# Bayesian Estimation of Simultaneous Regression Quantiles Using Hamiltonian Monte Carlo
## 1 Introduction  
Quantile regression, which is garnering increased attention from both theoretical and empirical perspectives [1; 2; 3; 4; 5], serves as a statistical cornerstone for modeling the intricate relationship between a set of predictor variables and a response variable [6; 7]. This method excels in estimating the conditional quantiles of the response variable, going beyond conventional mean estimation. By shedding light on the varying impacts of predictor variables across different segments of a distribution, quantile regression offers a nuanced understanding of complex relationships in data analysis. As researchers delve into unraveling the intricacies of variable interplay, the application of quantile regression emerges as an invaluable approach for capturing and interpreting diverse patterns. For instance, we can enumerate examples in various fields such as the economy [8; 9], ecology [10], epidemiology [11], survival analysis [12], and environmental economics [13; 14]. Furthermore, quantile regression is a form of statistical and econometric regression analysis that adds a particular method to estimating families of conditional quantile functions and supplements the exclusive focus of least squares techniques based on the estimation of conditional mean functions [15]. Originating from the seminal work of [16], quantile regression introduces a versatile method for estimating conditional quantiles. The authors demonstrated that any linear quantile of order \(\tau\) could be derived from minimizing a check function, \(\rho_{\tau}(u)=u(\tau-\mathds{1}_{u<0})\), where \(\mathds{1}\) represents the indicator function. In exclusively linear regression models, [PERSON] et al. [16] suggest a straightforward method to estimating conditional quantiles, consider the regression linear normal model, as the solution of the minimization problem that follows:\[\min_{\beta}\sum_{t}\rho_{\tau}(y_{t}-x^{\prime}_{t}\beta), \tag{1}\]  
where \(x_{t},t=1,\ldots,T\) denote a sequence of (row), \(x^{\prime}\) is the transposed of \(x\), \(K\)-vector of a known design matrix, and \(y_{t},t=1,\ldots,T\) are a random sample on the regression process \(u=y-x^{\prime}_{t}\beta\) having distribution function \(F\). Then, the \(\tau\)-th sample quantile, \(0<\tau<1\), may be defined as any solution of the minimization problem (1). In the framework of statistical research, Bayesian approaches have emerged as effective tools for investigating quantile regression [17; 18; 19; 20] after the notable developments presented in [21]. The authors utilize a likelihood function grounded in the asymmetric distribution of Laplace to advance the understanding of quantile regression. While both frequentist and Bayesian perspectives prove suitable for examining individual quantiles [22], the complexities of real-world applications often demand more nuanced exploration. To intricately capture the relationship between the response variable and explanatory variables, it becomes imperative to estimate regression quantiles at various orders. Although it is conceivable to perform separate estimations for these quantiles, this approach proves unsatisfactory; particularly in scenarios involving multiple explanatory variables, it provides estimators that do not respect the property of the non-crossing of the regression quantiles. To circumvent this limitation, a more favorable approach involves the simultaneous estimation of multiple regression quantiles [23; 24; 25; 26]. This ensures a coherent and consistent estimation procedure, particularly in situations characterized by the presence of multiple explanatory variables. In pursuit of refining the Bayesian quantile estimation paradigm, this article directs its attention to the groundwork laid by [PERSON] in the seminal work presented in [27]. The author's utilization of the Metropolis-Hastings within Gibbs algorithm proved instrumental in sampling unknown parameters from their full conditional distribution, offering a new methodological framework.