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Register a new userAn efficient algorithm for structured sparse quantile regression
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Added by
Ignace Loris
Publication date
2013
fields
Mathematical Statistics
and research's language is
English
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Quantile regression is studied in combination with a penalty which promotes structured (or group) sparsity. A mixed $ell_{1,infty}$-norm on the parameter vector is used to impose structured sparsity on the traditional quantile regression problem. An algorithm is derived to calculate the piece-wise linear solution path of the corresponding minimization problem. A Matlab implementation of the proposed algorithm is provided and some applications of the methods are also studied.
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As a competitive alternative to least squares regression, quantile regression is popular in analyzing heterogenous data. For quantile regression model specified for one single quantile level $tau$, major difficulties of semiparametric efficient estimation are the unavailability of a parametric efficient score and the conditional density estimation. In this paper, with the help of the least favorable submodel technique, we first derive the semiparametric efficient scores for linear quantile regression models that are assumed for a single quantile level, multiple quantile levels and all the quantile levels in $(0,1)$ respectively. Our main discovery is a one-step (nearly) semiparametric efficient estimation for the regression coefficients of the quantile regression models assumed for multiple quantile levels, which has several advantages: it could be regarded as an optimal way to pool information across multiple/other quantiles for efficiency gain; it is computationally feasible and easy to implement, as the initial estimator is easily available; due to the nature of quantile regression models under investigation, the conditional density estimation is straightforward by plugging in an initial estimator. The resulting estimator is proved to achieve the corresponding semiparametric efficiency lower bound under regularity conditions. Numerical studies including simulations and an example of birth weight of children confirms that the proposed estimator leads to higher efficiency compared with the Koenker-Bassett quantile regression estimator for all quantiles of interest.
Quantile regression models are a powerful tool for studying different points of the conditional distribution of univariate response variables. Their multivariate counterpart extension though is not straightforward, starting with the definition of multivariate quantiles. We propose here a flexible Bayesian quantile regression model when the response variable is multivariate, where we are able to define a structured additive framework for all predictor variables. We build on previous ideas considering a directional approach to define the quantiles of a response variable with multiple-outputs and we define noncrossing quantiles in every directional quantile model. We define a Markov Chain Monte Carlo (MCMC) procedure for model estimation, where the noncrossing property is obtained considering a Gaussian process design to model the correlation between several quantile regression models. We illustrate the results of these models using two data sets: one on dimensions of inequality in the population, such as income and health; the second on scores of students in the Brazilian High School National Exam, considering three dimensions for the response variable.
We consider the counting rate estimation of an unknown radioactive source, which emits photons at times modeled by an homogeneous Poisson process. A spectrometer converts the energy of incoming photons into electrical pulses, whose number provides a rough estimate of the intensity of the Poisson process. When the activity of the source is high, a physical phenomenon known as pileup effect distorts direct measurements, resulting in a significant bias to the standard estimators of the source activities used so far in the field. We show in this paper that the problem of counting rate estimation can be interpreted as a sparse regression problem. We suggest a post-processed, non-negative, version of the Least Absolute Shrinkage and Selection Operator (LASSO) to estimate the photon arrival times. The main difficulty in this problem is that no theoretical conditions can guarantee consistency in sparsity of LASSO, because the dictionary is not ideal and the signal is sampled. We therefore derive theoretical conditions and bounds which illustrate that the proposed method can none the less provide a good, close to the best attainable, estimate of the counting rate activity. The good performances of the proposed approach are studied on simulations and real datasets.
Model averaging considers the model uncertainty and is an alternative to model selection. In this paper, we propose a frequentist model averaging estimator for composite quantile regressions. In recent years, research on these topics has been added as a separate method, but no study has investigated them in combination. We apply a delete-one cross-validation method to estimate the model weights, and prove that the jackknife model averaging estimator is asymptotically optimal in terms of minimizing out-of-sample composite final prediction error. Simulations are conducted to demonstrate the good finite sample properties of our estimator and compare it with commonly used model selection and averaging methods. The proposed method is applied to the analysis of the stock returns data and the wage data and performs well.
With the availability of high dimensional genetic biomarkers, it is of interest to identify heterogeneous effects of these predictors on patients survival, along with proper statistical inference. Censored quantile regression has emerged as a powerful tool for detecting heterogeneous effects of covariates on survival outcomes. To our knowledge, there is little work available to draw inference on the effects of high dimensional predictors for censored quantile regression. This paper proposes a novel procedure to draw inference on all predictors within the framework of global censored quantile regression, which investigates covariate-response associations over an interval of quantile levels, instead of a few discrete values. The proposed estimator combines a sequence of low dimensional model estimates that are based on multi-sample splittings and variable selection. We show that, under some regularity conditions, the estimator is consistent and asymptotically follows a Gaussian process indexed by the quantile level. Simulation studies indicate that our procedure can properly quantify the uncertainty of the estimates in high dimensional settings. We apply our method to analyze the heterogeneous effects of SNPs residing in lung cancer pathways on patients survival, using the Boston Lung Cancer Survival Cohort, a cancer epidemiology study on the molecular mechanism of lung cancer.
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