Boundary or frontier estimation, and more generally, level sets estimation,
are recurrent functional estimation problems in statistics which are linked to outlier detection.
In biology, one is interested in estimating
reference curves,
that it to say curves which bound 90% (for instance) of the population.
Points outside this bound are considered as outliers compared to the reference
population. Here, reference curves are computed through nonparametric
regression quantile estimations (see Figure 1).
Figure 1: Reference curves for a given women population.
The sebum rate measured on the forhead of each is woman is reported against
the age. The curves represent various nonparametric estimates of the 10%
reference curves bounding 90% of the population.
Figure 2: Boundary estimation using a linear programming
approach: blue points (dots) are data points, the dot curve is the unknown boundary,
the continuous curve is the estimated boundary, the square points are points
on the estimated boundary and the triangle points are the additional points
actually used in the estimation (support vectors).
|
PhD theseshere |
| G. Stupfler
"Un modèle de Markov caché en assurance et Estimation de frontière et de point terminal", Université de Strasbourg, novembre 2011
[pdf].
|
| A. Lekina, "
Estimation non-paramétrique des quantiles extrêmes conditionnels
, Université Grenoble 1, octobre 2010
[pdf]. |
| L. Gardes,
"Estimation d'une fonction
quantile extrême", Université Montpellier 2, octobre 2003
[pdf]. |
| Frontier estimation with linear programming methods |
| A. Nazin & S. Girard.
"L1-optimal linear programming estimator for periodic frontier functions with Holder continuous derivative",
Automation and Remote Control, 75(12), 2152-2169, 2014.
[Associated technical report: pdf]. |
| G. Bouchard, S. Girard, A. Iouditski & A. Nazin.
"Some linear programming methods for frontier estimation", Applied
Stochastic Models in Business and Industry, 21(2), 175-185, 2005. |
| S. Girard, A. Iouditski & A. Nazin.
"L1-optimal frontier estimation via linear
programming", Automation and Remote Control, 66(12), 2000-2018, 2005.
[Associated technical report:
ps/pdf or
pdf]. |
| G. Bouchard, S. Girard, A. Iouditski & A. Nazin.
"Nonparametric estimation of the support boundary through linear
programming", Automation and Remote Control, 65(1), 58-64, 2004.
[Associated technical report:
ps/pdf or ps]. |
| Nonparametric regression on extreme observations |
|
S. Girard, G. Stupfler & A. Usseglio-Carleve,
"Functional estimation of extreme conditional expectiles",
Econometrics and Statistics, 21, 131--158, 2022.
[Associated technical report: pdf].
|
|
S. Girard, G. Stupfler & A. Usseglio-Carleve,
"Extreme Lp-quantile kernel regression",
In Advances in Contemporary Statistics and Econometrics, A. Daouia and A. Ruiz-Gazen (eds.), Springer, 197-219, 2021.
[Associated technical report:pdf].
|
|
A. Ag Ahmad, E. Deme, A. Diop & S. Girard.
"Estimation of the tail-index in a conditional location-scale family of heavy-tailed distributions", Dependence Modeling, 7, 394--417, 2019
[associated technical report: pdf].
|
| J. El Methni, L. Gardes & S. Girard.
"Kernel estimation of extreme regression risk measures",
Electronic Journal of Statistics, 12, 359--398, 2018.
[associated technical report: pdf].
|
|
L. Gardes & S. Girard. "On the estimation of the functional Weibull tail-coefficient", Journal of Multivariate Analysis, 146, 24--45, 2016.
[associated technical report: pdf].
|
|
S. Girard & S. Louhichi "On the strong consistency of the kernel estimator of extreme conditional quantiles",
In E. Ould-Said et al, editors,
Functional Statistics and Applications, Springer, pages 59--77, 2015.
[associated technical report: pdf].
|
| J. El Methni, L. Gardes & S. Girard.
"Nonparametric estimation of extreme risk measures from conditional heavy-tailed distributions", Scandinavian Journal of Statistics, 41, 988--1012, 2014.
[associated technical report: pdf].
|
|
A. Daouia, L. Gardes & S. Girard. "On kernel smoothing for extremal quantile regression", Bernoulli, 19, 2557--2589, 2013. [associated technical report: pdf].
|
| L. Gardes & S. Girard.
"Functional kernel estimators of large conditional quantiles",
Electronic Journal of Statistics, 6, 1715--1744, 2012.
[associated technical report:
pdf].
|
|
J. Carreau & S. Girard.
"Spatial extreme quantile estimation using a weighted log-likelihood approach",
Journal de la Société Française de Statistique, 152(3), 66--83, 2011.
|
| L. Gardes & S. Girard "Functional kernel estimators of conditional extreme quantiles" In F. Ferraty,
editor, Recent advances in functional data analysis and related topics, pages 135-140, Springer, Physica-Verlag, 2011.
[Associated technical report: pdf].
|
| A. Daouia, L. Gardes, S. Girard & A. Lekina. "Kernel estimators of extreme level curves",
Test, 20(2), 311--333, 2011.
[Associated technical report: pdf].
|
|
L. Gardes & S. Girard. "Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels",
Extremes, 13(2), 177--204, 2010.
[Associated technical report: pdf].
|
| L. Gardes, S. Girard & A. Lekina. "Functional nonparametric estimation of conditional extreme quantiles", Journal of Multivariate
Analysis, 101, 419--433, 2010.
[Associated technical report: pdf].
|
| L. Gardes & S. Girard. "A moving window approach for nonparametric estimation of the conditional tail index", Journal of Multivariate
Analysis, 99, 2368--2388, 2008.
[Associated technical report:
pdf]. |
| S. Girard & P. Jacob.
"A note on extreme values and kernel estimators of sample boundaries",
Statistics and Probability Letters, 64, 982--986, 2008.
[Associated technical report:
pdf].
|
| S. Girard & L. Menneteau.
"Smoothed extreme value estimators of non-uniform point processes boundaries with application to star-shaped supports estimation",
Communication in Statistics - Theory and Methods,
37(6), 881--897, 2008.
[Associated technical report:
pdf].
|
| S. Girard & L. Menneteau. "Central limit theorems for smoothed
extreme value estimates of point processes boundaries", Journal of
Statistical Planning and Inference, 135(2), 433-460, 2005.
[Associated technical report:
pdf]. |
| S. Girard & P. Jacob. "Extreme values and kernel estimates
of point processes boundaries", ESAIM: Probability and Statistics,
8, 150-168, 2004.
[Associated technical report:
pdf
ps]. |
| S. Girard & P. Jacob. "Projection estimates of point
processes boundaries", Journal of Statistical Planning and Inference,
116(1):1-15, 2003.
[Associated technical report:
pdf
ps]. |
| S. Girard & P. Jacob. "Extreme value and Haar series
estimates of point
process boundaries", Scandinavian Journal of Statistics,
30(2):369-384, 2003.
[Associated technical report:
pdf
ps]. |
| See also Statistics of extremes. |
| Frontier estimation using all the observations |
|
A. Daouia, S. Girard & A. Guillou.
"A Gamma-moment approach to monotonic boundaries estimation: with applications in econometric and nuclear fields", Journal of Econometrics, 178, 727--740, 2014.
[Associated technical report: pdf, detailed version including simulations:
pdf].
|
| S. Girard, A. Guillou & G. Stupfler. "Uniform strong consistency of a frontier estimator using kernel regression on high order moments", ESAIM: Probability and Statistics, 18, 642--666, 2014. [Associated technical report: pdf]
|
| S. Girard, A. Guillou & G. Stupfler. "Frontier estimation with kernel regression on high order moments", Journal of Multivariate Analysis,
116, 172--189, 2013. [Associated technical report: pdf, supplementary material: pdf].
|
| A. Daouia, L. Gardes & S. Girard "Nadaraya's estimates for large quantiles and free disposal support curves" In I. Van Keilegom and P. Wilson, editors, Exploring research
frontiers in contemporary statistics and econometrics, pages 1--22,
Springer, 2012.
[Associated technical report: pdf]. |
| S. Girard & P. Jacob.
"Frontier estimation with local polynomials and high power-transformed data",
Journal of Multivariate Analysis, 100, 1691--1705, 2009.
[Associated technical report: pdf]. |
| S. Girard & P. Jacob.
"Frontier estimation via kernel regression on high power-transformed data",
Journal of Multivariate Analysis, 99, 403--420, 2008.
[Associated technical report: pdf]. |
| L1 error in frontier estimation |
| J. Geffroy, S. Girard & P. Jacob. "Asymptotic normality of the
L1-error of a boundary estimate", Nonparametric Statistics,
18(1), 21-31, 2006.
[Associated technical report: pdf].
Warning: the published version of this technical paper contains
a lot of missprints. |
| S. Girard & P. Jacob. "Asymptotic normality of the L1-error
for Geffroy's estimate of point process
boundaries", Publications de l'Institut de Statistique de
l'Université de Paris, XLIX, 3-17, 2005.
[Associated technical report: ps]. |
| S. Girard. "On the asymptotic normality of the L1 error for Haar
series estimates of Poisson point processes boundaries", Statistics and
Probability Letters, 66, 81-90, 2004. |
| Dimension reduction in regression problems |
|
A. Chiancone, F. Forbes & S. Girard.
"Student Sliced Inverse Regression",
Computational Statistics and Data Analysis, 113, 441--456, 2017.
[Associated technical report: pdf].
|
|
S. Girard & J. Saracco. "An introduction to dimension reduction in nonparametric kernel regression" In D. Fraix-Burnet and D. Valls-Gabaud, editors, Regression methods for astrophysics, volume 66, pages 167--196, EDP Sciences, 2014.
[Associated technical report: pdf]. |
|
R. Coudret, S. Girard, & J. Saracco.
"A new sliced inverse regression method for multivariate response",
Computational Statistics and Data Analysis, 77, 285--299, 2014.
[Associated technical report: pdf]. |
|
M. Chavent, S. Girard, V. Kuentz, B. Liquet, T.M.N. Nguyen & J. Saracco.
"A sliced inverse regression approach for data stream",
Computational Statistics, 29, 1129--1152, 2014.
[Associated technical report: pdf]. |
| C. Bernard-Michel, S. Douté, M. Fauvel, L. Gardes & S. Girard. "Retrieval of Mars surface physical properties from OMEGA hyperspectral images using Regularized Sliced Inverse Regression", Journal of Geophysical Research - Planets, 114, E06005, 2009.
[Associated technical report:
pdf]. |
| C. Bernard-Michel, L. Gardes & S. Girard. "A Note on Sliced Inverse Regression with Regularizations", Biometrics, 64, 982--986, 2008. [Associated technical report: pdf]. |
| C. Bernard-Michel, L. Gardes & S. Girard.
"Gaussian Regularized Sliced Inverse Regression", Statistics and Computing, 19, 85--98, 2009.
[Associated technical report: pdf].
|
| See also Multivariate data analysis. |
| Regression quantiles and reference curves estimation |
| A. Gannoun, S. Girard, C. Guinot & J. Saracco. "Sliced
Inverse Regression in reference curves estimation", Computational Statistics and Data Analysis , 46(3):103-122, 2004.
[Associated technical report: pdf
ps]. |
| A. Gannoun, S. Girard, C. Guinot & J. Saracco. "Reference
ranges based
on nonparametric quantile regression", Statistics in Medicine,
21(20):3119-3135, 2002. |
| A. Gannoun, S. Girard, C. Guinot & J. Saracco. "Trois
méthodes non paramétriques pour l'estimation de courbes de
référence - Application l'analyse de propriétés biophysiques de la
peau", Revue de Statistique Appliquée, L(1), 65-89, 2002. |
| Miscellaneous |
| M. Allouche, S. Girard & E. Gobet,
"Generative model for fBm with deep ReLU neural networks", Journal of Complexity, 73, 101667, 2022.
[Associated technical report: pdf].
|
| A. Gannoun, S. Girard, C. Guinot & J. Saracco.
"Implémentation en C d'estimateurs non- paramétriques de quantiles conditionnels. Application au tracé de courbes de
référence", La revue
de Modulad, 31:59-70, 2004. |
