Probability Density Function Adjustment for Estimating Quantile Regression Coefficients

This study aims to improve the estimation of quantile regression coefficients by adjusting probability density functions using a selected τ-function that exhibits symmetric properties. The research focuses on five quantile levels Q(20)^th, Q(25)^th, Q(50)^th, Q(75)^th, and Q(80)^th and compares the proposed method with conventional multiple regression through simulation experiments under varying sample sizes and distributional conditions. Performance is evaluated using the mean absolute error (MAE) as the primary metric. The findings indicate that for small sample sizes (n=8, n=15), both multiple and quantile regression methods perform well, especially at lower quantiles (Q(20)^th to Q(50)^th). However, as sample sizes increase (n=50, n=100), quantile regression at higher quantiles (Q(50)^th, Q(75)^th, Q(80)^th) demonstrates superior estimation accuracy. In relation to kurtosis and skewness, the Q(50)^th and Q(80)^th quantiles are sensitive to distributional changes, effectively capturing transitions from high to normal kurtosis and central shifts in skewed distributions. The novelty of this research lies in the integration of the τ-function into the quantile regression framework, enhancing robustness and accuracy in coefficient estimation under non-normal conditions. This approach contributes to methodological advancements in regression analysis, particularly in applications involving non-standard data distributions.