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.     

Statistical inference for panel data semiparametric partially linear regression models with heteroscedastic errors

We consider a panel data semiparametric partially linear regression model with an unknown parameter vector for the linear parametric component, an unknown nonparametric function for the nonlinear component, and a one-way error component structure which allows unequal error variances (referred to as heteroscedasticity). We develop procedures to detect heteroscedasticity and one-way error component structure, and propose a weighted semiparametric least squares estimator (WSLSE) of the parametric component in the presence of heteroscedasticity and/or one-way error component structure. This WSLSE is asymptotically more efficient than the usual semiparametric least squares estimator considered in the literature. The asymptotic properties of the WSLSE are derived. The nonparametric component of the model is estimated by the local polynomial method. Some simulations are conducted to demonstrate the finite sample performances of the proposed testing and estimation procedures. An example of application on a set of panel data of medical expenditures in Australia is also illustrated.     

Thresholding projection estimators in functional linear models

We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows us to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits us to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove that these estimators are minimax and rates of convergence are given for some particular cases.     

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.     

Flexible modeling based on copulas in nonparametric median regression

Consider the model Y=m(X)+ε, where m(⋅)=med(Y|⋅) is unknown but smooth. It is often assumed that ε and X are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between ε and X by means of a copula model, i.e. (ε,X)∼C_θ(F_ε(⋅),F_X(⋅)), where C_θ is a copula function depending on an unknown parameter θ, and F_ε and F_X are the marginals of ε and X. Since many parametric copula families contain the independent copula as a special case, the so-obtained regression model is more flexible than the ‘classical’ regression model.We estimate the parameter θ via a pseudo-likelihood method and prove the asymptotic normality of the estimator, based on delicate empirical process theory. We also study the estimation of the conditional distribution of Y given X. The procedure is illustrated by means of a simulation study, and the method is applied to data on food expenditures in households.     

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.     

Berry-Esseen bounds for wavelet estimator in a regression model with linear process errors

In this paper, we derive the Berry-Esseen bounds of the wavelet estimator for a nonparametric regression model with linear process errors generated by φ-mixing sequences. As application, by the suitable choice of some constants, the convergence rate O(n^−1/6) of uniformly asymptotic normality of the wavelet estimator is obtained. Our results generalize some known results in the literature.     

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.     

Analysis of correlated binary data under partially linear single-index logistic models

Clustered data arise commonly in practice and it is often of interest to estimate the mean response parameters as well as the association parameters. However, most research has been directed to address the mean response parameters with the association parameters relegated to a nuisance role. There is relatively little work concerning both the marginal and association structures, especially in the semiparametric framework. In this paper, our interest centers on the inference of both the marginal and association parameters. We develop a semiparametric method for clustered binary data and establish the theoretical results. The proposed methodology is investigated through various numerical studies.     

Efficiency of a Liu-type estimator in semiparametric regression models

In this paper we consider the semiparametric regression model, y=Xβ+f+ε. Recently, Hu [11] proposed ridge regression estimator in a semiparametric regression model. We introduce a Liu-type (combined ridge-Stein) estimator (LTE) in a semiparametric regression model. Firstly, Liu-type estimators of both β and f are attained without a restrained design matrix. Secondly, the LTE estimator of β is compared with the two-step estimator in terms of the mean square error. We describe the almost unbiased Liu-type estimator in semiparametric regression models. The almost unbiased Liu-type estimator is compared with the Liu-type estimator in terms of the mean squared error matrix. A numerical example is provided to show the performance of the estimators.     

Functional-coefficient partially linear regression model

In this paper, the functional-coefficient partially linear regression (FCPLR) model is proposed by combining nonparametric and functional-coefficient regression (FCR) model. It includes the FCR model and the nonparametric regression (NPR) model as its special cases. It is also a generalization of the partially linear regression (PLR) model obtained by replacing the parameters in the PLR model with some functions of the covariates. The local linear technique and the integrated method are employed to give initial estimators of all functions in the FCPLR model. These initial estimators are asymptotically normal. The initial estimator of the constant part function shares the same bias as the local linear estimator of this function in the univariate nonparametric model, but the variance of the former is bigger than that of the latter. Similarly, initial estimators of every coefficient function share the same bias as the local linear estimates in the univariate FCR model, but the variance of the former is bigger than that of the latter. To decrease the variance of the initial estimates, a one-step back-fitting technique is used to obtain the improved estimators of all functions. The improved estimator of the constant part function has the same asymptotic normality property as the local linear nonparametric regression for univariate data. The improved estimators of the coefficient functions have the same asymptotic normality properties as the local linear estimates in FCR model. The bandwidths and the smoothing variables are selected by a data-driven method. Both simulated and real data examples related to nonlinear time series modeling are used to illustrate the applications of the FCPLR model.     

Estimating the inter-arrival time density of Markov renewal processes under structural assumptions on the transition distribution

We consider a stationary Markov renewal process whose inter-arrival time density depends multiplicatively on the distance between the past and present state of the embedded chain. This is appropriate when the jump size is governed by influences that accumulate over time. Then we can construct an estimator for the inter-arrival time density that has the parametric rate of convergence. The estimator is a local von Mises statistic. The result carries over to the corresponding semi-Markov process.     

Oracally efficient spline smoothing of nonlinear additive autoregression models with simultaneous confidence band

Under weak conditions of smoothness and mixing, we propose spline-backfitted spline (SBS) estimators of the component functions for a nonlinear additive autoregression model that is both computationally expedient for analyzing high dimensional large time series data, and theoretically reliable as the estimator is oracally efficient and comes with asymptotically simultaneous confidence band. Simulation evidence strongly corroborates with the asymptotic theory.     

Empirical likelihood for linear models under negatively associated errors

In this paper, we discuss the construction of the confidence intervals for the regression vector β in a linear model under negatively associated errors. It is shown that the blockwise empirical likelihood (EL) ratio statistic for β is asymptotically χ²-type distributed. The result is used to obtain an EL based confidence region for β.     

Using bimodal kernel for inference in nonparametric regression with correlated errors

For nonparametric regression model with fixed design, it is well known that obtaining a correct bandwidth is difficult when errors are correlated. Various methods of bandwidth selection have been proposed, but their successful implementation critically depends on a tuning procedure which requires accurate information about error correlation. Unfortunately, such information is usually hard to obtain since errors are not observable. In this article a new bandwidth selector based on the use of a bimodal kernel is proposed and investigated. It is shown that the new bandwidth selector is quite useful for the tuning procedures of various other methods. Furthermore, the proposed bandwidth selector itself proves to be quite effective when the errors are severely correlated.     

Nonparametric regression function estimation with surrogate data and validation sampling

This paper develops estimation approaches for nonparametric regression analysis with surrogate data and validation sampling when response variables are measured with errors. Without assuming any error model structure between the true responses and the surrogate variables, a regression calibration kernel regression estimate is defined with the help of validation data. The proposed estimator is proved to be asymptotically normal and the convergence rate is also derived. A simulation study is conducted to compare the proposed estimators with the standard Nadaraya-Watson estimators with the true observations in the validation data set and the complete observations, respectively. The Nadaraya-Watson estimator with the complete observations can serve as a gold standard, even though it is practically unachievable because of the measurement errors.     

On the block thresholding wavelet estimators with censored data

We consider block thresholding wavelet-based density estimators with randomly right-censored data and investigate their asymptotic convergence rates. Unlike for the complete data case, the empirical wavelet coefficients are constructed through the Kaplan-Meier estimators of the distribution functions in the censored data case. On the basis of a result of Stute [W. Stute, The central limit theorem under random censorship, Ann. Statist. 23 (1995) 422-439] that approximates the Kaplan-Meier integrals as averages of i.i.d. random variables with a certain rate in probability, we can show that these wavelet empirical coefficients can be approximated by averages of i.i.d. random variables with a certain error rate in L². Therefore we can show that these estimators, based on block thresholding of empirical wavelet coefficients, achieve optimal convergence rates over a large range of Besov function classes , p≥2, q≥1 and nearly optimal convergence rates when 1≤p<2. We also show that these estimators achieve optimal convergence rates over a large class of functions that involve many irregularities of a wide variety of types, including chirp and Doppler functions, and jump discontinuities. Therefore, in the presence of random censoring, wavelet estimators still provide extensive adaptivity to many irregularities of large function classes. The performance of the estimators is tested via a modest simulation study.     

Confidence intervals for marginal parameters under fractional linear regression imputation for missing data

Item nonresponse occurs frequently in sample surveys and other applications. Imputation is commonly used to fill in the missing item values in a random sample {Y_i;i=1,…,n}. Fractional linear regression imputation, based on the model with independent zero mean errors ?_i, is used to create one or more imputed values in the data file for each missing item Y_i, where {X_i,i=1,…,n}, is observed completely. Asymptotic normality of the imputed estimators of the mean μ=E(Y), distribution function θ=F(y) for a given y, and qth quantile θ_q=F^-1(q),0<q<1 is established, assuming that Y is missing at random (MAR) given X. This result is used to obtain normal approximation (NA)-based confidence intervals on μ,θ and θ_q. In the case of θ_q, a Bahadur-type representation and Woodruff-type confidence intervals are also obtained. Empirical likelihood (EL) ratios are also obtained and shown to be asymptotically scaled variables. This result is used to obtain asymptotically correct EL-based confidence intervals on μ,θ and θ_q. Results of a simulation study on the finite sample performance of NA-based and EL-based confidence intervals are reported.     

Tests for multiple regression based on simplicial depth

A general approach for developing distribution free tests for general linear models based on simplicial depth is applied to multiple regression. The tests are based on the asymptotic distribution of the simplicial regression depth, which depends only on the distribution law of the vector product of regressor variables. Based on this formula, the spectral decomposition and thus the asymptotic distribution is derived for multiple regression through the origin and multiple regression with Cauchy distributed explanatory variables. The errors may be heteroscedastic and the concrete form of the error distribution does not need to be known. Moreover, the asymptotic distribution for multiple regression with intercept does not depend on the location and scale of the explanatory variables. A simulation study suggests that the tests can be applied also to normal distributed explanatory variables. An application on multiple regression for shape analysis of fishes demonstrates the applicability of the new tests and in particular their outlier robustness.     

Model checking in errors-in-variables regression

This paper discusses a class of minimum distance tests for fitting a parametric regression model to a class of regression functions in the errors-in-variables model. These tests are based on certain minimized distances between a nonparametric regression function estimator and a deconvolution kernel estimator of the conditional expectation of the parametric model being fitted. The paper establishes the asymptotic normality of the proposed test statistics under the null hypothesis and that of the corresponding minimum distance estimators. We also prove the consistency of the proposed tests against a fixed alternative and obtain the asymptotic distributions for general local alternatives. Simulation studies show that the testing procedures are quite satisfactory in the preservation of the finite sample level and in terms of a power comparison.