An overview of model-robust regression

The proper combination of parametric and nonparametric regression procedures can improve upon the shortcomings of each when used individually. Considered is the situation where the researcher has an idea of which parametric model should explain the behavior of the data, but this model is not adequate throughout the entire range of the data. An extension of partial linear regression and two other methods of model-robust regression are developed and compared in this context. The model-robust procedures each involve the proportional mixing of a parametric fit to the data and a nonparametric fit to either the data or residuals. The emphasis of this work is on fitting in the small-sample situation, where nonparametric regression alone has well-known inadequacies. Performance is based on bias and variance considerations, and theoretical mean squared error formulas are developed for each procedure. An example is given that uses generated data from an underlying model with defined misspecification to provide graphical comparisons of the fits and to show the theoretical benefits of the model-robust procedures. Simulation results are presented which establish the accuracy of the theoretical formulas and illustrate the potential benefits of the model-robust procedures. Simulations are also used to illustrate the advantageous properties of a data-driven selector developed in this work for choosing the smoothing and mixing parameters. It is seen that the model-robust procedures (the final proposed method, in particular) give much improved fits over the individual parametric and nonparametric fits.     

Semiparametric location mixtures with distinct components

We consider a two-component location mixture model with symmetric components, one of which is assumed to be known, the other is unknown. We show identifiability under assumptions on the tails of the characteristic function for the true underlying mixture, and also construct asymptotically normal estimates. The model is an extension of the contamination model in Bordes et al. [Semiparametric estimation of a two-component mixture model when a component is known, Scand. J. Statist. 33 (2006), pp. 733–752], and also related to a location mixture of one symmetric density as in Bordes et al. [Semiparametric estimation of a two component mixture model, Ann. Statist. 34 (2006), pp. 1204–1232]. We show by simulation that estimating the additional location parameter leads to a slight loss of efficiency as compared with the contamination model.     

Efficiency of the generalized difference-based Liu estimators in semiparametric regression models with correlated errors

In this paper, a generalized difference-based estimator is introduced for the vector parameter β in the semiparametric regression model when the errors are correlated. A generalized difference-based Liu estimator is defined for the vector parameter β in the semiparametric regression model. Under the linear nonstochastic constraint Rβ=r, the generalized restricted difference-based Liu estimator is given. The risk function for the β?_GRD(η) associated with weighted balanced loss function is presented. The performance of the proposed estimators is evaluated by a simulated data set.     

Efficient Estimation of the Partly Linear Additive Hazards Model with Current Status Data

This paper focuses on efficient estimation, optimal rates of convergence and effective algorithms in the partly linear additive hazards regression model with current status data. We use polynomial splines to estimate both cumulative baseline hazard function with monotonicity constraint and nonparametric regression functions with no such constraint. We propose a simultaneous sieve maximum likelihood estimation for regression parameters and nuisance parameters and show that the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound. In addition, we show that rates of convergence for the estimators of nonparametric functions are optimal. We implement the proposed estimation through a backfitting algorithm on generalized linear models. We conduct simulation studies to examine the finite‐sample performance of the proposed estimation method and present an analysis of renal function recovery data for illustration.     

A Cox‐Aalen Model for Interval‐censored Data

The Cox‐Aalen model, obtained by replacing the baseline hazard function in the well‐known Cox model with a covariate‐dependent Aalen model, allows for both fixed and dynamic covariate effects. In this paper, we examine maximum likelihood estimation for a Cox‐Aalen model based on interval‐censored failure times with fixed covariates. The resulting estimator globally converges to the truth slower than the parametric rate, but its finite‐dimensional component is asymptotically efficient. Numerical studies show that estimation via a constrained Newton method performs well in terms of both finite sample properties and processing time for moderate‐to‐large samples with few covariates. We conclude with an application of the proposed methods to assess risk factors for disease progression in psoriatic arthritis.     

Efficient Tests Under a Weak Convergence Assumption

The asymptotic validity of tests is usually established by making appropriate primitive assumptions, which imply the weak convergence of a specific function of the data, and an appeal to the continuous mapping theorem. This paper, instead, takes the weak convergence of some function of the data to a limiting random element as the starting point and studies efficiency in the class of tests that remain asymptotically valid for all models that induce the same weak limit. It is found that efficient tests in this class are simply given by efficient tests in the limiting problem—that is, with the limiting random element assumed observed—evaluated at sample analogues. Efficient tests in the limiting problem are usually straightforward to derive, even in nonstandard testing problems. What is more, their evaluation at sample analogues typically yields tests that coincide with suitably robustified versions of optimal tests in canonical parametric versions of the model. This paper thus establishes an alternative and broader sense of asymptotic efficiency for many previously derived tests in econometrics, such as tests for unit roots, parameter stability tests, and tests about regression coefficients under weak instruments.     

Semiparametric estimation of employment duration models

There are a variety of economic areas, such as studies of employment duration and of the durability of capital goods, in which data on important variables typically are censored. The standard techinques for estimating a model from censored data require the distributions of unobservable random components of the model to be specified a priori up to a finite set of parameters, and misspecification of these distributions usually leads to inconsistent parameter estimates. However, economic theory rarely gives guidance about distributions and the standard estimation techniques do not provide convenient methods for identifying distributions from censored data. Recently, several distribution-free or semiparametric methods for estimating censored regression models have been developed. This paper presents the results of using two such methods to estimate a model of employment duration. The paper reports the operating characteristics of the semiparametric estimators and compares the semiparametric estimates with those obtained from a standard parametric model.     

A SEMIPARAMETRIC REGRESSION MODEL WITH MISSING COVARIATES IN CONTINUOUS-TIME CAPTURE-RECAPTURE STUDIES

Covariate data were missing when a semiparametric regression model was used to study bird abundance in the Mai Po Sanctuary, Hong Kong. This paper proposes an EM‐type algorithm to estimate the regression parameters for that study. Analytical calculation of the expectation in the EM method is difficult, or even impossible, especially when missing covariates are continuous. A Monte Carlo method is used in the EM algorithm to ease the calculation complexity. Asymptotic variances of the parameter estimates are also derived. Properties of the proposed estimators are assessed through numerical simulations and a real example.     

Semiparametric estimation for the dispersion parameter in the analysis of over- or underdispersed count data

This paper investigates several semiparametric estimators of the dispersion parameter in the analysis of over- or underdispersed count data when there is no likelihood available. In the context of estimating the dispersion parameter, we consider the double-extended quasi-likelihood (DEQL), the pseudo-likelihood and the optimal quadratic estimating (OQE) equations method and compare them with the maximum likelihood method, the method of moments and the extended quasi-likelihood through simulation study. The simulation study shows that the estimator based on the DEQL has superior bias and efficiency property for moderate and large sample size, and for small sample size the estimator based on the OQE equations outperforms the other estimators. Three real-life data sets arising in biostatistical practices are analyzed, and the findings from these analyses are quite similar to what are found from the simulation study.     

Testing Semiparametric Hypotheses in Locally Stationary Processes

In this paper, we investigate the problem of testing semiparametric hypotheses in locally stationary processes. The proposed method is based on an empirical version of the L₂‐distance between the true time varying spectral density and its best approximation under the null hypothesis. As this approach only requires estimation of integrals of the time varying spectral density and its square, we do not have to choose a smoothing bandwidth for the local estimation of the spectral density – in contrast to most other procedures discussed in the literature. Asymptotic normality of the test statistic is derived both under the null hypothesis and the alternative. We also propose a bootstrap procedure to obtain critical values in the case of small sample sizes. Additionally, we investigate the finite sample properties of the new method and compare it with the currently available procedures by means of a simulation study. Finally, we illustrate the performance of the new test in two data examples, one regarding log returns of the S&P 500 and the other a well‐known series of weekly egg prices.