Logistic Quantile Regression for Bounded Outcomes Using a Family of Heavy-Tailed Distributions

Abstract

Mean regression model could be inadequate if the probability distribution of the observed responses is not symmetric. Under such situation, the quantile regression turns to be a more robust alternative for accommodating outliers and misspecification of the error distribution, since it characterizes the entire conditional distribution of the outcome variable. This paper proposes a robust logistic quantile regression model by using a logit link function along the EM-based algorithm for maximum likelihood estimation of the p th quantile regression parameters in Galarza (Stat 6, 1, 2017). The aforementioned quantile regression (QR) model is built on a generalized class of skewed distributions which consists of skewed versions of normal, Student’s t, Laplace, contaminated normal, slash, among other heavy-tailed distributions. We evaluate the performance of our proposal to accommodate bounded responses by investigating a synthetic dataset where we consider a full model including categorical and continuous covariates as well as several of its sub-models. For the full model, we compare our proposal with a non-parametric alternative from the so-called quantreg R package. The algorithm is implemented in the R package lqr, providing full estimation and inference for the parameters, automatic selection of best model, as well as simulation of envelope plots which are useful for assessing the goodness-of-fit.

Appendix

Details of expectations in EM algorithm

The conditional distribution of the latent variable given the observed data $f(u_{i}|y_{i},\boldsymbol {\theta }^{(k)})$ will depend on the functional form of h(u_i|ν). Table 3 shows the conditional pdf of U given Y for specific choices of h(u_i|ν).

Table 3 Conditional distribution of U given Y for specific SKD distributions

In Table 3, $z_{i}= (y_{i}-\mathbf {x}_{i}^{\top }{\boldsymbol {\beta }}_{\!p})/\sigma$ and $\mathcal {F}(x|\alpha ,\lambda )$ represents the cdf of a Gamma (α,λ) distribution. Moreover, expressions for a and b are given by $a = \nu \phi \left (y_{i}|\mathbf {x}_{i}^{\top }{\boldsymbol {\beta }}_{\!p},\frac {\gamma ^{-1}\sigma ^{2}}{4{\xi _{i}^{2}}}\right )$ and $b = (1-\nu )\phi \left (y_{i}|\mathbf {x}_{i}^{\top }{\boldsymbol {\beta }}_{\!p},\frac {\sigma ^{2}}{4{\xi _{i}^{2}}}\right ).$ The notation TG(α,λ,t) represents a random variable with Gamma(α,λ) distribution truncated to the right at the value t. Finally, GIG(ν,a,b) denotes the Generalized Inverse Gaussian (GIG) distribution (see Barndorff-Nielsen and Shephard (2001) for more details).

Figure 10

Histograms of the residuals and fitted SKD densities for the median quantile regression model in Section 4.1. Plots provided by call the R Log.lqr function

Figure 11

An example of outputs from R lqr package for median quantile regression model in Section 4.1