Search

In a generalized linear model, the mean of the response variable is a possibly non-linear function of a linear combination of explanatory variables. When the nonlinear function is unknown and is estimated nonparametrically from the data, these models are known as single index models. Using the relation of generalized linear models with the exponential family model, this paper shows how to use a modified version of the empirical cumulant generating function to estimate the linear function of the explanatory variables with no need of smoothing techniques. The resulting estimator is consistent and normally distributed. Extensive simulations, partially reported here, show that the method works in practice. The method can also be seen as complementary to existing fully nonparametric methods. In fact, it can provide an initial value that can be used to fine tune a nonparametric estimator of the link function in the first step of the estimation. [ABSTRACT FROM AUTHOR]

This paper presents a comparative study of different estimations of the partially linear models based on the smoothing spline technique. Performance of this technique greatly depends on the selection of smoothing parameters. Many methods of selecting smoothing parameters such as an improved version of Akaike information criterion (AICc), generalized cross-validation (GCV), cross-validation (CV), Mallows' Cp criterion, risk estimation using classical pilots (REC) and local risk estimation (LRS) are developed in literature. The smoothing parameter selection has been discussed in respect to a smoothing spline implementation in predicting the partially linear model (PLM). To this end, a simulation study has been conducted to evaluate and compare the performance of six selection methods. In this connection, 1000 replications have been performed in simulation for sample sets with different sizes. The AICc method is recommended since it is stable and works well in all simulations. It performs better than other methods especially when the sample sizes are not large. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is two-fold. First, for the estimation or inference about the parameters of interest in semiparametric models, the commonly used plug-in estimation for infinite-dimensional nuisance parameter creates non-negligible bias, and the least favorable curve or under-smoothing is popularly employed for bias reduction in the literature. To avoid such strong structure assumptions on the models and inconvenience of estimation implementation, for the diverging number of parameters in a varying coefficient partially linear model, we adopt a bias-corrected empirical likelihood (BCEL) in this paper. This method results in the distribution of the empirical likelihood ratio to be asymptotically tractable. It can then be directly applied to construct confidence region for the parameters of interest. Second, different from all existing methods that impose strong conditions to ensure consistency of estimation when diverging the number of the parameters goes to infinity as the sample size goes to infinity, we provide techniques to show that, other than the usual regularity conditions, the consistency holds under moment conditions alone on the covariates and error with a diverging rate being even faster than those in the literature. A simulation study is carried out to assess the performance of the proposed method and to compare it with the profile least squares method. A real dataset is analyzed for illustration. [Copyright &y& Elsevier]

Drawing on linear model theory, we rigorously extend the notion of degrees of freedom to richly‐parameterised models, including linear hierarchical and random‐effect models, some smoothers and spatial models, and combinations of these. The number of degrees of freedom is often much smaller than the number of parameters. Our notion of degrees of freedom is compatible with similar ideas long associated with smoothers, but is applicable to new classes of models and can be interpreted using the projection theory of linear models. We use an example to illustrate the two applications of setting prior distributions for variances and fixing model complexity by fixing degrees of freedom. [ABSTRACT FROM PUBLISHER]

Suppose the expected value of a response variable Y may be written h(Xβ + γ(T)) where X and T are covariates, each of which may be vector-valued. β is an unknown parameter vector, γ is an unknown smooth function, and h is a known function In this article, we outline a method for estimating the parameter β, γ of this type of semiparametric model using a quasi-likelihood function Algorithms for computing the estimates are given and the asymptotic distribution theory for the estimators is developed The generalization of this approach to the case in which Y is a multivariate response is also considered The methodology is illustrated on two data sets and the results of a small Monte Carlo study are presented [ABSTRACT FROM AUTHOR]