Overview of the methods implemented in npbr
Presentation of the package npbr
npbr: A Package for Nonparametric Boundary Regression in R Abdelaati Daouia1 Thibault Laurent1,2 Hohsuk Noh3 1 TSE:
Toulouse School of Economics, University of Toulouse Capitole, France 2 CNRS 3 Sookmyung Women’s University
13th
Thibault LAURENT Nonparametric Boundary Regression in R
useR!2018 July 2018, Brisbane, Australia
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
What does “boundary regression” mean ?
OLS regression Boundary regression
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Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
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Annual sport records: data("records") European air controllers: data("air") French postal services: data("post") Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Constrained boundary regression ●
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Unconstrained: method = "u" Monotone: method = "m" Monotone and concave: method = "mc" Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Parametric model: polynomial estimators Model structure: ϕθ (x) = θ0 + θ1 x + . . . + θp x p ˆ = (θ ˆ0 , . . . , θ ˆp ) s.t. that ϕˆ Optimization problem: Find θ θ envelopes the full data and minimizes the area under its graph (Hall, 1998) 12
p=1 p=2 p=3
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Choose p which minimizes the AIC/BIC (Daouia, 2015) Easily implemented, but no constained version Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Nonparametric model: FDH, DEA Nonparametric: model structure is not specified a priori but is instead determined from data Example: Free Disposal Hull (Deprins, Simar and Tulkens, 1984) or Data Envelopment Analysis (Farrell, 1957) ●
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Very famous and popular in the economic litterature, but too sensitive to the extreme values Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Nonparametric model: Local maximum frontier estimators ˜ (x) = max yi 1I{|xi −x|≤h} (Gijbels and Peng, 2000) ϕ i=1,...,n
A data-driven rule for selecting h: package np (Li, Lin and Racine, 2013) ●
h = 500
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No constrained version and h should depend on x Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Nonparametric model: Localized linear fitting ˆ n,LL (x) = min{z : there exists θ s.t. yi ≤ z + θ(xi − x) for all ϕ i s.t. xi ∈ (x − h, x + h)}, (Hall et al., 1998) Optimal h: Hall and Park (2004) ●
h = 1000
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No constained version Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Nonparametric model: Quadratic (or Cubic) spline fitting Daouia et al. (2016) Denote a partition of [a, b] by a = t0 < t1 < . . . < tkn = b, with a = min xi and b = max xi by considering the j/kn th quantiles i
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tj = x[jn/kn ] of the distinct values of xi for j = 1, . . . , kn − 1 Let N = kn + 2 and π(x) = (π0 (x), . . . , πN−1 (x))> be the vector of normalized B-splines of order 2 (or 3) based on {tj } ˆ n (x) = π(x)> α ˆ , where α ˆ minimizes the same objective ϕ ˜ subject to the same envelopment constraints function as α and the additional monotonicity constraints π0 (tj )> α ≥ 0, j = 0, 1, . . . , kn , with π0 being the derivative of π. Number of inter-knot segments kn : AIC or BIC Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Quadratic spline fitting
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Unconstrained (kn = 3) Monotone (kn = 2) Monotone and concave (kn = 1)
Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Package on CRAN
On MAC OS: install the glpk library outside of R, using homebrew Toulouse School of Economics, CNRS (brew install glpk)
Thibault LAURENT
Nonparametric Boundary Regression in R
Overview of the methods implemented in npbr
Presentation of the package npbr
npbr functions Function dea_est loc_est loc_est_bw poly_est poly_degree dfs_momt dfs_pick rho_momt_pick kopt_momt_pick dfs_pwm_regul loc_max pick_est quad_spline_est quad_spline_kn cub_spline_est cub_spline_kn kern_smooth kern_smooth_bw Thibault LAURENT
Description DEA, FDH and linearized FDH Local linear fitting Bandwidth choice for local linear fitting Polynomial estimation Optimal polynomial degree selection Moment type estimation Pickands type estimation Conditional tail index estimation Threshold selection for moment/Pickands frontiers Nonparametric frontier regularization Local constant estimation Local extreme-value estimation Quadratic spline fitting Knot selection for quadratic spline fitting Cubic spline fitting Knot selection for cubic spline fitting Nonparametric kernel boundary regression Bandwidth choice for kernel boundary regression
Nonparametric Boundary Regression in R
Reference Farrell (1957), Deprins et al. (1984), Hall and Park (2002) Hall et al. (1998), Hall and Park (2004) Hall and Park (2004) Hall (1998) Daouia et al. (2015) Daouia et al. (2010), Dekkers et al. (1989) Daouia et al. (2010), Dekkers et al. (1989) Daouia et al. (2010), Dekkers et al. (1989) Daouia et al. (2010) Daouia et al. (2012) Gijbels and Peng (2000) Gijbels and Peng (2000) Daouia et al. (2015) Daouia et al. (2015) Daouia et al. (2015) Daouia et al. (2015) Parmeter and Racine (2013), Noh (2014) Parmeter and Racine (2013), Noh (2014) Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Usage poly_est(xtab, ytab, x, deg, control = list("tm_limit" = 700)) xtab: a numeric vector containing the observed inputs x1 , . . . , xn . ytab: a numeric vector of the same length as xtab containing the observed outputs y1 , . . . , yn . x: a numeric vector of evaluation points in which the estimator is to be computed. deg: an integer (polynomial degree). control: a list of parameters to the GLPK solver.
Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Example of use
require("npbr") data(air) x_air + R> + R> + R> + R>
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Toulouse School of Economics, CNRS
Overview of the methods implemented in npbr
Presentation of the package npbr
Conclusion
Available on CRAN Article published in Journal of Statistical Software (2017), http://dx.doi.org/10.18637/jss.v079.i09 Numerical illustrations given in the article: cubic spline fitting seems to be the best method whatever the shape of the data Hope to see you in UseR!2019, Toulouse, France, next year
Thibault LAURENT Nonparametric Boundary Regression in R
Toulouse School of Economics, CNRS
