Semiparametric Density Deconvolution

Abstract.  A new semiparametric method for density deconvolution is proposed, based on a model in which only the ratio of the unconvoluted to convoluted densities is specified parametrically. Deconvolution results from reweighting the terms in a standard kernel density estimator, where the weights are defined by the parametric density ratio. We propose that in practice, the density ratio be modelled on the log-scale as a cubic spline with a fixed number of knots. Parameter estimation is based on maximization of a type of semiparametric likelihood. The resulting asymptotic properties for our deconvolution estimator mirror the convergence rates in standard density estimation without measurement error when attention is restricted to our semiparametric class of densities. Furthermore, numerical studies indicate that for practical sample sizes our weighted kernel estimator can provide better results than the classical non-parametric kernel estimator for a range of densities outside the specified semiparametric class.     

Uniformly adaptive estimation for models with arma errors

A semiparametric estimator based on an unknown density isuniformly adaptive if the expected loss of the estimator converges to the asymptotic expected loss of the maximum liklihood estimator based on teh true density (MLE), and if convergence does not depend on either the parameter values or the form of the unknown density. Without uniform adaptivity, the asymptotic expected loss of the MLE need not approximate the expected loss of a semiparametric estimator for any finite sample I show that a two step semiparametric estimator is uniformly adaptive for the parameters of nonlinear regression models with autoregressive moving average errors.     

Uniformly adaptive estimation for models with arma errors

A semiparametric estimator based on an unknown density isuniformly adaptive if the expected loss of the estimator converges to the asymptotic expected loss of the maximum liklihood estimator based on teh true density (MLE), and if convergence does not depend on either the parameter values or the form of the unknown density. Without uniform adaptivity, the asymptotic expected loss of the MLE need not approximate the expected loss of a semiparametric estimator for any finite sample I show that a two step semiparametric estimator is uniformly adaptive for the parameters of nonlinear regression models with autoregressive moving average errors.     

Semiparametric efficient estimation for the auxiliary outcome problem with the conditional mean model

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz‐Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite‐sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.     

The central limit theorem under semiparametric random censorship models

We study integrals for arbitrary Borel-measurable functions with respect to a semiparametric estimator of the distribution function in the random censorship model. Based on a representation of these integrals, which is similar to the one given by Stute for Kaplan–Meier integrals, a central limit theorem is established which generalizes a corresponding result of the Cheng and Lin estimator. It is shown that the semiparametric integral estimator is at least as efficient as the corresponding Kaplan–Meier integral estimator in terms of asymptotic variance if the correct semiparametric model is used. Furthermore, a necessary and sufficient condition for a strict gain in efficiency is stated. Finally, this asymptotic result is confirmed in a small simulation study under moderate sample sizes.     

Semiparametric ARCH Models

This article introduces a semiparametric autoregressive conditional heteroscedasticity (ARCH) model that has conditional first and second moments given by autoregressive moving average and ARCH parametric formulations but a conditional density that is assumed only to be sufficiently smooth to be approximated by a nonparametric density estimator. For several particular conditional densities, the relative efficiency of the quasi-maximum likelihood estimator is compared with maximum likelihood under correct specification. These potential efficiency gains for a fully adaptive procedure are compared in a Monte Carlo experiment with the observed gains from using the proposed semiparametric procedure, and it is found that the estimator captures a substantial proportion of the potential. The estimator is applied to daily stock returns from small firms that are found to exhibit conditional skewness and kurtosis and to the British pound to dollar exchange rate.     

The Semiparametric Normal Variance-Mean Mixture Model

We study the normal variance-mean mixture model from a semiparametric point of view, i.e. we let the mixing distribution belong to a non-parametric family. The main results are consistency of the non-parametric maximum likelihood estimator and construction of an asymptotically normal and efficient estimator for the Euclidian part of the parameter. We study the model according to the theory outlined in the monograph by Bickel et al. (1993) and apply a general result (based on the theory of empirical processes) for semiparametric models from van der Vaart (1996) to prove asymptotic normality and efficiency of the proposed estimator.     

Semiparametric Efficient Estimation of the Mean of a Time Series in the Presence of Conditional Heterogeneity of Unknown Form

We obtain semiparametric efficiency bounds for estimation of a location parameter in a time series model where the innovations are stationary and ergodic conditionally symmetric martingale differences but otherwise possess general dependence and distributions of unknown form. We then describe an iterative estimator that achieves this bound when the conditional density functions of the sample are known. Finally, we develop a “semi-adaptive” estimator that achieves the bound when these densities are unknown by the investigator. This estimator employs nonparametric kernel estimates of the densities. Monte Carlo results are reported.     

Semiparametric Efficient Estimation of the Mean of a Time Series in the Presence of Conditional Heterogeneity of Unknown Form

Abstract

We obtain semiparametric efficiency bounds for estimation of a location parameter in a time series model where the innovations are stationary and ergodic conditionally symmetric martingale differences but otherwise possess general dependence and distributions of unknown form. We then describe an iterative estimator that achieves this bound when the conditional density functions of the sample are known. Finally, we develop a “semi-adaptive” estimator that achieves the bound when these densities are unknown by the investigator. This estimator employs nonparametric kernel estimates of the densities. Monte Carlo results are reported.     

Semiparametric estimation in copula models

The author recalls the limiting behaviour of the empirical copula process and applies it to prove some asymptotic properties of a minimum distance estimator for a Euclidean parameter in a copula model. The estimator in question is semiparametric in that no knowledge of the marginal distributions is necessary. The author also proposes another semiparametric estimator which he calls “rank approximate Z‐estimator” and whose asymptotic normality he derives. He further presents Monte Carlo simulation results for the comparison of various estimators in four well‐known bivariate copula models.     

样本选择模型的一个简单半参数估计量

本文发展了一个针对样本选择模型的两阶段半参数估计量,其首先在第一阶段基于对数欧几里得分布差异测度估计离散选择概率,进而在第二阶段利用非参数sieve方法估计一个包含参数和非参数部分的部分线性模型以得到模型参数的估计。相对于文献中已有的半参数估计量,该估计量的计算更加简便,且计算负担相对较小。我们说明了该半参数估计量的一致性和渐近正态性,同时给出了其渐近方差的计算公式。蒙特卡洛模拟结果符合我们的理论结论。     

Conditional Score Approach to Errors‐in‐Variable Current Status Data Under the Proportional Odds Model

Abstract. The conditional score approach is proposed to the analysis of errors‐in‐variable current status data under the proportional odds model. Distinct from the conditional scores in other applications, the proposed conditional score involves a high‐dimensional nuisance parameter, causing challenges in both asymptotic theory and computation. We propose a composite algorithm combining the Newton–Raphson and self‐consistency algorithms for computation and develop an efficient conditional score, analogous to the efficient score from a typical semiparametric likelihood, for building an asymptotic linear expression and hence the asymptotic distribution of the conditional‐score estimator for the regression parameter. Our proposal is shown to perform well in simulation studies and is applied to a zebrafish basal cell carcinoma data involving measurement errors in gene expression levels.     

Profile Hellinger distance estimation

The successful application of the Hellinger distance approach to fully parametric models is well known. The corresponding optimal estimators, known as minimum Hellinger distance (MHD) estimators, are efficient and have excellent robustness properties [Beran R. Minimum Hellinger distance estimators for parametric models. Ann Statist. 1977;5:445–463]. This combination of efficiency and robustness makes MHD estimators appealing in practice. However, their application to semiparametric statistical models, which have a nuisance parameter (typically of infinite dimension), has not been fully studied. In this paper, we investigate a methodology to extend the MHD approach to general semiparametric models. We introduce the profile Hellinger distance and use it to construct a minimum profile Hellinger distance estimator of the finite-dimensional parameter of interest. This approach is analogous in some sense to the profile likelihood approach. We investigate the asymptotic properties such as the asymptotic normality, efficiency, and adaptivity of the proposed estimator. We also investigate its robustness properties. We present its small-sample properties using a Monte Carlo study.     

Efficient Estimation in Heteroscedastic Partially Linear Varying Coefficient Models

This article considers statistical inference for the heteroscedastic partially linear varying coefficient models. We construct an efficient estimator for the parametric component by applying the weighted profile least-squares approach, and show that it is semiparametrically efficient in the sense that the inverse of the asymptotic variance of the estimator reaches the semiparametric efficiency bound. Simulation studies are conducted to illustrate the performance of the proposed method.     

A note on the asymptotic relative efficiency of estimators from the additive hazards model

In this note, the asymptotic variance formulas are explicitly derived and compared between the parametric and semiparametric estimators of a regression parameter and survival probability under the additive hazards model. To obtain explicit formulas, it is assumed that the covariate term including a regression coefficient follows a gamma distribution and the baseline hazard function is constant. The results show that the semiparametric estimator of the regression coefficient parameter is fully efficient relative to the parametric counterpart when the survival time and a covariate are independent, as in the proportional hazards model. Relative to a more realistic case of the parametric additive hazards model with a Weibull baseline, the loss of efficiency of the semiparametric estimator of survival probability is moderate.     

On Efficient Estimators of the Proportion of True Null Hypotheses in a Multiple Testing Setup

We consider the problem of estimating the proportion θ of true null hypotheses in a multiple testing context. The setup is classically modelled through a semiparametric mixture with two components: a uniform distribution on interval [0,1] with prior probability θ and a non‐parametric density f . We discuss asymptotic efficiency results and establish that two different cases occur whether f vanishes on a non‐empty interval or not. In the first case, we exhibit estimators converging at a parametric rate, compute the optimal asymptotic variance and conjecture that no estimator is asymptotically efficient (i.e. attains the optimal asymptotic variance). In the second case, we prove that the quadratic risk of any estimator does not converge at a parametric rate. We illustrate those results on simulated data.     

Discrete triangular associated kernel and bandwidth choices in semiparametric estimation for count data

This work deals with semiparametric kernel estimator of probability mass functions which are assumed to be modified Poisson distributions. This semiparametric approach is based on discrete associated kernel method appropriated for modelling count data; in particular, the famous discrete symmetric triangular kernels are used. Two data-driven bandwidth selection procedures are investigated and an explicit expression of optimal bandwidth not available until now is provided. Moreover, some asymptotic properties of the cross-validation criterion adapted for discrete semiparametric kernel estimation are studied. Finally, to measure the performance of semiparametric estimator according to each type of bandwidth parameter, some applications are realized on three real count data-sets from sociology and biology.     

Root n consistent and optimal density estimators for moving average processes

Abstract.  The marginal density of a first order moving average process can be written as a convolution of two innovation densities. Saavedra & Cao [Can. J. Statist. (2000), 28, 799] propose to estimate the marginal density by plugging in kernel density estimators for the innovation densities, based on estimated innovations. They obtain that for an appropriate choice of bandwidth the variance of their estimator decreases at the rate 1/ n . Their estimator can be interpreted as a specific U -statistic. We suggest a slightly simplified U -statistic as estimator of the marginal density, prove that it is asymptotically normal at the same rate, and describe the asymptotic variance explicitly. We show that the estimator is asymptotically efficient if no structural assumptions are made on the innovation density. For innovation densities known to have mean zero or to be symmetric, we describe improvements of our estimator which are again asymptotically efficient.     

A Semiparametric Model for Line Transect Sampling

This paper introduces an appealing semiparametric model for estimating wildlife abundance based on line transect data. The proposed method requires the existence of a parametric model and then improves the estimator using a kernel method. Properties of the resultant estimator are derived and an expression for the asymptotic mean square error (AMSE) of the estimator is given. Minimization of the AMSE leads to an explicit formula for an optimal choice of the smoothing parameter. Small-sample properties of the proposed estimator using the parametric half-normal model are investigated and compared with the classical kernel estimator using both simulations and real data. Numerical results show that improvements over the classical kernel estimator often can be realized even when the true density is far from the half-normal model.     

Multiplicative Adjustment Method for Semiparametric Regression with Mixing Dependent Data

In this article, we extend a semiparametric regression estimator with multiplicative adjustment to time series context. The asymptotic theory and results from a simulation study are discussed. Theoretical results and numerical comparison show that, in the time series case, the semiparametric estimator is better than the traditional local polynomial estimator in a wide neighbourhood around the true regression function.