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Estimation under L-moment condition models

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Estimation under L-moment condition models Estimation under L-moment condition models Alexis Decurninge, Michel Broniatowski LSTA, Universite Pierre et Marie Curie, Paris VI 29 August 2013 Estimation under L-moment condition models Plan 1 L-moments : properties 2 Moment and L-moment equations models 3 Minimum of ϕ-divergence estimators 4 Asymptotic properties Estimation under L-moment condition models L-moments : properties Denition Sample : X1,...,Xn real random variable of common distribution function F λr = 1 r r−1 k=0 (−1)k r − 1 k E[Xr−k:r] X1:n ≤ X2:n ≤ ... ≤ Xn:n : order statistics λ1 = E[X] : measure of location λ2 = E[X2:2 − X1:2] : measure of scale λ3 λ2 = E[X3:3−2X2:3+X1:3] E[X2:2−X1:2] : measure of skewness λ4 λ2 = E[X4:4−3X3:4+3X2:4−X1:4] E[X2:2−X1:2] : measure of kurtosis Existence since |x|dF(x) < ∞ Estimation under L-moment condition models L-moments : properties Denition (continuous distributions) If F is continuous : E[Xj:r] = r! (j − 1)!(r − j)! R xF(x)j−1 (1 − F(x))r−jdF(x) L-moments can then be written : λr = 1 0 F−1 (t)Lr(t)dt = 1 0 F−1 (t)dKr(t) with Lr shifted Legendre polynomials (orthogonal basis in L2([0, 1])) Lr(t) = r k=0 (−1)k r k 2 tr−k(1−t)k = r k=0 (−1)r−k r k r + k k tk Kr(t) = t 0 Lr−1(u)du Estimation under L-moment condition models L-moments : properties Denition (discrete distributions) L-moments for a multinomial of support x1 ≤ x2 ≤ ... ≤ xn and associated weights π1, ..., πn ( n i=1 πi = 1) λr = n i=1 w(r) i xi = n i=1 Kr i a=1 πa − Kr i−1 a=1 πa xi Estimation under L-moment condition models L-moments : properties Estimation of L-moments Unbiased estimator : U-statistics l(u) r = 1 n r 1≤i1<···xi be the empirical distribution for a sample x1, ..., xn. The plug-in estimator is then : ˆθ (0) n = arg inf θ∈Θ inf G∈M(0) θ (Fn ) Dϕ(Fn, G) But ; existence? quick computation? Estimation under L-moment condition models Minimum of ϕ-divergence estimators Minimum of ϕ-divergence : illustration Estimation under L-moment condition models Minimum of ϕ-divergence estimators Projection for the L-moment constraint model We choose to minimize the divergence between quantile measure in order to obtain linearity The model become Mθ(F) = {G ∈ M+|G−1 F−1 , 1 0 K(u)dG−1 (u) = f (θ)} and the estimator ˆθn = arg inf θ∈Θ inf G∈Mθ(Fn ) 1 0 ϕ dG−1 dF−1 n dF−1 n (u) Estimation under L-moment condition models Minimum of ϕ-divergence estimators Information of the quantile density The quantile density is a measure of sparsity Example : Weibull family f (x) = ν σ x σ ν−1 e−(x/σ)ν q(u) = Q (u) = σ ν(1−u) (− ln(1 − u))1/ν−1 Estimation under L-moment condition models Minimum of ϕ-divergence estimators Relation with transport Let Xn = {x1, . . . , xn} and µn the Lebesgue measure on Xn. If now Tn(x) = G−1 ◦ Fn(x) for x ∈ Xn : inf R K(Fn (x))dTn (x)=f (θ) Xn ϕ dTn dµn dµn(x) = inf1 0 K(u)dG−1(u)=f (θ) 1 0 ϕ dG−1 dF−1 n dF−1 n (u) Tn is a transport between the random variable X of distribution function Fn and Y of distribution function G : Tn(X) d = Y Estimation under L-moment condition models Minimum of ϕ-divergence estimators Dual representation Fenchel-Legendre transform of ϕ ∀t ∈ R, ψ(t) = sup x∈R {tx − ϕ(x)} Let µ be the Lebesgue measure Proposition Let θ ∈ Θ and F be xed. If there exists some T such that K(F(x))dT(x) = f (θ) and dT dµ ∈ int(dom(ϕ)) µ-a.s. inf R K(F(x))dT(x)=f (θ) ϕ dT dµ dµ = supξ∈Rl ξT f (θ) − R ψ(ξT K(F(x)))dµ Estimation under L-moment condition models Minimum of ϕ-divergence estimators Dual representation If ψ is derivable and there exists a solution ξ∗ of the dual problem which is an interior point of {ξ ∈ Rl s.t. R ψ(< ξ, K(F(x)) >)dµ < ∞}, then ξ∗ is the unique maximum checking : ψ (ξ∗T K ◦ F(x))K ◦ F(x)dµ = f (θ) and θ → ξ∗(θ) is continuous. Estimation under L-moment condition models Minimum of ϕ-divergence estimators Dual representation for χ2 -divergence With the χ2 -divergence ϕ(x) = (x−1)2 2 , ψ(t) = 1 2 t2 + t The solution ξ∗ of the dual problem is ξ∗ = Ω−1 f (θ) − K(F(x))dµ with Ω = K(F(x))K(F(x))T dµ And the estimator is then ˆθn = arg inf θ∈Θ f (θ) − K(Fn(x))dµ Ω−1 n f (θ) − K(Fn(x))dµ with Ωn = K(Fn(x))K(Fn(x))T dµ Estimation under L-moment condition models Asymptotic properties Asymptotic properties of the estimators under the model Theorem Let X1, ..., Xn be random samples coming from the same distribution F0. Let suppose that there exists θ0 such that F0 ∈ Mθ0 , θ0 is the unique solution of the equation f (θ) = f (θ0) f is continuous and Θ is compact the matrix Ω = K(F0(x))K(F0(x))T dx is non singular. Then with probability approaching one, ˆθn p → θ0 Estimation under L-moment condition models Asymptotic properties Asymptotic properties of the estimators under the model Asymptotic normality of the estimator : Theorem Let dene J0 = Jf (θ0) be the Jacobian of f with respect to θ in θ0 M = (JT 0 Ω−1 J0)−1 , H = MJT 0 Ω−1 , P = Ω−1 − Ω−1 J0MJT 0 Ω−1 Σ = [F0(min(x, y)) − F0(x)F0(y)]K (F0(x)).K (F0(y))dxdy Then, √ n ˆθn − θ0 ˆξn d → N(0, diag(HΣHT , PΣPT )) Estimation under L-moment condition models Asymptotic properties Further work Same asymptotic properties under misspecication Extension to multivariate case Estimation under L-moment condition models Asymptotic properties Radar power data Estimation under L-moment condition models Asymptotic properties Thermal noise vs impulsive noise Estimation under L-moment condition models Asymptotic properties Processing chain for adaptive detection H0 (no target) H1 (target) Decision D=0 : no target Noise (1 − Pfa) Miss (1 − Pdet) Decision D=1 : tar- get False alarm (Pfa) Detection (Pdet) Constraint : keep the false alarm constant with robustness to misspecication other targets Aim : estimation of H0 distribution