Abstract
Bayesian inference on quantile regression (QR) model with mixed discrete and non-ignorable missing covariates is conducted by reformulating QR model as a hierarchical structure model. A probit regression model is adopted to specify missing covariate mechanism. A hybrid algorithm combining the Gibbs sampler and the Metropolis-Hastings algorithm is developed to simultaneously produce Bayesian estimates of unknown parameters and latent variables as well as their corresponding standard errors. Bayesian variable selection method is proposed to recognize significant covariates. A Bayesian local influence procedure is presented to assess the effect of minor perturbations to the data, priors and sampling distributions on posterior quantities of interest. Several simulation studies and an example are presented to illustrate the proposed methodologies.
Highlights
Quantile regression (QR) (Hendricks and Koenker, 1992; Hallock and Koenker, 2001; Chernozhukov, 2005; Gaglianone et al, 2011; Cade and Noon, 2003) has become an important tool for quantifying the conditional quantile relationship between a response variable and some covariates, due to its merits, such as, few assumptions on the distribution of random errors except for requiring that random errors have a zero conditional quantile, and more robustness to outliers and heavy-tailed data than ordinary least squares regression
The aforementioned Gibbs sampler together with Type III prior for parameters β, α and γ is adopted to estimate parameters, and calculate G(ω0) and Bayesian local influence measures corresponding to Bayes factor (i.e., FICB,ej ), Kullback-Leibler divergence (i.e., SICD,ej ) and posterior mean distance (i.e., SICMh,ej ) of d(θ) = θ based on S0 = 7000 observations collected after 3000 burn-in iterations
This paper considers Bayesian inference on a quantile regression model in the presence of nonignorable missing covariates
Summary
Quantile regression (QR) (Hendricks and Koenker, 1992; Hallock and Koenker, 2001; Chernozhukov, 2005; Gaglianone et al, 2011; Cade and Noon, 2003) has become an important tool for quantifying the conditional quantile relationship between a response variable and some covariates, due to its merits, such as, few assumptions on the distribution of random errors except for requiring that random errors have a zero conditional quantile, and more robustness to outliers and heavy-tailed data than ordinary least squares regression. The main contribution of this paper includes that (i) a complicated QR model is considered by incorporating discrete and continuous and nonignorable missing covariates; (ii) a sequence of one-dimensional exponential family conditional distributions is adopted to specify the distribution of missing covariates because of discrete and continuous covariates involved; (iii) a sequence of one-dimensional probit regression models is employed to formulate nonignorable missingness covariates mechanism, which is easier to draw observations required for statistical inference from their corresponding conditional distributions than the widely used logistic regression models; (iv) a Bayesian adaptive LASSO procedure is developed to select covariates/explanatory variables in QR model and missingness covariates mechanism model; (v) Bayesian local influence analysis of Zhu et al (2011) is extended to check the plausibility of missingness covariates mechanism in the considered QR model. Technical details are presented in the Supplementary Materials (Wang and Tang, 2019)
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