Three different approaches to frontier estimation St´ ephane Girard INRIA Grenoble Rhˆ one-Alpes & LJK http://mistis.inrialpes.fr/people/girard/ March, 2014 Joint work with A. Guillou (Universit´ e de Strasbourg, France), A. Iouditski (Universit´ e de Grenoble, France), L. Menneteau (Universit´ e de Montpellier, France), A. Nazin (Institute of Control Sciences, Moscow, Russia) and G. Stupfler (Universit´ e Aix-Marseille, France). 1 St´ ephane Girard Three different approaches to frontier estimation
Outline Very brief overview of the literature: First frontier estimator Geffroy (ISUP, 1964) Piecewise polynomial estimators H¨ ardle, Park, Tsybakov (JMVA, 1995) Extreme-value estimators, Linear programming estimators, High order moments estimators. 2 St´ ephane Girard Three different approaches to frontier estimation
Framework Let ( X i , Y i ), 1 ≤ i ≤ n be n independent copies of a random pair ( X , Y ) such that their common distribution has a support S := { ( x , y ) ∈ Ω × R ; 0 ≤ y ≤ g ( x ) } where X has a density f X on the compact subset Ω ⊂ R d , Y | X = x has a density f ( . | x ) on [0 , g ( x )], g is a positive function, g ( x ) = sup { Y | X = x } . We address the problem of the estimation of g , called the frontier of S . 3 St´ ephane Girard Three different approaches to frontier estimation
Illustration Ω = [0 , 1] 4 St´ ephane Girard Three different approaches to frontier estimation
Sharp/non-sharp boundaries H¨ ardle, Park, Tsybakov (JMVA, 1995) assumed that, for all ( x , y ) ∈ S , f X ( x ) ≥ f min > 0, f ( y | x ) ≥ c ( g ( x ) − y ) α where c > 0 and α ≥ 0. Two cases arise: If α = 0 then f ( y | x ) ≥ c > 0 for all y ∈ [0 , g ( x )], this is the situation of a “sharp boundary”. If α > 0 then we may have f ( y | x ) → 0 as y → g ( x ), this is the situation of a “non-sharp boundary”. 5 St´ ephane Girard Three different approaches to frontier estimation
Geffroy’s estimator First frontier estimator Geffroy (ISUP, 1964), based on the extreme-values of the sample: Partition of Ω = [0 , 1] into equidistant k n intervals I n , r , r = 1 , . . . , k n , Maxima on each bin: Y ∗ n , r = max { Y i : X i ∈ I n , r } , Piecewise constant estimator: k n � I { x ∈ I n , r } Y ∗ ˆ g n ( x ) = n , r . r =1 6 St´ ephane Girard Three different approaches to frontier estimation
Illustration: Geffroy’s estimator 7 St´ ephane Girard Three different approaches to frontier estimation
Geffroy’s estimator � 1 Asymptotic behaviour of the L 1 − distance ∆ n := 0 | ˆ g n ( x ) − g ( x ) | dx . Theorem Assume that g is γ − Lipschitzian γ ∈ (0 , 1] and α = 0 (sharp boundary). If some conditions on ( k n ) hold, then ( n / k n )(∆ n − β n ) converges in distribution to a Gumbel r.v. with c.d.f ψ ( z ) = exp( − exp( − θ z )) where x ∈ [0 , 1] f X ( x ) f ( g ( x ) | x ) , θ = inf and β n is the solution of the equation � � � 1 log k n − n β n f X ( x ) f ( g ( x ) | x ) exp dx = 1 . k n 0 The rate of convergence ( n / k n ) is (up to a logarithmic factor) n γ/ (1+ γ ) . 8 St´ ephane Girard Three different approaches to frontier estimation
Piecewise polynomial estimators Proposed in H¨ ardle, Park, Tsybakov (JMVA, 1995) to deal with sharp or non-sharp boundaries ( α ≥ 0), smoother frontiers, i.e. for γ > 0, it is assumed that the ⌊ γ ⌋ th derivative of the frontier g is ( γ − ⌊ γ ⌋ ) − Lipschitzian. The estimator requires a partition I n , r , r = 1 , . . . , k n of Ω = [0 , 1]. On the r th bin, the estimator is defined as the polynomial of degree ⌊ γ ⌋ covering all the points and with smallest surface. k n � g θ ˆ n ( x ) = I { x ∈ I n , r } P n , r ( x ; θ n , r ) . r =1 � θ n , r = arg min P n , r ( x ; θ ) dx s.t. P n , r ( X i ; θ ) ≥ Y i , X i ∈ I n , r . θ I n , r Note that if γ ∈ (0 , 1] then ⌊ γ ⌋ = 0 and we find back Geffroy’s estimator. 9 St´ ephane Girard Three different approaches to frontier estimation
Piecewise polynomial estimators Theorem Under the above assumptions, and for a well chosen partition, piecewise polynomial estimators have the optimal rate of convergence for the L 1 − error, that is n γ/ (1+( α +1) γ ) . In the case where α = 0 (sharp boundary) and γ ∈ (0 , 1], Geffroy’s estimator has the optimal rate of convergence. In practice, the estimators are biased downward and discontinuous. The choice of the partition ( k n ) is also an issue. 10 St´ ephane Girard Three different approaches to frontier estimation
Illustration: Piecewise linear estimator 11 St´ ephane Girard Three different approaches to frontier estimation
Contributions Extreme-value estimator (smoothed, bias correction, sharp boundary, pointwise asymptotic normality) Linear programming estimator (smoothed, no partition of Ω, sharp boundary, strong L 1 − consistency) High order moments estimator (smoothed, no partition of Ω, non-sharp boundary, pointwise asymptotic normality, strong L ∞ − consistency) 12 St´ ephane Girard Three different approaches to frontier estimation
1. Extreme-value estimator Support S = { ( x , y ) ∈ Ω × R ; 0 ≤ y ≤ g ( x ) } with Ω ⊂ R d . Geffroy’s estimator. k n � g (0) I { x ∈ I n , r } Y ∗ ˆ n ( x ) = n , r . r =1 where { I n , r , r = 1 , . . . , k n } is a partition of Ω and n , r = max { Y i : X i ∈ I n , r } . Y ∗ Bias correction. Assume that Y | X = x is uniformly distributed on [0 , g ( x )] (sharp boundary). k n � g (1) n , r (1 + N − 1 ˆ n ( x ) = I { x ∈ I n , r } Y ∗ n , r ) , r =1 where N n , r is the number of X i ∈ I n , r . 13 St´ ephane Girard Three different approaches to frontier estimation
Extreme-value estimator Smoothing � g (2) g (1) ˆ n ( x ) = R d K h n ( x − t )ˆ n ( t ) dt where K h n ( u ) = h − d n K ( u / h n ), K is d − dimensional density with compact support and h n is a smoothing parameter. Nonparametric regression over the extreme-values of the sample: � k n � g (2) n , r (1 + N − 1 K h n ( x − t ) dt Y ∗ ˆ n ( x ) = n , r ) I n , r r =1 G & Menneteau (JSPI, 2005), Menneteau (ESAIM, 2008) Theorem Assume that g is γ − Lipschitzian, γ ∈ (0 , 1] . Under some conditions on the ( h n ) and ( k n ) sequences, for all ( x 1 , ..., x p ) ⊂ Ω , the random vector � � nh d / 2 k − 1 / 2 g (2) (ˆ n ( x j ) − g ( x j )) : 1 ≤ j ≤ p n n is asymptotically centred Gaussian with diagonal covariance matrix. 14 St´ ephane Girard Three different approaches to frontier estimation
Extreme-value estimator Choosing h n ≍ n − 1 / ( γ + d ) and k n ≍ n d / ( γ + d ) , the rate of convergence is n γ/ ( d + γ ) , up to logarithmic factors. Optimal L 1 − rate of convergence for sharp boundaries ( α = 0) and γ − Lipschitzian frontiers, γ ∈ (0 , 1]. The rate of convergence of this extreme-value estimator is no more optimal for smoother frontier functions ( γ > 1). The approximation of g ( x ) by a constant value Y ∗ n , r for x ∈ I n , r is not precise enough. 15 St´ ephane Girard Three different approaches to frontier estimation
Illustration: Extreme-value estimator 16 St´ ephane Girard Three different approaches to frontier estimation
Contributions Extreme-value estimator (smoothed, bias correction, sharp boundary, pointwise asymptotic normality) Linear programming estimator (smoothed, no partition of Ω, sharp boundary, strong L 1 − consistency) High order moments estimator (smoothed, no partition of Ω, non-sharp boundary, pointwise asymptotic normality, strong L ∞ − consistency) 17 St´ ephane Girard Three different approaches to frontier estimation
2. Linear programming estimator Support S = { ( x , y ) ∈ [0 , 1] × R ; 0 ≤ y ≤ g ( x ) } , where g is γ − Lipschitzian, γ ∈ (0 , 1]. The estimator is a linear combination of kernel functions: n � g n ( x ) = ˆ α i K h n ( x − X i ) . i =1 The coefficients ( α i ) i =1 ,..., n are obtained by minimizing the surface of the estimated support: � n � min ˆ g n ( x ) dx = min α i , R i =1 under the following constraints : for all i = 1 , . . . , n ˆ g n ( X i ) ≥ Y i (the sample is below the estimated frontier) α i ≥ 0 (the estimated frontier function is positive) n ( X i ) | ≤ c 0 h γ − 1 | ˆ g ′ (Lipschitz constraint) n Linear Programming (LP) problem. 18 St´ ephane Girard Three different approaches to frontier estimation
Linear programming estimator Assume that Y | X = x is uniformly distributed on [0 , g ( x )] Remark 1. Joint distribution of the sample Σ n = ( X i , Y i ) i =1 ,..., n : n � g ( X i ) 1 P (Σ n | g ) = · g ( X i ) I { 0 ≤ Y i ≤ g ( X i ) } , C g i =1 � with C g = R g ( x ) dx . g n = � n Log-likelihood. Since C ˆ i =1 α i , we have n n � � L ( α ) = log P (Σ n | ˆ g n ) = − n log α i + log I { Y i ≤ ˆ g n ( X i ) } . i =1 i =1 The (LP) problem can be read as the maximization of the log-likelihood n ( X i ) | ≤ c 0 h γ − 1 under the additional constraints | ˆ g ′ , i = 1 , . . . , n . n 19 St´ ephane Girard Three different approaches to frontier estimation
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