Sobre la incorrecta delimitación del mercado afectado

2.2.1 Problem setting

Let Φ be a continuous map defined on a subset X of Rd with values in Rk. We note B(X ) the Borel σ-field of X and F (X ) the set of finite measures on X . Let µ0 ∈ F (X ) be an unknown measure satisfying the constraint y =R Φdµ0. Assume we observe a perturbed version yobs of y:

yobs = Z

X

2.2. THE AMEM ESTIMATE 33 where ε is an error term supposed bounded in norm from above by some positive constant η, representing the maximal noise level. Based on the data yobs, we aim at reconstructing the measure µ0 with a maximum entropy procedure. In image analysis this measure may be viewed as the intensity at each pixel of the image, blurred by an unknown filter. Other applications in seismic tomography can be found in [FLLn06], while we discuss an application to Econometry in Section 2.3.2.

For two probability measures ν, µ, we recall that the relative entropy of ν with respect to µ is given by K(ν|µ) = Z X log dν dµ  dν + µ(X ) − ν(X ) if ν  µ, K(ν|µ) = +∞ otherwise.

We denote by KY the closed ball of Rk centered at the observation yobs and of radius η. The true measure µ0 is known to satisfy the moment condition R Φdµ0 ∈ KY, however, the map Φ being unknown, we consider the approximate moment condition

Z

X

Φm(x)dµ0(x) ∈ KY. (2.2)

Moreover, we note Mm = {µ ∈ F (X ) : R Φmdµ} the set of finite measures satisfying the approximate moment condition. Let us now explain the construction of the AMEM estimator. Let X1, . . . , Xn be a discretization of the space X , for which the associated empirical measure Pn = _n1Pn_i=1δXi is assumed to converge weakly to some distribution PX having full support

on X . The Xi’s may be i.i.d. realizations of a random variable X with distribution PX, or a deterministic design, in which case PX is known by the statistician. We search for an estimator of µ0 that can be written as a weighted version of the empirical measure Pn

Pn(w) := 1 n n X i=1 wiδXi,

for some vector w = (w1, ..., wn)t∈ Rn. Moreover, we want the estimator to satisfy to approxi- mate moment condition. Let W = (W1, ..., Wn)t be a vector of n i.i.d. realizations drawn from a measure ν and consider the random measure Pn(W ). Seeing each weighted measure Pn(w) as a realization of Pn(W ), the measure ν⊗n can be interpreted as a prior distribution on the parameter w. The posterior distribution ν∗ is defined as the probability measure minimizing the relative entropy K(.|ν⊗n) under the constraint that the approximate moment condition (2.2) holds in mean,

Eν∗[P_n(W )] ∈ M_m.

The estimator ˆµm,n is obtained as the expectation of Pn(W ) under ν∗, ˆ µm,n = Eν∗[P_n(W )] = 1 n n X i=1 Eν∗(W_i)δ_X i.

The existence of ν∗ requires the feasibility of the problem, i.e. the existence of a vector δ in the convex hull of the support of ν⊗n such that R ΦmdPn(δ) ∈ KY. It is shown in [LP08] that

34 CHAPTER 2. QUADRATIC APPROXIMATE MAXIMUM ENTROPY ON THE MEAN under the Assumptions of Theorem 2.2.1, this condition tends to be verified with probability 1 as m → ∞ and n → ∞. Hence for m and n large enough, the AMEM estimate ˆµm,n is well defined with high probability, and asymptotically with probability 1.

2.2.2 Convergence of the AMEM estimate

We recall that for ν a probability measure on R, Λν and Λ∗ν denote respectively the log- Laplace and Cramer transforms of ν, given by

Λν(s) = log Z R esxdν(x) and Λ∗_ν(s) = sup u∈R {su − Λν(u)}, s ∈ R. We define the functional

µ 7→ Iν(µ|PX) = Z X Λ∗_ν  dµ dPX  dPX if µ  PX, Iν(µ|PX) = +∞ otherwise,

which is the f-divergence of µ with respect to PX associated to the convex function Λ∗ν. We note Cb the set of continuous bounded functions on X . For all g ∈ Cb, we denote by | . |g the semi-norm defined for µ ∈ F (X ) by |µ|g =

 R gdµ

. We recall that the family of semi-norms {| . |_g, g ∈ Cb} defines the weak topology: a sequence {µn}n∈N converges weakly toward µ if, and only if, limn→∞|µn− µ|g = 0, for all g ∈ Cb.

We make the following assumptions.

A2.1. The minimization problem is feasible, i.e., there exists a continuous function g0 defined on the convex hull of the support of ν such that R Φg0 dPX ∈ KY.

A2.2. The function Λ00_ν is bounded by a constant K > 0.

A2.3. The approximating sequence Φm converges to Φ in L2(PX). Its rate of convergence is given by

kΦm− Φk_L2 :=

p

EkΦm(X) − Φ(X)k2 = O(ϕ−1m ), for some growing sequence {ϕm}m∈N.

A2.4. The function G : x 7→ sup_m∈NkΦm(x)k is square integrable: R G2dPX < ∞. A2.5. For all m ∈ N, the components of Φm are linearly independent.

We are now in a position to state our main result.

Theorem 2.2.1 (Convergence of the AMEM estimate) Suppose that A2.1 and A2.2 hold and let µ∗be the minimizer of the functional µ 7→ Iν(µ|PX) subject to the constraintR Φdµ ∈ KY.

• The AMEM estimate ˆµm,n is given by

dˆµm,n(x) = Λ0ν(ˆvtm,nΦm(x))dPn(x), where ˆvm,n minimizes over Rk, Hm,n(v) = PnΛν(vtΦm) − infy∈KY v

2.3. APPLICATIONS 35