Semiparametric Quantile Regression Model Research Articles

Smoothed quantile residual life regression analysis with application to the Korea HIV/AIDS cohort study

BackgroundThe residual life of a patient with human immunodeficiency virus (HIV) is of major interest to patients and their physicians. While existing analyses of HIV patient survival focus mostly on data collected at baseline, residual life analysis allows for dynamic analysis based on additional data collected over a period of time. As survival times typically exhibit a right-skewed distribution, the median provides a more useful summary of the underlying distribution than the mean. In this paper, we propose an efficient inference procedure that fits a semiparametric quantile regression model assessing the effect of longitudinal biomarkers on the residual life of HIV patients until the development of dyslipidemia, a disease becoming more prevalent among those with HIV.MethodsFor estimation of model parameters, we propose an induced smoothing method that smooths nonsmooth estimating functions based on check functions. For variance estimation, we propose an efficient resampling-based estimator. The proposed estimators are theoretically justified. Simulation studies are used to evaluate their finite sample performances, including their prediction accuracy. We analyze the Korea HIV/AIDS cohort study data to examine the effects of CD4 (cluster of differentiation 4) cell count on the residual life of HIV patients to the onset of dyslipidemia.ResultsThe proposed estimator is shown to be consistent and normally distributed asymptotically. Under various simulation settings, our estimates are approximately unbiased. Their variances estimates are close to the empirical variances and their computational efficiency is superior to that of the nonsmooth counterparts. Two measures of prediction performance indicate that our method adequately reflects the dynamic character of longitudinal biomarkers and residual life. The analysis of the Korea HIV/AIDS cohort study data shows that CD4 cell count is positively associated with residual life to the onset of dyslipidemia but the effect is not statistically significant.ConclusionsOur method enables direct prediction of residual lifetimes with a dynamic feature that accommodates data accumulated at different times. Our estimator significantly improves computational efficiency in variance estimation compared to the existing nonsmooth estimator. Analysis of the HIV/AIDS cohort study data reveals dynamic effects of CD4 cell count on the residual life to the onset of dyslipidemia.

Instrumental variable quantile regression under random right censoring

Summary This paper studies a semiparametric quantile regression model with endogenous variables and random right censoring. The endogeneity issue is solved using instrumental variables. It is assumed that the structural quantile of the logarithm of the outcome variable is linear in the covariates and censoring is independent. The regressors and instruments can be either continuous or discrete. The specification generates a continuum of equations of which the quantile regression coefficients are a solution. Identification is obtained when this system of equations has a unique solution. Our estimation procedure solves an empirical analogue of the system of equations. We derive conditions under which the estimator is asymptotically normal and prove the validity of a bootstrap procedure for inference. The finite sample performance of the approach is evaluated through numerical simulations. An application to the national Job Training Partnership Act study illustrates the method.

A Bayesian Semi-parametric Quantile Regression Approach for Joint Modeling of Longitudinal Ordinal and Continuous Responses

Quantile regression (QR) models are one of the methods for longitudinal data analysis. When responses seemto be skew and asymmetric due to outliers and heavy-tails, QR models may work suitably. This paper developes the semi-parametric quantile regression model for analyzing longitudinal continuous and ordinal mixed responses. The latent variable model and some threshold parameters are used to perform the quantile regression model’s ordinal part. The error of the latent variable model has Asymmetric Laplace (AL) distribution. The error term’s distribution is assumed to be AL distribution to model the continuous responses. The correlations of longitudinal responses belong to the same individual and those of mixed continuous and ordinal responses are considered using a random-effects approach. The regression spline is used to approximate the non-parametric part of the model. The parameter estimation procedure is performed under aBayesian paradigm using the Gibbs sampling method. A simulation study is performed to demonstrate the proposed model’s performance where the relative biases, standard errors, and root of MSEs of estimated parameters are decreased in the semi- parametric QR joint model when the number of subjects is increased. In our application, it was found that the mother’s age and her child’s age have significant effects on reading ability, and antisocial behavior depends on the child’s gender.

Nonparametric and Semiparametric Quantile Regression via a New MM Algorithm

Quantile regression is a popular method with a wide range of scientific applications, but the computation for quantile regression is challenging. Hunter and Lange proposed an MM algorithm for solving optimization problems in parametric quantile regression models. For nonparametric and semiparametric quantile regression, their algorithm can be applied to estimate unknown quantile functions in a pointwise manner. However, the resulting estimates may suffer from drawbacks like nonsmooth with discontinuous points and unstable at extreme quantile levels. To remedy the above issues, we propose a new MM algorithm and show that it yields continuous, smoother, and faster estimated quantile functions. We systematically study the new MM algorithm using the local linear quantile regression model. We prove that the proposed algorithm preserves the monotone descent property in an asymptotic sense. We then extend it to some popular nonparametric and semiparametric quantile regression models. For semiparametric models, we propose new efficient backfitting algorithms based on the new MM algorithm. Compared to traditional backfitting algorithms, the new procedures can significantly reduce computational costs for fully iterative backfitting. The performance of the proposed algorithms is demonstrated via extensive simulation studies and a real data example. Supplementary materials for this article are available online.

Ownership Structure, Corporate Governance, and Performance of Listed Companies—An Empirical Application of a Semi-Parametric Quantile Regression Model

China’s listed companies have different ownership characteristics and market environments from those of other countries and thus exhibit vastly different changes. From the existing corporate life cycle perspective, companies differ in their different development stages, which makes each factor’s effect dynamic. How to adjust the governance mechanism to the requirements of the company’s stage of development is an urgent issue in sustainable corporate governance. To address the above issues, we establish a semi-parametric quantile regression model to analyze the relationship between the ownership structure and corporate performance based on the data of listed companies on the Shanghai Stock Exchange between 2013 and 2021. Moreover, corporate governance measures taken at different stages of the corporate life cycle are discussed to see whether they effectively improve corporate governance. We conclude that there are non-linear effects of ownership structure while dynamic changes in corporate governance mechanisms exist. Companies should be concerned about the non-linear effects of ownership structures while considering the company’s life cycle and choosing appropriate governance measures. The results will help develop a sustainable development strategy to ensure that the company can improve its profitability and mitigate agency problems.

Nonparametric inference on smoothed quantile regression process

This paper studies the global estimation in semiparametric quantile regression models. For estimating unknown functional parameters, an integrated quantile regression loss function with penalization is proposed. The first step is to obtain a vector-valued functional Bahadur representation of the resulting estimators, and then derive the asymptotic distribution of the proposed infinite-dimensional estimators. Furthermore, a resampling approach that generalizes the minimand perturbing technique is adopted to construct confidence intervals and to conduct hypothesis testing. Extensive simulation studies demonstrate the effectiveness of the proposed method, and applications to the real estate dataset and world happiness report data are provided.

Flexible Bayesian quantile curve fitting with shape restrictions under the Dirichlet process mixture of the generalized asymmetric Laplace distribution

We propose a flexible Bayesian semiparametric quantile regression model based on Dirichlet process mixtures of generalized asymmetric Laplace distributions for fitting curves with shape restrictions. The generalized asymmetric Laplace distribution exhibits more flexible tail behaviour than the frequently used asymmetric Laplace distribution in Bayesian quantile regression. In addition, nonparametric mixing over the shape and scale parameters with the Dirichlet process mixture extends its flexibility and improves the goodness of fit. By assuming the derivatives of the regression functions to be the squares of the Gaussian processes, our approach ensures that the resulting functions have shape restrictions such as monotonicity, convexity and concavity. The introduction of shape restrictions prevents overfitting and helps obtain smoother and more stable estimates of the quantile curves, especially in the tail quantiles for small and moderate sample sizes. Furthermore, the proposed shape‐restricted quantile semiparametric regression model deals with sparse estimation for regression coefficients using the horseshoe+ prior distribution, and it is extended to cases with group‐specific curve estimation and censored data. The usefulness of the proposed models is demonstrated using simulated datasets and real applications.

Does the Impact of Carbon Price Determinants Change with the Different Quantiles of Carbon Prices? Evidence from China ETS Pilots

Since carbon price volatility is critical to the risk management of the CO2 emissions trading market, research has focused on energy prices and macroeconomic drivers which cause changes in carbon prices and make the carbon market more volatile than other markets. However, they have ignored whether the impact of carbon price determinants changes when the carbon price is at different levels. To fill this gap, this paper applies a semiparametric quantile regression model to explore the effects of energy prices and macroeconomic drivers on carbon prices at different quantiles. The model combines the advantages of parameter estimation, nonparametric estimation and quantile regression to describe the nonlinear relationship between carbon price and its fundamentals, which do not need to make any assumptions about the random error. Carbon prices are high–tailed and exhibit higher kurtosis, the traditional models which tend to assume that data are normally distributed can’t perform well. Furthermore, the semiparametric model doesn’t need to assume that the data are normally distributed. Therefore, the semiparametric model can effectively model the data. Some new evidence from China’s emission trading scheme (ETS) pilots shows that energy prices and macroeconomic drivers have different effects on carbon prices at high or low quantiles. First, the negative impact of coal prices on carbon prices was greater at the lower quantile of carbon prices in the Shenzhen ETS pilot. However, the effects of coal prices were positive in the Beijing ETS pilot, which may be attributed to great demand for coal. Second, oil prices had greater negative effects on carbon prices at higher quantiles in Beijing and Hubei ETS pilots. This can be attributed to the fact that businesses use less oil when carbon prices are high. For the Shenzhen ETS pilot, the effects of oil prices were positive. Third, natural gas prices have a stronger effect on carbon prices as quantiles increased in the Beijing and Hubei ETS pilots. Lastly, the effects of macroeconomic drivers on carbon prices at low quantiles were stronger in the Shenzhen ETS pilots and higher at the medium quantiles in Beijing and Hubei ETS pilots. These findings suggest that the impact of determinants on the carbon prices at different levels is not constant. Ignoring this issue will lead to a missed warning about the risks of the carbon market. This study will be of positive significance for China’s emission trading scheme (ETS) pilots, in order to accurately monitor the effects of carbon prices determinants and effectively avoid carbon market risks.

Predicting pollution incidents through semiparametric quantile regression models

In this paper we present a method to forecast pollution episodes using measurements of the pollutant concentration along time. Specifically, we use a backfitting algorithm with local polynomial kernel smoothers to estimate a semiparametric additive quantile regression model. We also propose a statistical hypothesis test to determine critical values, i.e., the values of the concentration that are significant to forecast the pollution episodes. This test is based on a wild bootstrap approach modified to suit asymmetric loss functions, as occurs in quantile regression. The validity of the method was checked with both simulated and real data, the latter from $${\hbox {SO}}_{2}$$ emissions from a coal-fired power station located in Northern Spain.

Bayesian semiparametric quantile regression modeling for estimating earthquake fatality risk

This paper develops a Bayesian semiparametric quantile regression model for count data. The count responses are converted to continuous responses through the “jittered” method and a transform function. A Bayesian semiparametric quantile regression modeling approach is then developed. The error distribution in the quantile regression model is assumed to be a mixture of asymmetric Laplace distributions constructed with Dirichlet process. Historical death tolls of China caused by earthquakes from 1969 to 2006 are used for fitting, and a parametric model is employed for model comparison. The results of model comparison show that the proposed semiparametric quantile regression model outperforms the parametric model. The empirical analysis illustrates that the impact of earthquake magnitude on death tolls is significant. Moreover, the impact of the magnitude is more pronounced on higher percentiles of death tolls.

Fuzzy Quantile Linear Regression Model Adopted with a Semi-Parametric Technique based on Fuzzy Predictors and Fuzzy Responses

As an extension to statistical techniques, fuzzy regression analysis estimates the relationship between fuzzy independent variables and non-fuzzy and/or fuzzy dependent variables. However, facing skewed or non-normal data may lead to vague results if the conventional least square models are used. However, there are many situations in applications of intelligence systems where regression components are fuzzy quantities instead of exact values. The present work is aimed to propose a fuzzy semi-parametric quantile regression model with fuzzy predictors, a fuzzy smooth function, exact coefficients and fuzzy responses. The proposed method is based on a new class of novel kernel-based signed-distance measures in space of fuzzy numbers. Main features of the proposed signed-distance including its robustness were studied. It was shown that the proposed signed-distance measure is more reasonable and effective than a common signed-distance which extensively has been used in fuzzy set theory. This is the reason why we decided to apply the proposed signed-distance to extend the classical quantile regression model. Then, a hybrid algorithm was suggested to evaluate optimal bandwidth, non-fuzzy regression coefficients and fuzzy smooth function at a specific quantile level. The proposed method was then compared with some common fuzzy regression models based on two data sets including a simulation study and an applied example. The numerical results clearly indicated that the proposed fuzzy semi-parametric quantile linear regression model performs better than the other available methods in terms of some common goodness-of-fit criteria in cases where the data set includes skewness and/or some outlier data point. The main advantages of the proposed method are as follows: 1-it is not so sensitive to the outlier data points and/or skewed data, 2- it possesses better fitting effect compared to other existing fuzzy regression models. Thus, the proposed quantile regression model could be successfully applied in many practical studies of fuzzy regression model in real-world applications.

Semiparametric quantile regression with random censoring

This paper considers estimation and inference in semiparametric quantile regression models when the response variable is subject to random censoring. The paper considers both the cases of independent and dependent censoring and proposes three iterative estimators based on inverse probability weighting, where the weights are estimated from the censoring distribution using the Kaplan–Meier, a fully parametric and the conditional Kaplan–Meier estimators. The paper proposes a computationally simple resampling technique that can be used to approximate the finite sample distribution of the parametric estimator. The paper also considers inference for both the parametric and nonparametric components of the quantile regression model. Monte Carlo simulations show that the proposed estimators and test statistics have good finite sample properties. Finally, the paper contains a real data application, which illustrates the usefulness of the proposed methods.

Maternal breastfeeding and children's cognitive development

Do children with lower test scores benefit more from breastfeeding than those with higher scores? In this paper, I examine the distributional effects of maternal breastfeeding on the cognitive test scores of 11,544 children who were born in 2000 and 2001 in the United Kingdom using a semiparametric quantile regression model. I find evidence that maternal breastfeeding has larger positive impacts on children with lower test scores. Effects for children below the 20th percentile are about 2–2.5 times greater than those for children above the 80th percentile. I also find that these distributional effects are larger when the duration of breastfeeding is extended. One policy implication is that a public policy aims at promoting breastfeeding might narrow a disparity in children's cognition.

Urban house price surfaces near a World Heritage Site: Modeling conditional price and spatial heterogeneity

In housing price regression, a large bundle of non-separable structural and location characteristics, potentially affecting prices nonlinearly, constitute the relevant set of predictors. Spatial subcenters and complex spatial association structures may, therefore, exist or, stated differently, horizontal market segmentation might be prevalent. Moreover, it is not unlikely for the housing price generating market mechanisms to vary across different parts of the conditional price distribution. This can ultimately cause disparate price segments to exhibit varying functional relationships through different subsets of characteristics and lead to vertical market segmentation. In order to take nonlinearity, horizontal and vertical market segmentation into account within the scope of housing price regressions, we propose incorporating a semiparametric approach into the quantile regression framework. In our empirical application, we investigate rental data from the German city of Regensburg, which contains an Old Town on the World Heritage List. Focusing on location effects exerted by the World Heritage Site, we illustrate how statements about horizontal and vertical market segmentation can be derived from a semiparametric quantile regression model based on empirical evidence and economic reasoning.

Partially linear additive quantile regression in ultra-high dimension

We consider a flexible semiparametric quantile regression model for analyzing high dimensional heterogeneous data. This model has several appealing features: (1) By considering different conditional quantiles, we may obtain a more complete picture of the conditional distribution of a response variable given high dimensional covariates. (2) The sparsity level is allowed to be different at different quantile levels. (3) The partially linear additive structure accommodates nonlinearity and circumvents the curse of dimensionality. (4) It is naturally robust to heavy-tailed distributions. In this paper, we approximate the nonlinear components using B-spline basis functions. We first study estimation under this model when the nonzero components are known in advance and the number of covariates in the linear part diverges. We then investigate a nonconvex penalized estimator for simultaneous variable selection and estimation. We derive its oracle property for a general class of nonconvex penalty functions in the presence of ultra-high dimensional covariates under relaxed conditions. To tackle the challenges of nonsmooth loss function, nonconvex penalty function and the presence of nonlinear components, we combine a recently developed convex-differencing method with modern empirical process techniques. Monte Carlo simulations and an application to a microarray study demonstrate the effectiveness of the proposed method. We also discuss how the method for a single quantile of interest can be extended to simultaneous variable selection and estimation at multiple quantiles.

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.

Semiparametric Hierarchical Composite Quantile Regression

In biological, medical, and social sciences, multilevel structures are very common. Hierarchical models that take the dependencies among subjects within the same level are necessary. In this article, we introduce a semiparametric hierarchical composite quantile regression model for hierarchical data. This model (i) keeps the easy interpretability of the simple parametric model; (ii) retains some of the flexibility of the complex non parametric model; (iii) relaxes the assumptions that the noise variances and higher-order moments exist and are finite; and (iv) takes the dependencies among subjects within the same hierarchy into consideration. We establish the asymptotic properties of the proposed estimators. Our simulation results show that the proposed method is more efficient than the least-squares-based method for many non normally distributed errors. We illustrate our methodology with a real biometric data set.

Bayesian Quantile Regression for Censored Data

In this paper we propose a semiparametric quantile regression model for censored survival data. Quantile regression permits covariates to affect survival differently at different stages in the follow-up period, thus providing a comprehensive study of the survival distribution. We take a semiparametric approach, representing the quantile process as a linear combination of basis functions. The basis functions are chosen so that the prior for the quantile process is centered on a simple location-scale model, but flexible enough to accommodate a wide range of quantile processes. We show in a simulation study that this approach is competitive with existing methods. The method is illustrated using data from a drug treatment study, where we find that the Bayesian model often gives smaller measures of uncertainty than its competitors, and thus identifies more significant effects.

A Bayesian Semiparametric Quantile Regression Model for Longitudinal Data with Application to Insurance Company Costs

This article examines the average cost function for property and casualty insurers. Cost function describes the relationship between a firm’s minimum production cost and outputs. The comparison of cost functions could shed light on the relative cost efficiency of individual firms, which is of interest to many market participants and has been given extensive attention in the insurance industry. To identify and compare the cost function of insurers, common practice is to assume an identical functional form between cost and outputs and to rank insurers according to the center (usually mean) of the cost distribution. Such assumption could be misleading because insurers tend to adopt different technologies that are reflected by the cost function in their production process. We find that the cost distribution is skewed with a heavy tail. Therefore, the center-based comparison could lead to biased inference regarding an insurer’s efficiency in operation. To address these issues, we propose Bayesian semiparametric quantile regression approach to model the longitudinal data on production cost of insurance companies. Particularly we formulate the quantile regression using the asymmetric Laplace distribution; the effects of various firm characteristics on the quantiles of cost are specified as a linear function; the firm-specific relation between cost and multiple outputs is captured by a single-index function based on a spline basis where the coefficients are assumed to have a Dirichlet process prior. The single-index formulation with splines renders flexibility in modeling the nonlinear cost-output relation and the use of Dirichlet process leads to a natural clustering of insurers with similar cost efficiency. The method is applied to data on US property casualty insurers from the National Association of Insurance Commissioners (NAIC). The analysis of average cost at different quantiles indicates that better insights on efficiency are gained by comparing the whole cost distribution. A comparison of the model results with an external financial strength ratings for property casualty insurers provides interesting insights on which part of the cost distribution are perhaps weighted more by the rating agency.

Semiparametric modeling and estimation of heteroscedasticity in regression analysis of cross-sectional data

We consider the problem of modeling heteroscedasticity in semiparametric regression analysis of cross-sectional data. Existing work in this setting is rather limited and mostly adopts a fully nonparametric variance structure. This approach is hampered by curse of dimensionality in practical applications. Moreover, the corresponding asymptotic theory is largely restricted to estimators that minimize certain smooth objective functions. The asymptotic derivation thus excludes semiparametric quantile regression models. To overcome these drawbacks, we study a general class of location-dispersion regression models, in which both the location function and the dispersion function are semiparametrically modeled. We establish unified asymptotic theory which is valid for many commonly used semiparametric structures such as the partially linear structure and single-index structure. We provide easy to check sufficient conditions and illustrate them through examples. Our theory permits non-smooth location or dispersion functions, thus allows for semiparametric quantile heteroscedastic regression and robust estimation in semiparametric mean regression. Simulation studies indicate significant efficiency gain in estimating the parametric component of the location function. The results are applied to analyzing a data set on gasoline consumption.