Conditional quantile estimation based on optimal quantization: From theory to practice

Small-sample properties of a nonparametric estimator of conditional quantiles based on optimal quantization, that was recently introduced (Charlier et?al., 2015), are investigated. More precisely, (i) the practical implementation of this estimator is discussed (by proposing in particular a method to properly select the corresponding smoothing parameter, namely the number of quantizers) and (ii) its finite-sample performances are compared to those of classical competitors. Monte Carlo studies reveal that the quantization-based estimator competes well in all cases and sometimes dominates its competitors, particularly when the regression function is quite complex. A real data set is also treated. While the main focus is on the case of a univariate covariate, simulations are also conducted in the bivariate case.

[1]  Gillis Pagés,et al.  A space quantization method for numerical integration , 1998 .

[2]  Victor Chernozhukov,et al.  Conditional Quantile Processes Based on Series or Many Regressors , 2011, Journal of Econometrics.

[3]  R. Koenker,et al.  Penalized triograms: total variation regularization for bivariate smoothing , 2004 .

[4]  M. C. Jones,et al.  Local Linear Quantile Regression , 1998 .

[5]  Karen Gonzalez,et al.  Numerical method for optimal stopping of piecewise deterministic Markov processes , 2009, 0903.2114.

[6]  Keming Yu,et al.  Quantile regression: applications and current research areas , 2003 .

[7]  Adam Krzyżak,et al.  Rates of convergence for partitioning and nearest neighbor regression estimates with unbounded data , 2006 .

[8]  Pin T. Ng,et al.  Quantile smoothing splines , 1994 .

[9]  V. Chernozhukov,et al.  QUANTILE AND PROBABILITY CURVES WITHOUT CROSSING , 2007, 0704.3649.

[10]  Keming Yu,et al.  Local linear spatial quantile regression , 2009 .

[11]  Gilles Pagès,et al.  Optimal quadratic quantization for numerics: the Gaussian case , 2003, Monte Carlo Methods Appl..

[12]  Aurélie Fischer,et al.  Quantization and clustering with Bregman divergences , 2010, J. Multivar. Anal..

[13]  M. Wand,et al.  EXACT MEAN INTEGRATED SQUARED ERROR , 1992 .

[14]  Adam Krzyzak,et al.  A Distribution-Free Theory of Nonparametric Regression , 2002, Springer series in statistics.

[15]  Max H. Farrell,et al.  Optimal convergence rates, Bahadur representation, and asymptotic normality of partitioning estimators☆ , 2013 .

[16]  Jianqing Fan,et al.  Robust Non-parametric Function Estimation , 1994 .

[17]  So K Kb.  EFFICIENT SEMIPARAMETRIC ESTIMATION OF A PARTIALLY LINEAR QUANTILE REGRESSION MODEL , 2003 .

[18]  B. Efron.  The jackknife, the bootstrap, and other resampling plans , 1987 .

[19]  Young K. Truong,et al.  ROBUST NONPARAMETRIC FUNCTION ESTIMATION , 1994 .

[20]  Probal Chaudhuri,et al.  Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation , 1991 .

[21]  D. Paindaveine,et al.  Conditional quantile estimation through optimal quantization , 2014, 1405.2781.

[22]  Stephen Portnoy,et al.  Bivariate quantile smoothing splines , 1998 .

[23]  Joel L. Horowitz,et al.  Nonparametric Estimation of an Additive Quantile Regression Model , 2004 .

[24]  S. Graf,et al.  Foundations of Quantization for Probability Distributions , 2000 .

[25]  K. ..  KERNEL AND NEAREST NEIGHBOR ESTIMATION OF A CONDITIONAL QUANTILE by , 2008 .

[26]  S. Girard,et al.  Reference curves based on non‐parametric quantile regression , 2002, Statistics in medicine.

[27]  R. Koenker,et al.  Regression Quantiles , 2007 .