Estimation Of Semiparametric Models Research Articles

Thin plate spline model under skew-normal random errors: estimation and diagnostic analysis for spatial data

Expected Maximization (EM) algorithm is often used for estimation in semiparametric models with non-normal observations. However, the EM algorithm’s main disadvantage is its slow convergence rate. In this paper, we propose the Laplace approximation to maximize the marginal likelihood, given a non-linear function assumed as a spline random effect for a skew-normal thin plate spline model. For this, we used automatic differentiation to get the derivatives and provide a numerical evaluation of the Hessian matrix. Comparative simulations and applications between the EM algorithm for thespatial dimension and Laplace approximation were carried out to illustrate the proposed method’s performance. We show that the Laplace approximation is an efficient method, has flexibility to express log-likelihood in a semiparametric model and obtain a fast estimation process for non-normal models. In addition, a local influence analysis was carried out to evaluate the estimation sensitivity.

Quantile estimation of semiparametric model with time-varying coefficients for panel count data.

Panel count data frequently occurs in follow-up studies, such as medical research, social sciences, reliability studies, and tumorigenicity experiences. This type data has been extensively studied by various statistical models with time-invariant regression coefficients. However, the assumption of invariant coefficients may be violated in some reality, and the temporal covariate effects would be of great interest in research studies. This motivates us to consider a more flexible time-varying coefficient model. For statistical inference of the unknown functions, the quantile regression approach based on the B-spline approximation is developed. Asymptotic results on the convergence of the estimators are provided. Some simulation studies are presented to assess the finite-sample performance of the estimators. Finally, two applications of bladder cancer data and US flight delay data are analyzed by the proposed method.

Control variables approach to estimate semiparametric models of mismeasured endogenous regressors with an application to U.K. twin data

We study the identification and estimation of semiparametric models with mismeasured endogenous regressors using control variables that ensure the conditional covariance restriction on endogenous regressors and unobserved causes. We provide a set of sufficient conditions for identification, which control for both endogeneity and measurement error. We propose a sieve-based estimator and derive its asymptotic properties. Given the sieve approximation, our proposed estimator is easy to implement as weighted least squares. Monte Carlo simulations illustrate that our proposed estimator performs well in the finite samples. In an empirical application, we estimate the return to education on earnings using U.K. twin data, in which self-reported education is potentially measured with error and is also correlated with unobserved factors. Our approach utilizes the twin’s reported education as a control variable to obtain consistent estimates. We find that a one-year increase in education leads to an 11% increase in hourly wage. The estimate is significantly higher than those from OLS and IV approaches which are potentially biased. The application underscores that our proposed estimator is useful to correct for both endogeneity and measurement error in estimating returns to education.

Estimation of semiparametric models with errors following a scale mixture of Gaussian distributions

In this paper, we consider a semiparametric regression model where the error follows a scale mixture of Gaussian distributions. The purpose is to estimate the target function which is assumed to belong to some class of functions using the EM algorithm and approximations via P-splines and B-splines. We illustrate the proposed methodology through several simulation studies. Other forms of function approximation are also studied, namely Fourier and wavelet expansions.

Shrinkage and penalized estimation in semi-parametric models with multicollinear data

ABSTRACTIn this paper, we consider estimation techniques based on ridge regression when the matrix appears to be ill-conditioned in the partially linear model using kernel smoothing. Furthermore, we consider that the coefficients can be partitioned as where is the coefficient vector for main effects, and is the vector for ‘nuisance’ effects. We are essentially interested in the estimation of when it is reasonable to suspect that is close to zero. We suggest ridge pretest, ridge shrinkage and ridge positive shrinkage estimators for the above semi-parametric model, and compare its performance with some penalty estimators. In particular, suitability of estimating the nonparametric component based on the kernel smoothing basis function is also explored. Monte Carlo simulation study is used to compare the relative efficiency of proposed estimators, and a real data example is presented to illustrate the usefulness of the suggested methods. Moreover, the asymptotic properties of the proposed estimators are obtained.

Semiparametric estimation with missing covariates

This paper considers estimation in semiparametric models when some of the covariates are missing at random. The paper proposes an iterative estimator based on inverse probability weighting and local linear estimation of the nonparametric component. The resulting estimator is very general and can be used in the context of semiparametric maximum likelihood, quasi likelihood and robust estimation. The paper establishes the asymptotic normality of the estimator using both nonparametric and parametric estimation of the unknown probability weights. Two general examples illustrate the theory and Monte Carlo simulations show that the proposed estimator has good finite sample properties.

Shrinkage Estimation of Semiparametric Model with Missing Responses for Cluster Data

This paper simultaneously investigates variable selection and imputation estimation of semiparametric partially linear varying-coefficient model in that case where there exist missing responses for cluster data. As is well known, commonly used approach to deal with missing data is complete-case data. Combined the idea of complete-case data with a discussion of shrinkage estimation is made on different cluster. In order to avoid the biased results as well as improve the estimation efficiency, this article introduces Group Least Absolute Shrinkage and Selection Operator (Group Lasso) to semiparametric model. That is to say, the method combines the approach of local polynomial smoothing and the Least Absolute Shrinkage and Selection Operator. In that case, it can conduct nonparametric estimation and variable selection in a computationally efficient manner. According to the same criterion, the parametric estimators are also obtained. Additionally, for each cluster, the nonparametric and parametric estimators are derived, and then compute the weighted average per cluster as finally estimators. Moreover, the large sample properties of estimators are also derived respectively.

Asymptotic optimality of Hodges-Lehmann inverse rank likelihood estimators

Hodges and Lehmann proposed using rank test statistics evaluated at inversely transformed data to construct estimating equations. This paper reframes the resulting estimates in terms of a rank likelihood. In the context of a model of the form Y = h( e; z, β), where Y is a response, z a vector of predictors and e is a random error, this approach corresponds to computing the inverse e = g( Y, z; β) by solving Y = h( e; z, β) for e and using the distribution of the ranks of independent e’s as a likelihood. The properties of the resulting estimators have been developed in many important contexts. This paper will review and extend asymptotic optimality properties of Hodges Lehmann estimators in semiparametric models. In particular, the paper will establish semiparametric optimality of the estimate obtained from a Gaussian linear model. Moreover, it will be shown that the Hodges-Lehmann estimate obtained from the exponential likelihood is asymptotically minimax for the semiparametric accelerated failure time model with increasing hazard rates, and it will be shown that a uniform (Wilcoxon) score estimate applied to log Yi, 1 ≤ i ≤ n, is asymptotically minimax for an accelerated failure time model with increasing logit rate. References to recently developed software in R is provided.

Estimation in semiparametric models with missing data

This paper considers the problem of parameter estimation in a general class of semiparametric models when observations are subject to missingness at random. The semiparametric models allow for estimating functions that are non-smooth with respect to the parameter. We propose a nonparametric imputation method for the missing values, which then leads to imputed estimating equations for the finite dimensional parameter of interest. The asymptotic normality of the parameter estimator is proved in a general setting, and is investigated in detail for a number of specific semiparametric models. Finally, we study the small sample performance of the proposed estimator via simulations.

Impact of unknown covariance structures in semiparametric models for longitudinal data: An application to Wisconsin diabetes data

Semiparametric models are becoming increasingly attractive for longitudinal data analysis. Often there is lack of knowledge of the covariance structure of the response variable. Although it is still possible to obtain consistent estimators for both parametric and nonparametric components of a semipatrametric model by assuming an identity structure for the covariance matrix, the resulting estimators may not be efficient. We conducted extensive simulation studies to investigate the impact of an unknown covariance structure on estimators in semiparametric models for longitudinal data. In some situations the loss of efficiency could be substantial. A two-step estimator is thus proposed to improve the efficiency. Our study was motivated by a population based data analysis to examine the temporal relationship between systolic blood pressure and urinary albumin excretion.

Estimation of Semiparametric Models in the Presence of Endogeneity and Sample Selection

We analyze a semiparametric model for data that suffer from the problems of sample selection, where some of the data are observed for only part of the sample with a probability that depends on a selection equation, and of endogeneity, where a covariate is correlated with the disturbance term. The introduction of nonparametric functions in the model permits great flexibility in the way covariates affect response variables. We present an efficient Bayesian method for the analysis of such models that allows us to consider general systems of outcome variables and endogenous regressors that are continuous, binary, censored, or ordered. Estimation is by Markov chain Monte Carlo (MCMC) methods. The algorithm we propose does not require simulation of the outcomes that are missing due to the selection mechanism, which reduces the computational load and improves the mixing of the MCMC chain. The approach is applied to a model of women’s labor force participation and log-wage determination. Data and computer code used in this article are available online.

Asymptotics of estimators in semi-parametric model under NA samples

This paper is concerned with the heteroscedastic semi-parametric regression model y i = x i β + g ( t i ) + σ i e i , 1 ⩽ i ⩽ n , where σ i 2 = f ( u i ) , ( x i , t i , u i ) are fixed design points, g and f are unknown functions, and random errors e i are negatively associated (NA) random variables. The strong consistency and asymptotic normality for least-squares estimators and weighted least-squares estimators of β are studied. In addition, the strong consistency for the estimators of f ( · ) and g ( · ) is investigated. Some results on independent random error settings are extended.

Efficient estimation of Banach parameters in semiparametric models

Consider a semiparametric model with a Euclidean parameter and an infinite-dimensional parameter, to be called a Banach parameter. Assume: (a) There exists an efficient estimator of the Euclidean parameter. (b) When the value of the Euclidean parameter is known, there exists an estimator of the Banach parameter, which depends on this value and is efficient within this restricted model. Substituting the efficient estimator of the Euclidean parameter for the value of this parameter in the estimator of the Banach parameter, one obtains an efficient estimator of the Banach parameter for the full semiparametric model with the Euclidean parameter unknown. This hereditary property of efficiency completes estimation in semiparametric models in which the Euclidean parameter has been estimated efficiently. Typically, estimation of both the Euclidean and the Banach parameter is necessary in order to describe the random phenomenon under study to a sufficient extent. Since efficient estimators are asymptotically linear, the above substitution method is a particular case of substituting asymptotically linear estimators of a Euclidean parameter into estimators that are asymptotically linear themselves and that depend on this Euclidean parameter. This more general substitution case is studied for its own sake as well, and a hereditary property for asymptotic linearity is proved.

On the construction of efficient estimators in semiparametric models

This paper deals with the construction of efficient estimators in semiparametric models without the sample splitting technique. Schick (1987) gave sufficient conditions using the leave-one-out technique for a construction without sample splitting. His conditions are stronger and more cumbersome to verify than the necessary and sufficient conditions for the existence of efficient estimators which suffice for the construction based on sample splitting. In this paper we use a conditioning argument to weaken Schick′s conditions. We shall then show that in a large class of semiparametric models and for properly chosen estimators of the score function the resulting weaker conditions reduce to the minimal conditions for the construction with sample splitting. In other words, in these models efficient estimators can be constructed without sample splitting under the same conditions as those used for the construction with sample splitting. We demonstrate our results by constructing an efficient estimator using these ideas in a semiparametric additive regression model.

Application of convolution theorems in semiparametric models with non-i.i.d. data

A useful approach to asymptotic efficiency for estimators in semiparametric models is the study of lower bounds on asymptotic variances via convolution theorems. Such theorems are often applicable in models in which the classical assumptions of independence and identical distributions fail to hold, but to date, much of the research has focused on semiparametric models with independent and identically distributed (i.i.d.) data because tools are available in the i.i.d. setting for verifying pre-conditions of the convolution theorems. We develop tools for non-i.i.d. data that are similar in spirit to those for i.i.d. data and also analogous to the approaches used in parametric models with dependent data. This involves extending the notion of the tangent vector figuring so prominently in the i.i.d. theory and providing conditions for smoothness, or differentiability, of the parameter of interest as a function of the underlying probability measures. As a corollary to the differentiability result we obtain sufficient conditions for equivalence, in terms of asymptotic variance bounds, of two models. Regularity and asymptotic linearity of estimators are also discussed.

On Optimal B-Robust Influence Functions in Semiparametric Models

Bounded influence functions are used for robust estimation in semiparametric models. In this paper, we generalize Hampel's variational problem to semiparametric models and define the optimal B-robust influence function as the one solving the variational problem. We identify the lowest bounds for influence functions and establish the existence and uniqueness of the optimal influence functions in general semiparametric models. Explicit optimal influence functions are given for a special case. Examples are provided to illustrate the procedures for calculating the optimal influence functions and for constructing the corresponding optimal estimators.

NONPARAMETRIC AND SEMIPARAMETRIC METHODS FOR ECONOMIC RESEARCH

Abstract. Developments in the vast and growing literatures on nonparametric and semiparametric statistical estimation are reviewed. The emphasis is on useful methodology rather than statistical properties for their own sake. Some empirical applications to economic data are described. The paper deals separately with nonparametric density estimation, nonparametric regression estimation, and estimation of semiparametric models.

Curve fitting by polynomial-trigonometric regression

SUMMARY We consider a method of estimating an unknown regression curve by regression on a combination of low-order polynomial terms and trigonometric terms. Estimation based on trigonometric functions alone is known to suffer from bias problems at the boundaries due to the periodic nature of the fitted functions. We show that these boundary problems are alleviated by adding low-order polynomial terms. The utility of the method is illustrated with examples, and asymptotic results show that the estimators are competitive with other nonparametric procedures. Applications to estimation in semiparametric models are also considered with particular emphasis on the case of analysis of covariance when the relationship between the response and the covariate is difficult to model parametrically.

Introduction à la théorie des bornes d'efficacité semiparamétriques

As in parametric models, one can define an efficiency bound for estimators in semi-parametric models, which is analogous to the Cramer-Rao bound. This paper surveys the available results: computation of the bound by mean of projection, possibility and construction of semi-parametric estimators, efficient estimation in semi-parametric models. Numerous examples are discussed and a new class of estimators in censored regression models is introduced.

On the Strong Mixing Property for Linear Sequences

On the Strong Mixing Property for Linear Sequences