Asian Journal of Probability and Statistics

Quantile regression provides a robust alternative to mean regression, but its performance may deteriorate under heavy-tailed errors and outliers. The Cauchit link, based on the Cauchy distribution, offers additional resistance to extreme values. This study compares the performance stability of two Cauchit quantile regression models. The first proposed model is expressed as: \(Q_y(\tau)=\frac{1}{\pi}\left\{\tan ^{-1}\left(\beta_0^{(\tau)}+\beta_1^{(\tau)} x_i\right)\right\}+\frac{1}{2} . .\) The second Cauchit quantile regression model was proposed because the first model was unable to handle outliers adequately or transform the estimates into a ratio bounded between -1 and 1. The second model was derived by taking the inverse of \(\pi\) on the right-hand side of the first model, yielding: \(Q_y(\tau)=\pi\left\{\tan ^{-1}\left(\beta_0^{(\tau)}+\beta_1^{(\tau)} x_i\right)\right\}+\frac{1}{2}, .\) Using Monte Carlo simulation with a sample size of 3,000, both models were evaluated using skewness, kurtosis, residual mean-square error, standard deviation, the Akaike information criterion, pseudo-R, and comparisons of the median and mean. The first proposed model performed poorly, whereas the second model showed greater stability and robustness to outliers. These findings provide practical guidance for selecting a Cauchit quantile regression specification for data characterised by non-normal errors and outliers.