Quantity quantiles linear regression

We show that the definition of the θth sample quantile as the solution to a minimization problem introduced by Koenker and Bassett (Econometrica 46(1):33–50, 1978) can be easily extended to obtain an analogous definition for the θth sample quantity quantile widely investigated and applied in the Italian literature. The key point is the use of the first-moment distribution of the variable instead of its distribution function. By means of this definition we introduce a linear regression model for quantity quantiles and analyze some properties of the residuals. In Sect. 4 we show a brief application of the methodology proposed. This research was partially supported by Fondo d’Ateneo per la Ricerca anno 2005—Università degli Studi di Milano-Bicocca. The paper is the result of the common work of the authors; in particular M. Zenga has written Sects. 1 and 5 while P. Radaelli has written the remaining sections.     

On nonparametric regression estimators based on regression quantiles

In the ciassical regression model Y_i=h(x_i) + ? _i, i=1,…,n, Cheng (1984) introduced linear combinations of regression quantiles as a new class of estimators for the unknown regression function h(x). The asymptotic properties studied in Cheng (1984) are reconsidered. We obtain a sharper scrong consistency rate and we improve on the conditions for asymptotic normality by proving a new result on the remainder term in the Bahadur representation for regression quantiles.     

Distribution-free confidence intervals for quantiles in small samples

Estimators for quantiles based on linear combinations of order statistics have been proposed by Harrell and Davis(1982) and kaigh and Lachenbruch (1982). Both estimators have been demonstrated to be at least as efficient for small sample point estimation as an ordinary sample quantile estimator based on one or two order statistics: Distribution-free confidence intervals for quantiles can be constructed using either of the two approaches. By means of a simulation study, these confidence intervals have been compared with several other methods of constructing confidence intervals for quantiles in small samples. For the median, the Kaigh and Lachenbruch method performed fairly well. For other quantiles, no method performed better than the method which uses pairs of order statistics.     

Monotone support vector quantile regression

Quantile regression (QR) models have received a great deal of attention in both the theoretical and applied statistical literature. In this paper we propose support vector quantile regression (SVQR) with monotonicity restriction, which is easily obtained via the dual formulation of the optimization problem. We also provide the generalized approximate cross validation method for choosing the hyperparameters which affect the performance of the proposed SVQR. The experimental results for the synthetic and real data sets confirm the successful performance of the proposed model.     

Dimension reduction via local rank regression

Outer product of gradients (OPG) achieves dimension reduction via estimating the gradients of the regression function. In this paper, we propose two novel OPG estimators via local rank regression: the rank OPG estimator and the Walsh-average OPG estimator. Both proposals guard against a wide range of error distributions, and are safe alternatives to existing OPG estimators based on local linear regression or local L1 regression. The effectiveness of the new proposals are demonstrated via extensive numerical studies.     

Composite kernel quantile regression

The composite quantile regression (CQR) has been developed for the robust and efficient estimation of regression coefficients in a liner regression model. By employing the idea of the CQR, we propose a new regression method, called composite kernel quantile regression (CKQR), which uses the sum of multiple check functions as a loss in reproducing kernel Hilbert spaces for the robust estimation of a nonlinear regression function. The numerical results demonstrate the usefulness of the proposed CKQR in estimating both conditional nonlinear mean and quantile functions.     

Efficient quantile regression for heteroscedastic models

Quantile regression (QR) provides estimates of a range of conditional quantiles. This stands in contrast to traditional regression techniques, which focus on a single conditional mean function. Lee et al. [Regularization of case-specific parameters for robustness and efficiency. Statist Sci. 2012;27(3):350–372] proposed efficient QR by rounding the sharp corner of the loss. The main modification generally involves an asymmetric ?₂ adjustment of the loss function around zero. We extend the idea of ?₂ adjusted QR to linear heterogeneous models. The ?₂ adjustment is constructed to diminish as sample size grows. Conditions to retain consistency properties are also provided.     

Local regression when the responses are Interval-Censored

Conditional expectation imputation and local-likelihood methods are contrasted with a midpoint imputation method for bivariate regression involving interval-censored responses. Although the methods can be extended in principle to higher order polynomials, our focus is on the local constant case. Comparisons are based on simulations of data scattered about three target functions with normally distributed errors. Two censoring mechanisms are considered: the first is analogous to current-status data in which monitoring times occur according to a homogeneous Poisson process; the second is analogous to a coarsening mechanism such as would arise when the response values are binned. We find that, according to a pointwise MSE criterion, no method dominates any other when interval sizes are fixed, but when the intervals have a variable width, the local-likelihood method often performs better than the other methods, and midpoint imputation performs the worst. Several illustrative examples are presented.     

Quantile regression via iterative least squares computations

We present an estimating framework for quantile regression where the usual L ₁-norm objective function is replaced by its smooth parametric approximation. An exact path-following algorithm is derived, leading to the well-known ‘basic’ solutions interpolating exactly a number of observations equal to the number of parameters being estimated. We discuss briefly possible practical implications of the proposed approach, such as early stopping for large data sets, confidence intervals, and additional topics for future research.     

Symmetric quantiles and their applications

To develop estimators with stronger efficiencies than the trimmed means which use the empirical quantile, Kim (1992) Kim, S. J. 1992. The metrically trimmed means as a robust estimator of location. Annals of Statistics, 20: 1534–1547. [Crossref], [Web of Science ®] [Google Scholar] and Chen & Chiang (1996) Chen, L. A. and Chiang, Y. C. 1996. Symmetric type quantile and trimmed means for location and linear regression model. Journal of Nonparametric Statistics, 7: 171–185. [Taylor & Francis Online] [Google Scholar], implicitly or explicitly used the symmetric quantile, and thus introduced new trimmed means for location and linear regression models, respectively. This study further investigates the properties of the symmetric quantile and extends its application in several aspects. (a) The symmetric quantile is more efficient than the empirical quantiles in asymptotic variances when quantile percentage α is either small or large. This reveals that for any proposal involving the α th quantile of small or large α s, the symmetric quantile is the right choice; (b) a trimmed mean based on it has asymptotic variance achieving a Cramer-Rao lower bound in one heavy tail distribution; (c) an improvement of the quantiles-based control chart by Grimshaw & Alt (1997) Grimshaw, S. D. and Alt, F. B. 1997. Control charts for quantile function values. Journal of Quality Technology, 29: 1–7. [Taylor & Francis Online] [Google Scholar] is discussed; (d) Monte Carlo simulations of two new scale estimators based on symmetric quantiles also support this new quantile.     

Bayesian composite quantile regression

One advantage of quantile regression, relative to the ordinary least-square (OLS) regression, is that the quantile regression estimates are more robust against outliers and non-normal errors in the response measurements. However, the relative efficiency of the quantile regression estimator with respect to the OLS estimator can be arbitrarily small. To overcome this problem, composite quantile regression methods have been proposed in the literature which are resistant to heavy-tailed errors or outliers in the response and at the same time are more efficient than the traditional single quantile-based quantile regression method. This paper studies the composite quantile regression from a Bayesian perspective. The advantage of the Bayesian hierarchical framework is that the weight of each component in the composite model can be treated as open parameter and automatically estimated through Markov chain Monte Carlo sampling procedure. Moreover, the lasso regularization can be naturally incorporated into the model to perform variable selection. The performance of the proposed method over the single quantile-based method was demonstrated via extensive simulations and real data analysis.     

Confidence intervals for lognormal regression and a non-parametric alternative

Approximate confidence intervals are given for the lognormal regression problem. The error in the nominal level can be reduced to O(n ^?2), where n is the sample size. An alternative procedure is given which avoids the non-robust assumption of lognormality. This amounts to finding a confidence interval based on M-estimates for a general smooth function of both ? and F, where ? are the parameters of the general (possibly nonlinear) regression problem and F is the unknown distribution function of the residuals. The derived intervals are compared using theory, simulation and real data sets.     

Composite quantile regression for varying-coefficient single-index models

ABSTRACT

The varying-coefficient single-index model (VCSIM) is a very general and flexible tool for exploring the relationship between a response variable and a set of predictors. Popular special cases include single-index models and varying-coefficient models. In order to estimate the index-coefficient and the non parametric varying-coefficients in the VCSIM, we propose a two-stage composite quantile regression estimation procedure, which integrates the local linear smoothing method and the information of quantile regressions at a number of conditional quantiles of the response variable. We establish the asymptotic properties of the proposed estimators for the index-coefficient and varying-coefficients when the error is heterogeneous. When compared with the existing mean-regression-based estimation method, our simulation results indicate that our proposed method has comparable performance for normal error and is more robust for error with outliers or heavy tail. We illustrate our methodologies with a real example.     

Gibbs sampling methods for Bayesian quantile regression

This paper considers quantile regression models using an asymmetric Laplace distribution from a Bayesian point of view. We develop a simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the resulting Gibbs sampler can be accomplished by sampling from either normal or generalized inverse Gaussian distribution. We also discuss some possible extensions of our approach, including the incorporation of a scale parameter, the use of double exponential prior, and a Bayesian analysis of Tobit quantile regression. The proposed methods are illustrated by both simulated and real data.     

Penalized weighted composite quantile regression in the linear regression model with heavy-tailed autocorrelated errors

In this paper, a penalized weighted composite quantile regression estimation procedure is proposed to estimate unknown regression parameters and autoregression coefficients in the linear regression model with heavy-tailed autoregressive errors. Under some conditions, we show that the proposed estimator possesses the oracle properties. In addition, we introduce an iterative algorithm to achieve the proposed optimization problem, and use a data-driven method to choose the tuning parameters. Simulation studies demonstrate that the proposed new estimation method is robust and works much better than the least squares based method when there are outliers in the dataset or the autoregressive error distribution follows heavy-tailed distributions. Moreover, the proposed estimator works comparably to the least squares based estimator when there are no outliers and the error is normal. Finally, we apply the proposed methodology to analyze the electricity demand dataset.     

Hierarchically penalized quantile regression

In many regression problems, predictors are naturally grouped. For example, when a set of dummy variables is used to represent categorical variables, or a set of basis functions of continuous variables is included in the predictor set, it is important to carry out a feature selection both at the group level and at individual variable levels within the group simultaneously. To incorporate the group and variables within-group information into a regularized model fitting, several regularization methods have been developed, including the Cox regression and the conditional mean regression. Complementary to earlier works, the simultaneous group and within-group variables selection method is examined in quantile regression. We propose a hierarchically penalized quantile regression, and show that the hierarchical penalty possesses the oracle property in quantile regression, as well as in the Cox regression. The proposed method is evaluated through simulation studies and a real data application.     

Jackknife method for intermediate quantiles

Quantile function plays an important role in statistical inference, and intermediate quantile is useful in risk management. It is known that Jackknife method fails for estimating the variance of a sample quantile. By assuming that the underlying distribution satisfies some extreme value conditions, we show that Jackknife variance estimator is inconsistent for an intermediate order statistic. Further we derive the asymptotic limit of the Jackknife-Studentized intermediate order statistic so that a confidence interval for an intermediate quantile can be obtained. A simulation study is conducted to compare this new confidence interval with other existing ones in terms of coverage accuracy.