respect to LS-based method, which is shown to possess an expression that is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Moreover, we demonstrate that the new proposed estimator has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error. Even in the worst case scenarios, the ARE owns a lower bound equalling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric function respectively, versus the LS-based estimators. Finally, some Monte Carlo simulations and a real data analysis are conducted to illustrate the finite sample performance of the estimators.     

A Note on the Role of the Propensity Score for Estimating Average Treatment Effects

Hahn [Hahn, J. (1998). On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica 66:315-331] derived the semiparametric efficiency bounds for estimating the average treatment effect (ATE) and the average treatment effect on the treated (ATET). The variance of ATET depends on whether the propensity score is known or unknown. Hahn attributes this to “dimension reduction.” In this paper, an alternative explanation is given: Knowledge of the propensity score improves upon the estimation of the distribution of the confounding variables.     

Semiparametric models: a generalized self-consistency approach

Summary. In semiparametric models, the dimension d of the maximum likelihood problem is potentially unlimited. Conventional estimation methods generally behave like O ( d ³). A new O ( d ) estimation procedure is proposed for a large class of semiparametric models. Potentially unlimited dimension is handled in a numerically efficient way through a Nelson–Aalen-like estimator. Discussion of the new method is put in the context of recently developed minorization–maximization algorithms based on surrogate objective functions. The procedure for semiparametric models is used to demonstrate three methods to construct a surrogate objective function: using the difference of two concave functions, the EM way and the new quasi-EM (QEM) approach. The QEM approach is based on a generalization of the EM-like construction of the surrogate objective function so it does not depend on the missing data representation of the model. Like the EM algorithm, the QEM method has a dual interpretation, a result of merging the idea of surrogate maximization with the idea of imputation and self-consistency. The new approach is compared with other possible approaches by using simulations and analysis of real data. The proportional odds model is used as an example throughout the paper.     

A mixture model for dimension reduction

The existence of a dimension reduction (DR) subspace is a common assumption in regression analysis when dealing with high-dimensional predictors. The estimation of such a DR subspace has received considerable attention in the past few years, the most popular method being undoubtedly the sliced inverse regression. In this paper, we propose a new estimation procedure of the DR subspace by assuming that the joint distribution of the predictor and the response variables is a finite mixture of distributions. The new method is compared through a simulation study to some classical methods.     

The conditional cumulative distribution function in single functional index model

ABSTRACT

We consider the estimation of the conditional cumulative distribution function of a scalar response variable Y given a Hilbertian random variable X when the observations are linked via a single-index structure. We establish the pointwise and the uniform almost complete convergence (with the rate) of the kernel estimate of this model. As an application, we show how our result can be applied in the prediction problem via the conditional median estimate. Also, the choice of the functional index via the cross-validation procedure is also discussed but not attacked.     

The stochastic approximation method for the estimation of a multivariate probability density

We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a probability density, including the one introduced by Hall and Patil [1994. On the efficiency of on-line density estimators. IEEE Trans. Inform. Theory 40, 1504–1512]. We study the properties of these estimators and compare them with Rosenblatt's nonrecursive estimator. It turns out that, for pointwise estimation, it is preferable to use the nonrecursive Rosenblatt's kernel estimator rather than any recursive estimator. A contrario, for estimation by confidence intervals, it is better to use a recursive estimator rather than Rosenblatt's estimator.     

Variance Reduction in Smoothing Splines

Abstract.  We develop a variance reduction method for smoothing splines. For a given point of estimation, we define a variance-reduced spline estimate as a linear combination of classical spline estimates at three nearby points. We first develop a variance reduction method for spline estimators in univariate regression models. We then develop an analogous variance reduction method for spline estimators in clustered/longitudinal models. Simulation studies are performed which demonstrate the efficacy of our variance reduction methods in finite sample settings. Finally, a real data analysis with the motorcycle data set is performed. Here we consider variance estimation and generate 95% pointwise confidence intervals for the unknown regression function.     

On the Validity of the Bootstrap in Non‐Parametric Functional Regression

Abstract. We consider the functional non‐parametric regression model Y= r( χ )+?, where the response Y is univariate, χ is a functional covariate (i.e. valued in some infinite‐dimensional space), and the error ? satisfies E(? | χ ) = 0. For this model, the pointwise asymptotic normality of a kernel estimator of r (·) has been proved in the literature. To use this result for building pointwise confidence intervals for r (·), the asymptotic variance and bias of need to be estimated. However, the functional covariate setting makes this task very hard. To circumvent the estimation of these quantities, we propose to use a bootstrap procedure to approximate the distribution of . Both a naive and a wild bootstrap procedure are studied, and their asymptotic validity is proved. The obtained consistency results are discussed from a practical point of view via a simulation study. Finally, the wild bootstrap procedure is applied to a food industry quality problem to compute pointwise confidence intervals.     

An Adaptive Two-stage Estimation Method for Additive Models

Abstract.  In this paper, a two-stage estimation method for non-parametric additive models is investigated. Differing from Horowitz and Mammen's two-stage estimation, our first-stage estimators are designed not only for dimension reduction but also as initial approximations to all of the additive components. The second-stage estimators are obtained by using one-dimensional non-parametric techniques to refine the first-stage ones. From this procedure, we can reveal a relationship between the regression function spaces and convergence rate, and then provide estimators that are optimal in the sense that, better than the usual one-dimensional mean-squared error (MSE) of the order n ^−4/5 , the MSE of the order n ^{− 1} can be achieved when the underlying models are actually parametric. This shows that our estimation procedure is adaptive in a certain sense. Also it is proved that the bandwidth that is selected by cross-validation depends only on one-dimensional kernel estimation and maintains the asymptotic optimality. Simulation studies show that the new estimators of the regression function and all components outperform the existing estimators, and their behaviours are often similar to that of the oracle estimator.     

Generalized additive models for longitudinal data

We introduce a class of models for longitudinal data by extending the generalized estimating equations approach of Liang and Zeger (1986) to incorporate the flexibility of nonparametric smoothing. The algorithm provides a unified estimation procedure for marginal distributions from the exponential family. We propose pointwise standard-error bands and approximate likelihood-ratio and score tests for inference. The algorithm is formally derived by using the penalized quasilikelihood framework. Convergence of the estimating equations and consistency of the resulting solutions are discussed. We illustrate the algorithm with data on the population dynamics of Colorado potato beetles on potato plants.     

Generalized Self-Consistency: Multinomial logit model and Poisson likelihood

A generalized self-consistency approach to maximum likelihood estimation (MLE) and model building was developed in Tsodikov [2003. Semiparametric models: a generalized self-consistency approach. J. Roy. Statist. Soc. Ser. B Statist. Methodology 65(3), 759–774] and applied to a survival analysis problem. We extend the framework to obtain second-order results such as information matrix and properties of the variance. Multinomial model motivates the paper and is used throughout as an example. Computational challenges with the multinomial likelihood motivated Baker [1994. The Multinomial–Poisson transformation. The Statist. 43, 495–504] to develop the Multinomial–Poisson (MP) transformation for a large variety of regression models with multinomial likelihood kernel. Multinomial regression is transformed into a Poisson regression at the cost of augmenting model parameters and restricting the problem to discrete covariates. Imposing normalization restrictions by means of Lagrange multipliers [Lang, J., 1996. On the comparison of multinomial and Poisson log-linear models. J. Roy. Statist. Soc. Ser. B Statist. Methodology 58, 253–266] justifies the approach. Using the self-consistency framework we develop an alternative solution to multinomial model fitting that does not require augmenting parameters while allowing for a Poisson likelihood and arbitrary covariate structures. Normalization restrictions are imposed by averaging over artificial “missing data” (fake mixture). Lack of probabilistic interpretation at the “complete-data” level makes the use of the generalized self-consistency machinery essential.     

Rank reducible varying coefficient model

We propose in this article a novel dimension reduction method for varying coefficient models. The proposed method explores the rank reducible structure of those varying coefficients, hence, can do dimension reduction and semiparametric estimation, simultaneously. As a result, the new method not only improves estimation accuracy but also facilitates practical interpretation. To determine the structure dimension, a consistent BIC criterion is developed. Numerical experiments are also presented.     

Dimension reduction in estimating equations with covariates missing at random

To estimate parameters defined by estimating equations with covariates missing at random, we consider three bias-corrected nonparametric approaches based on inverse probability weighting, regression and augmented inverse probability weighting. However, when the dimension of covariates is not low, the estimation efficiency will be affected due to the curse of dimensionality. To address this issue, we propose a two-stage estimation procedure by using the dimension-reduced kernel estimation in conjunction with bias-corrected estimating equations. We show that the resulting three estimators are asymptotically equivalent and achieve the desirable properties. The impact of dimension reduction in nonparametric estimation of parameters is also investigated. The finite-sample performance of the proposed estimators is studied through simulation, and an application to an automobile data set is also presented.