Abstract
Missing data often appear as a practical problem while applying classical models in the statistical analysis. In this paper, we consider a semiparametric regression model in the presence of missing covariates for nonparametric components under a Bayesian framework. As it is known that Gaussian processes are a popular tool in nonparametric regression because of their flexibility and the fact that much of the ensuing computation is parametric Gaussian computation. However, in the absence of covariates, the most frequently used covariance functions of a Gaussian process will not be well defined. We propose an imputation method to solve this issue and perform our analysis using Bayesian inference, where we specify the objective priors on the parameters of Gaussian process models. Several simulations are conducted to illustrate effectiveness of our proposed method and further, our method is exemplified via two real datasets, one through Langmuir equation, commonly used in pharmacokinetic models, and another through Auto-mpg data taken from the StatLib library.
Highlights
In nonparametric regression, the objective is to find relationships between response and covariates without assuming the parametric form of a regression function
We present a detailed result in Table S of Supplementary S.3 (Bishoyi et al, 2019) to assess the performance of our proposed methods under misspecification of covariance functions for the Gaussian process (GP) prior in Model (4.1) based on missing values of xi (MSEx), predicted mean squared error (PMSE) and deviance information criterion (DIC) values
We have considered the problem of imputation of missing covariates for the nonparametric part in a semiparametric regression under the Bayesian framework
Summary
The objective is to find relationships between response and covariates without assuming the parametric form of a regression function. Faes et al (2011) developed a nonparametric model based on spline basis functions, where covariates are missing They carried out inference using variational Bayes approximations (cf., Beal (2003)) and showed that in the case of missing covariates, variational Bayes approximations produce multimodality in the posterior distributions where the one-to-one mapping does not exist for the unknown function. There appear several studies focusing on GP models with inputs subject to some measurement uncertainty (Girard and Murray-Smith (2003), Quinonero-Candela and Roweis (2003) and Damianou and Lawrence (2015)) They often developed a two-stage procedure for estimating such GP models either using variational Bayesian methods or expectation maximization procedures, wherein the first stage, they estimated the model parameters only for complete cases and in the second step, they alternately updated model parameters and adjusted estimates of missing input points.
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