A Convex Combination Method for Quantile Regression with Interval Data

Quantile regression is a regression method that modelling a relationship between quantile of variable response and one or more variable predictors. Quantile regression has advantages that linear regression does not have; it is robust against outliers and can model heteroscedasticity data. The parameters of quantile regression can be estimated using the Bayesian method. The Bayesian method is a data analysis tool derived based on the Bayesian inference principle. Bayesian inference is the process of studying data analysis inductively with the Bayes theorem. To estimate regression parameters with Bayesian inference, it is necessary to find the posterior distribution of the regression parameters where the posterior distribution is proportional to the product of the prior distribution and its likelihood function. Since the calculation of the posterior distribution analytically is difficult to do if more parameters are estimated, the Markov Chain Monte Carlo (MCMC) method is proposed. The use of the Bayesian method in quantile regression has advantages, namely the use of MCMC has the advantages of obtaining sample parameter values from an unknown posterior distribution, using computationally efficient, and easy to implement. Yu and Moyeed (2001) introduced Bayesian quantile regression using the likelihood function of errors with an Asymmetric Laplace Distribution (ALD) and found that minimizing parameter estimates in quantile regression is the same as maximizing the likelihood function of errors with an Asymmetric Laplace Distribution (ALD). The method used to estimate quantile regression parameters is Gibbs sampling from the ALD, which is a combination of the exponential and normal distributions. To find the parameters of the regression model by sampling the posterior distribution found in this thesis. The results obtained from Gibbs sampling are a sample sequence of estimated parameters. After obtaining the sample sequences, the sample lines are averaged to obtain an estimated regression parameter. The case study in this thesis discusses the effect of risk factors from motor vehicle insurance customers on the size of claims submitted by customers.

Bayesian analysis of a Tobit quantile regression model

Variational Bayesian inference for interval regression with an asymmetric Laplace distribution

This study aims to construct the model for the length of hospital stay for patients with COVID-19 using quantile regression and Bayesian quantile approaches. The quantile regression models the relationship at any point of the conditional distribution of the dependent variable on several independent variables. The Bayesian quantile regression combines the concept of quantile analysis into the Bayesian approach. In the Bayesian approach, the Asymmetric Laplace Distribution (ALD) distribution is used to form the likelihood function as the basis for formulating the posterior distribution. All 688 patients with COVID-19 treated in M. Djamil Hospital and Universitas Andalas Hospital in Padang City between March-July 2020 were used in this study. This study found that the Bayesian quantile regression method results in a smaller 95% confidence interval and higher value than the quantile regression method. It is concluded that the Bayesian quantile regression method tends to yield a better model than the quantile method. Based on the Bayesian quantile regression method, it investigates that the length of hospital stay for patients with COVID-19 in West Sumatra is significantly influenced by Age, Diagnoses status, and Discharge status.

This chapter looks at several methods for quantile and expectile regression that fall within the VGLM/VGAM framework. The following main categories are described: LMS-type quantile regression methods, the classical method (based on a loss or check function) and its connection with the asymmetric Laplace distributions (ALD), and expectile regression. A parallelism assumption for the ALD and ER allows for one solution to the quantile-crossing problem. The location parameter of the ALD can be modelled using link functions, therefore responses such as counts can be potentially handled. A second solution to the quantile-crossing problem is called the ‘onion’ method, which is likened to estimating successive layers of an onion. The VGAM package is used to illustrate the models.

This study used a quantile regression model and a non-homogeneous regression model to calibrate probabilistic forecasts of wind speed. These techniques were applied to the forecasts of wind speed over Pyeongchang area using 51-member European Centre for Medium-Range Weather Forecast (ECMWF). Reliability analysis was carried out by using rank histogram to identify the statistical consistency of ensemble forecasts and corresponding observations. The performances were evaluated by rank histogram, mean absolute error, root mean square error and continuous ranked probability score. The results showed that the forecasts of quantile regression and non-homogeneous regression models performed better than the raw ensemble forecasts. However, the differences of prediction skills between quantile regression and nonhomogeneous regression models were insignificant.

In this paper, portfolio selection problem is studied under Asymmetric Laplace Distribution(ALD) framework. Asymmetric Laplace distribution is able to capture tail-heaviness, skewness, andleptokurtosis observed in empirical financial data that cannot be explained by traditional Gaussiandistribution. Under Asymmetric Laplace distribution framework, portfolio selection methods basedon di erent risk measures are discussed. Moreover, we derived the Expectation-Maximization (EM)procedure for parameter estimation of Asymmetric Laplace distribution. Performance of the proposedmethod is illustrated via extensive simulation studies. Two real data examples are complemented toconfirm that the Asymmetric Laplace distribution based portfolio selection models are effcient.

Introduction: Quantile regression is a valuable alternative for survival data analysis, enabling flexible evaluations of covariate effects on survival outcomes with intuitive interpretations. It offers practical computation and reliability. However, challenges arise when applying quantile regression to censored data, particularly for upper quantiles. The minimum distance approach, utilizing dual-kernel estimation and the inverse cumulative distribution function, shows promise in addressing these challenges, especially with higher-dimensional covariates. Methods: This study contrasts two methods within the realm of quantile linear regression for survival analysis: check-based modeling and the minimum distance approach. Effectiveness is assessed across various scenarios through comprehensive simulation. Results: The simulation results showed that using the quantile regression model with the minimum distance approach reduces the percentage of root mean square error in parameter estimation compared to the quantile regression models based on the check loss function. Additionally, a larger sample size and reduced censoring percentage led to decreased root mean square error in parameter estimation. Conclusion: The research highlights the benefits of using the minimum distance approach for quantile regression. It reduces errors, improves model predictions, captures patterns, and optimizes parameters even with complete data. However, this approach has limitations. The accuracy of estimated quantiles can be influenced by the choice of distance metric and weighting scheme. The assumption of independence between censoring mechanism and survival time may not hold in real-world scenarios. Additionally, dealing with large datasets can be computationally complex.

This study applies quantile regression to model the Body Mass Index (BMI) of 152 students from Delta State Polytechnic in Delta State, Nigeria. The BMI serves as the response variable, while skin fold (SK), feeding habit (FH), feeding frequency (FF), and frequency of exercising (FE) are considered as explanatory variables. A comparison is made between the results obtained from the classical linear regression model and quantile regression models at different quantiles (p = 0.1, 0.25, 0.5, 0.75, and 0.9) to examine the impact of the variables on the students' BMI and assess if the relationships differ across quantiles. The findings reveal significant differences between the classical linear regression and quantile regression models, emphasizing the importance of quantile regression in capturing the nuances of the relationship between variables at different points of the BMI distribution. The study highlights the limitations of the classical linear regression model in providing a comprehensive understanding of the data and underscores the value of quantile regression in enhancing our insights into the relationship between BMI and the considered factors. This research contributes to the broader literature on quantile regression and its applications in exploring BMI and related health issues among students

This paper studies the connections among quantile regression, the asymmetric Laplace distribution, maximum likelihood and maximum entropy. We show that the maximum likelihood problem is equivalent to the solution of a maximum entropy problem where we impose moment constraints given by the joint consideration of the mean and median. Using the resulting score functions we propose an estimator based on the joint estimating equations. This approach delivers estimates for the slope parameters together with the associated “most probable” quantile. Similarly, this method can be seen as a penalized quantile regression estimator, where the penalty is given by deviations from the median regression. We derive the asymptotic properties of this estimator by showing consistency and asymptotic normality under certain regularity conditions. Finally, we illustrate the use of the estimator with a simple application to the U.S. wage data to evaluate the effect of training on wages.

Objective: The purpose of this study was to explore the full distribution of children’s lead poisoning and identify “high risk” locations or areas in the neighborhood of the inner city of Syracuse (NY, USA), using quantile regression models. Methods: Global quantile regression (QR) and geographically weighted quantile regression (GWQR) were applied to model the relationships between children’s lead poisoning and three environmental factors at different quantiles (25th, 50th, 75th, and 90th). The response variable was the incident rate of children’s blood lead level ≥ 5 µg/dL in each census block, and the three predictor variables included building year, town taxable values, and soil lead concentration. Results: At each quantile, the regression coefficients of both global QR and GWQR models were (1) negative for both building year and town taxable values, indicating that the incident rate of children lead poisoning reduced with newer buildings and/or higher taxable values of the houses; and (2) positive for the soil lead concentration, implying that higher soil lead concentration around the house may cause higher risks of children’s lead poisoning. Further, these negative or positive relationships between children’s lead poisoning and three environmental factors became stronger for larger quantiles (i.e., higher risks). Conclusions: The GWQR models enabled us to explore the full distribution of children’s lead poisoning and identify “high risk” locations or areas in the neighborhood of the inner city of Syracuse, which would provide useful information to assist the government agencies to make better decisions on where and what the lead hazard treatment should focus on.

We used a quantile regression under asymmetric Laplace distribution for predicting stock returns. Specifically, we apply this method to the classical capital asset pricing model (CAPM) to estimate the beta coefficient which measure risk in the portfolios management analysis at given levels of quantile. Quantile regression estimation is equivalent to the parametric case where the error term is asymmetrically Laplace distributed. Finally, we use the method to measures the volatility of a portfolio relative to the market.

This paper proposes a new approach to analyze stock return asymmetry and quantiles. We also present a new scale mixture of uniform (SMU) representation for the asymmetric Laplace distribution (ALD). The use of the SMU for a probability distribution is a data augmentation technique that simplifies the Gibbs sampler of the Bayesian Markov chain Monte Carlo algorithms. We consider a stochastic volatility (SV) model with an ALD error distribution. With the SMU representation, the full conditional distribution for some parameters is shown to have closed form. It is also known that the ALD can be used to obtain the coefficients of quantile regression models. This paper also considers a quantile SV model by fixing the skew parameter of the ALD at specific quantile level. Simulation study shows that the proposed methodology works well in both SV and quantile SV models using Bayesian approach. In the empirical study, we analyze index returns of the stock markets in Australia, Japan, Hong Kong, Thailand, and the UK and study the effect of S&P 500 on these returns. The results show the significant return asymmetry in some markets and the influence by S&P 500 in all markets at all quantile levels. Copyright © 2014 John Wiley & Sons, Ltd.

Statiscal Downscaling (SD) was a method to model the relationship between local scale data as response variable and global scale data as predictor variable. The response variable was the rainfall and the predictor variable was the global circulation model (GCM) ouput. In general, GCM output had a large dimension and multicollinearity so the principal component analysis was used to solve this problem. This study modeled the rainfall with the selected principal component using quantile regression. This functional relationship could be parametric or nonparametric relationship. In this case, nonparametric functional relationship used spline to accommodate the extreme value with the quantile regression. The result showed that the quantile spline model was better than the quantile polynomial regression model especially in predicting the extreme values in the 90 th and 95 th quantile with correlation values of 0.95 and 0.93.

The adoption of quantile regression has become increasingly prevalent for addressing heteroscedastic data. In this work, we consider the issue of measuring the uncertainty for the shrinkage model selection procedure in quantile regression models. Two measures of uncertainty, i.e. the nested model confidence set (NMCS) and a LogP measure for quantile regression, are established by using the wild bootstrap method to overcome the so-called bootstrap validity problem, noting that the parametric bootstrap-based model confidence set approach in the literature fails to work under the quantile regression setting. The asymptotic properties of the coverage probability of the NMCS, as well as the consistency of the LogP measure, are also developed for quantile regression models. Simulations and a real data example are devoted to illustrating the finite sample performance and potential applications.