D G Zheng

F J Efimov

W Bejarano

Perruquetti

On the interval estimation for nonlinear singular system

This paper investigates the interval observer design problem for a class of nonlinear singular systems with uncertainties in the state and in the output. Although the asymptotic estimation of the state for this class of systems may be not possible, it has been shown that, under suitable assumptions, an interval observer can be designed to provide the upper and the lower estimations of the real state. Moreover, this technique can be applied to unobservable nonlinear singular systems to obtain the interval estimation. The proposed result can be easily extended to deal with the nonlinear singular systems with parameter uncertainty as well.

I. INTRODUCTION

Singular system approach was introduced to model a large class of systems [START_REF] Campbell | Singular systems of differential equations II[END_REF], [START_REF] Campbell | Singular systems of differential equations I[END_REF]. In [START_REF] Yip | Solvability, controllability, and observability of continuous descriptor systems[END_REF], the authors have studied the solvability, controllability and observability concepts for singular systems with regular matrix pencil. The algebraic duality between controllability and observability for singular systems with regular matrix pencil is proven by using the Schwartz distribution framework in [START_REF] Cobb | Controllability, observability, and duality in singular systems[END_REF]. In [START_REF] Hou | Casual observability of descriptor systems[END_REF], the concept of casual observability was proposed for singular systems. The strong observability and strong detectability of a general class of singular linear systems with unknown inputs are recently tackled in [START_REF] Bejarano | Observability and detectability analysis of singular linear systems with unknown inputs[END_REF], [START_REF] Bejarano | Observability and detectability of singular linear systems with unknown inputs[END_REF] by converting the singular system into a regular one with unknown inputs and algebraic constraints. The observability problem for nonlinear singular system has been treated in [START_REF] Bejarano | State reconstruction of nonlinear differential-algebraic systems with unknown inputs[END_REF], [START_REF] Bejarano | Observation of nonlinear differential-algebraic systems with unknown inputs[END_REF].

Concerning the observer design, a Luenberger-like observer has been proposed in [START_REF] Paraskevopoulos | Observers for singular systems[END_REF] for linear singular systems. [START_REF] Darouach | Design of observers for descriptor systems[END_REF] gave necessary and sufficient condition for the existence of a reduced order observer for the linear singular systems with known inputs, and the result was extended to treat the linear singular systems with unknown inputs in [START_REF] Darouach | Reduced-order observer design for descriptor systems with unknown inputs[END_REF]. In [START_REF] Hou | Observer design for descriptor systems[END_REF], a generalized observer was studied by involving the derivative of input and output. For the nonlinear singular system, [START_REF] Kaprielian | An observer for a nonlinear descriptor system[END_REF] studied an observer for a class of nonlinear singular systems in which the system was linearized around the equilibrium point. The same technique was used in [START_REF] Boutayeb | Observers design for non linear descriptor systems[END_REF] to study a reduced order observer for a class of nonlinear singular systems. And recently, the technique of regularization by applying the geometrical differential method to the nonlinear singular systems was introduced in [START_REF] Boutat | Observers design for a class of nonlinear singular systems[END_REF]. Gang Most of the cited references are for the asymptotic estimation of the state for singular systems without uncertainties. The observer design becomes complicated when considering the systems with uncertain terms in the state and in the measurement. In this situation, the exact estimation may be not possible, and one solution is to provide the upper and lower bound estimation of the admissible values for the state by applying the theory of set-membership or interval estimation [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF], [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF], [START_REF] Perruquetti | On practical stability with the settling time via vector norms[END_REF], [START_REF] Perruquetti | Connecting wasewskis conditions with mmatrices: Application to constrained stabilization[END_REF] Concerning the interval observer design for singular system, the only work existing in the literature is [START_REF] Efimov | Interval estimation for systems with time delays and algebraic constraints[END_REF], where the problem of interval observer design is addressed for a class of linear singular systems with delays. In this work, the studied singular system was decoupled into two parts x = (x 1 , x 2 ) T : the dynamical one and the algebraic one. The authors then proposed an interval observer by restrictively assuming that x 2 can be fully described as the function of x 1 . Thus the algebraic equation can be removed and the studied singular system is equivalent to a regular one. Without imposing this restrictive condition, this paper proposes a method to design an interval observer for a class of nonlinear singular systems with uncertainties. By imposing the observability rank condition for the linear part and assuming some boundedness properties of the studied system, this paper shows that an interval observer for this class of uncertain nonlinear singular systems always exists, which provides the upper and the lower estimations of the system's state.

In this paper, the following notation is used. R denotes the set of real numbers and R + = {x ∈ R : x ≥ 0}. aRb represents the element-wise relation R (a and b are vectors or matrices): for example a < b (vectors) means ∀i : a i < b i . For a matrix A ∈ R m×n , define A + = max{0, A }1 and A -= A + -A . For a vector x ∈ R n , define x + = max{0, x} and x -= x +x. For a matrix (function) A the symbol A i denotes its i th column, for a vector (function) b the symbol b i denotes its corresponding element. A matrix A ∈ R n×n is called Metzler if all its elements outside the main diagonal are nonnegative. Due to the limitation of pages, the proofs of the results are omitted.

II. PROBLEM STATEMENT AND BACKGROUNDS

A. Problem statement

Consider the following uncertain nonlinear singular system:

Σ ξ : Ē ξ = Āξ + f (ξ , u) + v(t) y = Cξ + w(t) (1)
where ξ ∈ R n is the state whose initial value belongs to a compact set I 0 (ξ (t 0 )) = [ξ (t 0 ), ξ (t 0 )]; y ∈ R p and u ∈ R m are respectively the output and the input. Ē ∈ R n×n , Ā ∈ R n×n , C ∈ R p×n , and the vector field f represents the nonlinear term with the appropriate dimension. w(t) and v(t) are the disturbance in the output and in the model, respectively. When the matrix Ē is nonsingular, then (1) can be written as:

ξ = Ē-1 Āξ + Ē-1 f (ξ , u) + Ē-1 v(t) y = Cξ + w(t) (2)
which becomes a classical regular system. We can cite lots of existing methods to design an observer for the system without the uncertainties, if it is observable. For the case with uncertainties in the state and in the output, an interval observer was proposed in [START_REF] Raïssi | Interval state estimation for a class of nonlinear systems[END_REF] for the observable system (2) to give the upper and lower estimation of the real state with the only requirement of the uncertainty boundedness. This method is extended in [START_REF] Zheng | Interval state estimation for uncertain nonlinear systems[END_REF] to treat the case even if the system is not observable. When the matrix Ē is singular, (1) represents a large class of nonlinear singular systems with uncertainties in the state and in the output. This paper is devoted to designing an interval observer for this larger class of uncertain nonlinear singular systems. Before this, let us firstly recall the basic background on comparison systems which will be used for interval observer design.

III. ASSUMPTIONS AND PRELIMINARY RESULTS

Concerning the observability for singular systems, there exist several different definitions in the literature, including observability, R-observability and Impulse-observability [START_REF] Yip | Solvability, controllability, and observability of continuous descriptor systems[END_REF], [START_REF] Dai | Singular Control Systems[END_REF]. Generally speaking, they characterize the state reconstruction ability from different aspects: R-observability defines the ability to estimate the reachable set of the studied system; Impulse-observability corresponds to the ability to estimate the impulse term of the studied system and the observability covers both mentioned abilities to estimate all states of the studied system.

In this paper, for the uncertain nonlinear singular system Σ ξ , we are interested in the interval estimation of the (non-impulsive) trajectory of the state ξ (t) with the known information of y(t) and u(t) for t > 0. Therefore, this paper considers the following definition of observability adopted from Definition 1 in [START_REF] Bejarano | Observability and detectability of singular linear systems with unknown inputs[END_REF].

Definition 1: System Σ ξ is observable if y(ξ 1 ,t) = y(ξ 2 ,t) for all ξ 1 , ξ 2 ∈ R n and t > 0 implies ξ 1 (0 + ) = ξ 2 (0 + ).
In what follows, we make the following assumption for the original system Σ ξ .

Assumption 1: For the triple ( Ē, Ā, C) of system Σ ξ defined in [START_REF] Bejarano | Observability and detectability of singular linear systems with unknown inputs[END_REF], it is assumed that the following rank conditions:

rank Ē C = n (3) and rank s Ē - Ā C = n, ∀s ∈ C (4)
are satisfied.

It is obvious that, even if the rank conditions ( 3) and ( 4) are assumed to be satisfied, the studied uncertain nonlinear singular system Σ ξ might not be observable, due to the nonlinear term f (ξ , u) and the uncertainties (v, w). And this motivates the study of the interval estimation treated in this paper. Without loss of generalities, for the singular matrix Ē ∈ R n×n , it is assumed that rank Ē = q < n. Due to the rank condition (3), there exists a non-singular matrix

P = P 1 P 2 P 3 P 4 ∈ R (p+n)×(p+n) such that P Ē C = I n 0 ,
which is equivalent to:

P 1 Ē + P 2 C = I n P 3 Ē + P 4 C = 0 (5)
with P 1 ∈ R n×n , P 2 ∈ R n×p , P 3 ∈ R p×n and P 4 ∈ R p×p . Based on the above result, we have the following lemma. Lemma 1: Suppose Assumption 1 is satisfied for the triple ( Ē, Ā, C) of Σ ξ . Then there always exist matrices K ∈ R n×p , L ∈ R n×p and an invertible matrix Q ∈ R n×n such that the following matrix:

R = QNQ -1 (6)
is Hurwitz and Metzler, where

N = (P 1 + KP 3 ) Ā + L C, (7)
with P 1 and P 3 being defined in [START_REF] Boutat | Observers design for a class of nonlinear singular systems[END_REF].

With the above deduced matrix Q which transforms the Hurwitz matrix N to the Hurwitz and Metzler matrix R, since it is non-singular, then we can choose x = Qξ as a diffeomorphism, with which system Σ ξ in (1) can be rewritten as follows:

Σ x : E ẋ = Ax + f (x, u) + v(t) y = Cx + w(t) (8)
where

E = ĒQ -1 , A = ĀQ -1 , C = CQ -1 and f (x, u) = f (Q -1
x, u). For the transformed system (8), we can state the following lemma. Lemma 2: Suppose Assumption 1 is satisfied for the triple ( Ē, Ā, C) of Σ ξ . Then for the transformed system Σ x (E, A,C) defined in [START_REF] Campbell | Singular systems of differential equations II[END_REF], there always exist a matrix K ∈ R n×p , two invertible matrices P = P 1 P 2 P 3 P 4 ∈ R (p+n×(p+n) and Q ∈ R n×n , such that the following equality is satisfied:

QP 24 C + QP 13 E = I n
where the matrices P 13 and P 24 are defined as follows:

P 13 = P 1 + KP 3 P 24 = P 2 + KP 4 (9)
Since the initial condition ξ 0 for (1) is supposed to be located into a certain interval I(ξ 0 ) = [ξ 0 , ξ 0 ], then using the diffeomorphism x = Qξ , the initial condition x 0 is also known within a certain interval I(x 0 ) = [x 0 , x 0 ]. Thus the interval estimation of ξ in (1) is equivalent to estimate the interval of x in (8) by using the knowledge of (u, y).

IV. MAIN RESULTS

This section will firstly analyze the boundedness properties by making some assumptions for (1), then propose an interval observer to estimate the state.

A. Interval estimation

Prior to introduce the interval observer for (1), let us firstly make the following assumption which is necessary in the sequel.

Assumption 2: For the studied system (1), it is assumed that:

1) the state ξ (t) is bounded under the bounded input u(t) ∈ C ∞ , i.e. ξ (t) ∈ Ω ⊂ R n for a given u(t) ∈ U ⊂ R m for all t ≥ 0 where Ω and U are two given compact sets; 2) the function f (ξ , u) for all ξ ∈ Ω and u ∈ U is locally Lipschtiz w.r.t (ξ , u); 3) the disturbances w(t) and v(t) are bounded, and the derivative of w(t) is bounded for all t ≥ 0, i.e. there exist constants v, v, w, w, w d , w d such that v ≤ v(t) ≤ v, w ≤ w(t) ≤ w, and w d ≤ ẇ(t) ≤ w d . Remark 1: Since system (8) was transformed from the system (1) by applying the diffeomorphism x = Qξ : Ω → X , therefore if Assumption 2 is fulfilled for (1), then the state x(t) of ( 8) is bounded as well for the bounded input u(t), i.e. x(t) ∈ X ⊂ R n for all t ≥ 0 where X represents the compact set defined via the diffeomorphism. Moreover, since 8), thus it is locally Lipschtiz w.r.t (x, u) for all x ∈ X and u ∈ U .

f (x, u) = f (Q -1 x, u) in (
As we have showed in Section III, if Assumption 1 and Assumption 2 are satisfied, then there exist the matrices K, L, and the non-singular matrices P and Q such that the matrix N defined in [START_REF] Campbell | Singular systems of differential equations I[END_REF] is Hurwitz, and the matrix R defined in [START_REF] Boutayeb | Observers design for non linear descriptor systems[END_REF] is Hurwitz and Metzler. For the sake of simplicity, note

Q P 13 = QP 13 Q P 24 = QP 24 Q L = QL
where P 13 and P 24 are defined in [START_REF] Cobb | Controllability, observability, and duality in singular systems[END_REF]. Then one can design the following two dynamics:

ż = Rz + (RQ P 24 -Q L ) y + ∆(z, z, u) x = z + Q P 24 y (10) and ż = Rz + (RQ P 24 -Q L ) y + ∆(z, z, u) x = z + Q P 24 y (11) with ∆(z, z, u) = ∆ f (z, z, u) + ∆ v + ∆ w + ∆ w d ∆ f (z, z, u) = Q + P 13 f (z, z, u) -Q - P 13 f (z, z, u) ∆ v = Q + P 13 v -Q - P 13 v ∆ w = Q + L w -Q - L w ∆ w d = Q - P 24 w d -Q + P 24 w d and ∆(z, z, u) = ∆ f (z, z, u) + ∆ v + ∆ w + ∆ w d ∆ f (z, z, u) = Q + P 13 f (z, z, u) -Q - P 13 f (z, z, u) ∆ v = Q + L v -Q - L v ∆ w = Q + L w -Q - L w ∆ w d = Q - P 24 w d -Q + P 24 w d By noting Γ(x, x, x, u) = ∆(z, z, u) -Q P 13 f (x, u) -Q P 13 v(t) -Q L w(t) + Q P 24 ẇ(t) (12)
and Γ(x, x, x, u) = -∆(z, z, u)

+ Q P 13 f (x, u) + Q P 13 v(t) +Q L w(x, u) -Q P 24 ẇ(t) (13)
we have the following corollary. Corollary 1: For Γ(x, x, x, u) and Γ(x, x, x, u) defined in ( 12) and ( 13) with x ∈ X and u ∈ U , we have Γ(x, x, x, u) ≥ 0 and Γ(x, x, x, u) ≥ 0 Moreover, there exist positive constants l Γ , l Γ , l Γ , l Γ and l Γ , l Γ such that for a chosen submultiplicative norm • the following inequalities:

Γ(x, x, x, u) ≤ l Γ x -x + l Γ x -x + l Γ Γ(x, x, x, u) ≤ l Γ x -x + l Γ x -x + l Γ
are satisfied for all t ≥ 0. Theorem 1: Suppose Assumption 1 and Assumption 2 are satisfied. Then for any initial state x(t 0 ) of ( 8) belongs to a certain interval I(x(t 0 )) = [x(t 0 ), x(t 0 )], systems ( 10) and ( 11) form an interval observer for [START_REF] Campbell | Singular systems of differential equations II[END_REF] such that the following inequality:

x(t) ≤ x(t) ≤ x(t), (14)
holds for all t ≥ t 0 . Mover, if there exist positive definite symmetric matrices S, M and a positive scalar µ such that the following LMI is satisfied:

R T S + SR + α µ I + M S S -1 µ I 0 ( 15
)
where

R = diag{R, R}, α = 2 max{l 2 Γ , l 2 Γ , l 2 Γ , l 2 Γ , l 2 Γ , l 2
Γ }, then the variables x(t) and x(t) are bounded for all t ≥ 0.

Corollary 2: For system (8) with the initial state ξ (t 0 ) ∈ [ξ (t 0 ), ξ (t 0 )], if Assumption 1 and Assumption 2 are satisfied, then there exists a non-singular matrix Q such that the interval estimation of ξ in (1) is given as follows:

Q -1 + x -Q -1 -x ≤ ξ ≤ Q -1 + x -Q -1 -x (16)
where x and x are the states of the proposed interval observer defined in [START_REF] Dai | Singular Control Systems[END_REF] and [START_REF] Darouach | Design of observers for descriptor systems[END_REF].

Let us remark that Assumption 2 imposed the state boundedness ξ (t) ∈ Ω ⊂ R n under a given bounded input u(t) ∈ U ⊂ R m for all t ≥ 0, and this property is sometimes difficult to be checked for general uncertain nonlinear singular systems, therefore the proposed result could not be applied. However, if the boundedness property of the state in Assumption 2 is satisfied only for t ∈ [0, T ] with T being a finite time, i.e. ξ (t) ∈ Ω ⊂ R n under a given bounded input u(t) ∈ U ⊂ R m for all t ∈ [0, T ], then Theorem 1 is still valid during this finite time T . Therefore the proposed result can be relaxed by the following corollary.

Corollary 3: Suppose Assumption 1 and Assumption 2 are valid only for t ∈ [0, T ] with T being a finite time. Then for any initial state x(t 0 ) of ( 8) belongs to a certain interval [START_REF] Dai | Singular Control Systems[END_REF] and [START_REF] Darouach | Design of observers for descriptor systems[END_REF] give an interval estimation of x(t) of (8) for t ∈ [t 0 , T ], i.e. x(t) ≤ x(t) ≤ x(t) for all t ∈ [t 0 , T ]. And the interval estimation of ξ (t) of ( 1) for all t ∈ [t 0 , T ] can be still obtained by [START_REF] Hou | Observer design for descriptor systems[END_REF].

I(x(t 0 )) = [x(t 0 ), x(t 0 )], systems

B. Design procedure

For system Σ ξ Ē, Ā, C defined in (1), supposed that Assumption 1 and Assumption 2 are both satisfied for all t ≥ 0 (or for all t ∈ [0, T ]), then the interval estimation of the state ξ for (1) can be obtained via the following procedure: Step1: Due to the satisfaction of the rank condition (3) in Assumption 1, determine an invertible matrix P = P 1 P 2 P 3 P 4 such that P Ē C = I n 0 ;

Step2: Due to the satisfaction of the rank condition (4) in Assumption 1, choose the matrices K, L such that

N = L C + (P 1 + KP 3 ) Ā is Hurwitz; Step3: Choose an invertible matrix Q such that R = QNQ -1
is Hurwitz and Metzler; Step4: Transform system Σξ Ē, Ā, C into Σ x (E, A,C) by applying the diffeomorphism x = Qξ ; Step5: If Assumption 2 is satisfied for all t ≥ 0 (or for all t ∈ [0, T ]), calculate the bound of the function f in (8); Step6: Design interval observer [START_REF] Dai | Singular Control Systems[END_REF] and [START_REF] Darouach | Design of observers for descriptor systems[END_REF] for Σ x (E, A,C), which yields x(t) ≤ x(t) ≤ x(t) for all t ≥ 0 (or for all t ∈ [0, T ]); Step7: Finally we obtain the interval estimation of ξ (t) for all t ≥ 0 (or for all t ∈ [0, T ]):

Q -1 + x -Q -1 -x ≤ ξ ≤ Q -1 + x -Q -1 -x.
V. ILLUSTRATIVE EXAMPLE Let us consider the following uncertain nonlinear singular system:

Ē ξ = Āξ + f (ξ , u) + v(t) y = Cξ + w(t) (17)
with to check that the studied uncertain nonlinear singular system may b not observable with respect to the output y, thus an asymptotic observer could not be designed. The following applies the proposed method to design an interval observer to get the upper and the lower estimations of the state.

Ē =   0 1 0 0 0 1 0 0 0   , Ā =   20 0 0 1 8 0 1 0 1   and C =   1 0 0   T f (ξ , u) =    sin( ξ 3 32 ) u sin( 4ξ 2 -ξ 3 240 ) -0.75 sin(
Step

P Ē C = I 3×3 0 .
Step 2: Therefore, we can choose:

K = [0, -1, -6] T , L = [-2, -19, 5] T such that N = L C + (P 1 + KP 3 ) Ā =   -2 0 0 0 0 -1 0 8 -6   with negative eigenvalues (-2, -4, -2).
Step 3: Hence, we can find the following invertible matrix:

Q =   1 0 0 0 1 -0.25 0 0 1   such that R = QNQ -1 =   -2 0 0 0 -2 0 0 8 -4 
 which is Hurwitz and Metzler.

Step 4: So, with the transformation x = Qξ , system [START_REF] Kaprielian | An observer for a nonlinear descriptor system[END_REF] can be written as:

E ẋ = Ax + f (x, u) + v(t) y = Cx + w(t) (18)
with

E = ĒQ -1 =   0 1 0.25 0 0 1 0 0 0   , A = ĀQ -1 =   20 0 0 1 8 2 1 0 1   , C = CQ -1 = 1 0 0 and f (x, u) = f (Q -1 x, u) =   sin( x 3 32 ) u sin( x 2 60 ) -0.75 sin( x 2 x 3 30 )   .
Step 5: Let us now compute the bounding function for f (x, u) in [START_REF] Paraskevopoulos | Observers for singular systems[END_REF]. To this end, define the following two functions:

Product(x, x) = min{x 2 x 3 , x 2 x 3 , x 2 x 3 , x 2 x 3 } max{x 2 x 3 , x 2 x 3 , x 2 x 3 , x 2 x 3 } , sin(x, x) sin(x, x) = sin(x) sin(x)
corresponding to the interval of the product x 2 x 3 for x = [x 2 x 3 ] T with x ≤ x ≤ x and the interval of the function sin(x) for a scalar x with x ≤ x ≤ x (for |x| ≤ π/2). Then we have the following bounding functions for f :

f 1 (x, x) = sin(x/32, x/32) f 1 (x, x) = sin(x/32, x/32) f 2 (x, x) f 2 (x, x) = Product u u , sin(x/60, x/60) sin(x/60, x/60) f 3 (x, x)
= -0.75sin(Product(x/30, x/30)) f 3 (x, x) = -0.75sin(Product(x/30, x/30))

Take l Γ = l Γ = l Γ = l Γ = l Γ = l Γ = 1, then we have α = 2. By solving the LMI defined in [START_REF] Hou | Casual observability of descriptor systems[END_REF], we find µ

= 1, S = diag{S 1 , S 1 }, M = diag{M 1 , M 1 } with S 1 =   1.6 0 0 0 3 0.4 0 0.4 0.5   , M 1 =   3.5 0 0 0 4.3 0.2 0 0.2 3.4   ,
thus all conditions of Theorem 1 have been verified.

Step 6: Then we can design the interval observer for the transformed system [START_REF] Paraskevopoulos | Observers for singular systems[END_REF]. The results of the interval estimations of the state x for this system are depicted in Fig. 123, from which we can see that the proposed observer can provide both the upper and the lower estimations of x. Step 7: Then, by applying

Q -1 + x -Q -1 -x ≤ ξ ≤ Q -1 + x -Q -1 -x
we obtain the upper and the lower estimation of the real state ξ for the uncertain nonlinear singular systems [START_REF] Kaprielian | An observer for a nonlinear descriptor system[END_REF] (see Fig. 456).

VI. CONCLUSION

When treating the uncertain nonlinear singular system while the asymptotic estimation is not possible, this paper is devoted to designing an interval observer for the studied uncertain system. By imposing the rank conditions and assuming the boundedness of the uncertainties (and the derivative of the uncertainty) in the state and in the output, we proved that an interval observer can always be synthesized to provide the upper and the lower estimations of the real state. The efficiency of the proposed method is highlighted by an illustrative example.

4ξ 2 ξ 3 

 3 , w(t) = 0.1 sin(t). It is straightforward

1 :P 3 =

 13 Let us firstly check the rank conditions imposed in Assumption 1. Since rank Ē C = 3 and rank s Ē -Ā C = 3, ∀s ∈ Cthus Assumption 1 is fulfilled. Moreover, the input is chosen as u(t) = 0.5 sin(t) such that Assumption 2 is at least satisfied for t ∈ [0, T ] with T being a finite time (see Fig.1-3 for T = 5).Then one can find an invertible matrix P = 0 0 1 and P 4 = 0 such that

Upper estimation of x 1 Lower estimation of x 1 Fig. 1 .

 111 Fig. 1. The interval estimation for x 1 .

Upper estimation of x 2 Lower estimation of x 2 Fig. 2 .Lower estimation of x 3 Fig. 3 .

 22233 Fig. 2. The interval estimation for x 2 .

1 Upper estimation of ξ 1 Lower estimation of ξ 1

 111

Fig. 4 .

 4 Fig. 4. The interval estimation for ξ 1 .

2 Upper estimation of ξ 2 Lower estimation of ξ 2 Fig. 5 .

 2225 Fig. 5. The interval estimation for ξ 2 .

3 Upper estimation of ξ 3 Lower estimation of ξ 3 Fig. 6 .

 3336 Fig. 6. The interval estimation for ξ 3 .

  Zheng, D, Efimov and Wilfrid Perruquetti are with Project Non-A, INRIA Lille-Nord Europe, 40 avenue Halley and CRIStAL UMR CNRS 9189, Ecole Centrale de Lille, 59650 Villeneuve d'Ascq, France. Francisco Javier Bejarano is with SEPI, ESIME Ticomn, IPN, Av. San Jos Ticomn 600, C.P. 07340, Mexico City, Mexico. This paper was supported by Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council and FEDER through the Contrat de Projets Etat Region (CPER) CIA 2007-2013, and also supported by ARCIR Project ESTIREZ, Nord-Pas de Calais Regional Council, France. F.J. Bejarnano acknowledges as well the support of Proyecto SIP 20151040, Mexico.

It means each element of A + is the maximum value between 0 and the maximum value of A at the same position.


Mário S Alvim

Miguel E Andrés

Catuscia Palamidessi

Entropy and Attack Models in Information Flow

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

In recent years, there has been a growing interest in considering the quantitative aspects of Information Flow, partly because often the a priori knowledge of the secret information can be represented by a probability distribution, and partly because the mechanisms to protect the information may use randomization to obfuscate the relation between the secrets and the observables.

Several works in literature use an Information Theoretic approach to model the problem and define the leakage in a quantitative way, see for example [START_REF] Zhu | Anonymity vs. information leakage in anonymity systems[END_REF][START_REF] Clark | Quantitative information flow, relations and polymorphic types[END_REF][START_REF] Malacaria | Assessing security threats of looping constructs[END_REF][START_REF] Malacaria | Lagrange multipliers and maximum information leakage in different observational models[END_REF][START_REF] Moskowitz | Quasi-anonymous channels[END_REF][START_REF] Moskowitz | Covert channels and anonymizing networks[END_REF][START_REF] Chatzikokolakis | Anonymity protocols as noisy channels[END_REF]. The idea is that the system is seen as a channel. The input represents the secret, the output represents the observable, and the correlation between the input and output (mutual information) represents the information leakage. The worst case leakage corresponds then to the capacity of the channel, which is by definition the maximum mutual information that can be obtained by varying the input distribution.

In the works mentioned above, the notion of mutual information is based on Shannon entropy, which (because of its mathematical properties) is the most established measure of uncertainty. From the security point of view, this measure corresponds to a particular model of attack and a particular way of estimating the security threat (vulnerability of the secret). Other notions have been considered, and argued to be more appropriate for security in certain scenarios. These include: Rényi min-entropy [START_REF] Cachin | Entropy Measures and Unconditional Security in Cryptography[END_REF][START_REF] Smith | On the foundations of quantitative information flow[END_REF], Bayes risk [START_REF] Chatzikokolakis | On the Bayes risk in information-hiding protocols[END_REF], guessing entropy [START_REF] Massey | Guessing and entropy[END_REF], and marginal guesswork [START_REF] Pliam | On the incomparability of entropy and marginal guesswork in brute-force attacks[END_REF]. Köpf and Basin discuss the relation between bruteforce guessing attacks and entropy in [START_REF] Köpf | An information-theoretic model for adaptive side-channel attacks[END_REF], in the context of information flow induced by a deterministic program, and define the information leakage as difference between the input entropy and the conditional one, namely the entropy based on the a priori input distribution, and the entropy of the a posteriori distribution (i.e. after observing teh output), respectively. One of their main results is that, in their framework, the notion of leakage under the various notions of attacks considered in their paper is always non-negative.

In this talk, we extend the analysis of Köpf and Basin to the probabilistic scenario, and we consider also other notions of entropy, including the family of entropies proposed by Rényi [START_REF] Rényi | On Measures of Entropy and Information[END_REF]. We argue that in the probabilistic case the notion of information leakage needs to be revised. In fact, when the same secret can give different observables (according to a probability distribution), the difference between a priori and a posteriori entropy may be negative. This is due to the fact that the notion of entropy uses the probability distribution in two different ways: for averaging and for representing the belief of the attacker. While the leakage should depend on the difference induced by the belief change due to the observation, the averaging probability should remain the same. (A similar concern has also inspired the works of [START_REF] Michael | Belief in information flow[END_REF] and [START_REF] Hamadou | Reconciling belief and vulnerability in information flow[END_REF].) In order to avoid the unnatural consequence of a negative leakage, we propose to base the notion of leakage directly on the (more primitive) notion of mutual information. We consider some cases of entropy, in particular the Rényi's entropies, for which the corresponding notion of mutual information has been investigated in [START_REF] Csiszár | Generalized cutoff rates and Rényi's information measures[END_REF], and we show that in this way the property of non-negativeness is ensured.


Jean-Charles Bricongne
email: jean-charles.bricongne@univ-tours.fr

Professors George Borjas

Lionel Fontagné

Harold Demsetz No

Hal Varian No

Francis X Diebold No

James Hamilton No

David N Weil No

Stephen Cecchetti No

Robert Engle No

Robert James Hodrick Yes

Orley Ashenfelter Yes

Kenneth S Rogoff Yes

Jeffrey Marc

Wooldridge Yes

Christopher Sims Yes

Jeremy Stein No

David K Levine No

William D Nordhaus Yes

Robert J Barro No

Janet L Yellen No

Michael L Katz No

Lawrence Christiano No

Charles Engel No

Michael Woodford No

Robert J Shiller No

Bennett Mccallum No

Steven N Durlauf No

John B Taylor Yes

Maurice Obstfeld No

Christopher Ruhm No

Michael C Jensen No

Jeffrey Alexander Frankel No

Robert Gibbons Yes

Joshua D Angrist No

Carl Walsh Yes

Lawrence Katz Yes

Ross Levine No

Charles F Manski No

William Poole

Do prizes in economics affect productivity? A mix of motivation and disappointment *

Keywords: awards, diff-in-diff methodology, iterative diff-in-diff, John Bates Clark Medal, productivity

Economists' productivity may evolve after an important award such as the John Bates Clark Medal or the "Nobel Prize". Yet, when establishing this stylized fact through a diff-in-diff methodology by comparing the treatment group performance of award winners with the one of the control group made of other contenders, one should take into account the fact that these may also alter their productivity due to disappointment.

To challenge this hypothesis, using the John Bates Clark Medal as the event likely to impact economists' productivity, two different methods are used. The first one uses a standard diff-in-diff approach, with a treatment group made of American economists of high enough standard, using as a criterion the belonging to the American Econometrics Association, and a control group made of non-American economists of comparable standard, not fulfilling the criteria (being American, or at least established in the US) to be in a position to hope for this reward. This first approach gives mixed results.

The second approach sets an innovative method to build a new treatment and control group. This latest method may be called iterative diff-in-diff: it consists in selecting, into the initial sample for the treatment group, the members of the sample who fit most the expected result, in the case of the present article, the decrease of their productivity. When comparing this sub-sample with the rest of the initial treatment sample, one tries to find a criterion, in line with the initial question raised (in the present case, the fact of attributing a high value to honors, proxied by the position of the honors section in their CV), which differentiates the two sub-samples.

Introduction

Do prizes in economics, such as the John Bates Clark or the Nobel Prize, affect productivity? All outcomes are possible, depending on the factors at play:  An increase in productivity, due to a rise in motivation or in the financing of projects.

 A decrease in productivity, due to the fact that those who were awarded the prize think they have reached their goal and prefer to allocate their time in a different way, for instance, by delivering more speeches, investigating other areas or taking on responsibilities other than research. This decrease in productivity may be incidental, in that the economists had already reached the peak of their productivity and it would have declined anyway. However, assessing this effect is difficult and would suppose that the people awarding the prizes know when the peak of a career has been reached. This is a strong assumption given the information asymmetry between those who grant the prizes and the contenders.

 No change in productivity: even if the awarding of the prize leads to gains in terms of reputation and financing (amount of the prizes, or increased financing of projects), economists do not change their behavior. Concerning the rise in financing, this may be perceived as a temporary shock as opposed to a permanent change, by analogy with the permanent income hypothesis.

The question raised for prize winners also concerns other contenders, since these may be disappointed and then adapt their productivity over time. Thus, one of the goals of this article is to see if the control group used by [START_REF] Chan | Does the John Bates Clark Medal Boost Subsequent Productivity And Citation Success?[END_REF] for instance is adequate, in the sense that not only the economists being awarded the John Bates Clark Medal may have an improved productivity, but also in the sense that economists who failed in obtaining it may be disappointed and see their productivity declining: the magnitude of supplementary productivity between the treatment and the control group may be thus overestimated, because it would be due both to an increased productivity of the former and a lessened productivity of the latter.

The impact of these prizes cannot be assessed directly using dummies alone because many factors (age, year, etc.) can have an influence. We thus need to consider a control group and a diff-in-diff methodology.

The paper is organized as follows: after discussing the construction of the treatment and control group, the existing literature is reviewed. Two kinds of diff-in-diff are then performed: the "usual" method, which is then challenged with an innovative iterative one. The last section concludes.

Methodology: construction of the treatment and control group and need to control for variables such as age

Construction of the treatment and control group

The treatment group is made of the economists who did not receive the John Bates Clark, but who met the conditions to receive it (being American / established in the USA before the age of 40), and with a sufficient level to be able to claim it. The members of the Econometric Society are good candidates to meet this later condition.

To construct the control group, we need economists who have characteristics that are comparable with those of the treatment group. For that purpose, we include economists of the American Econometric Society (this indicator being mixed with the ranking in RePEc in terms of publications), who are not American, and not established in the USA when they were between 35 and 40, which implies a detailed scrutiny of their careers through their CVs, to check this condition. This explains why this control group is rather restricted, since only non-American members of the Econometric Society, who were not in the USA between 35 and 40, and who supplied enough information in their CVs to check this condition, are included.

Need to control for the age of the economists

The age of economists seems to impact their productivity. In focusing on the John Bates Clark Medal, and studying the productivity before and after the age of 40, this factor is controlled for in the analysis.

This stylized fact is illustrated by the following graphs, using a broader sample than the one used in this article, since the construction of this graph does not demand to have characteristics from CVs. The sample fuelling this graph is made of economists who were either awarded some economic prizes such as the Nobel Prize, the John Bates Clark Medal, the Frisch Medal and the IZA reward, economists who are among the best 358 ranked in the RePEc classification (we select only some of these economists, using the additional condition of age), coupled with the fact of being fellows of the Econometric Society. We can see in this first graph that the there is a peak in the age distribution at around 55 years.

We use another indicator to confirm the influence of age on the productivity of the economists in the sample: we calculate the ratio between the actualized score (provided by RePEc) and the nonactualized score of each economist.

We then draw a graph connecting this ratio with the age of the economists (Cf. Graph 2).

Graph 2: Ratio of the actualized score/non-actualized score (average=1)

Source: RePEc, author's calculations

This ratio decreases up to the age of around 65, and then stabilizes below the average (around 0.9).

The average ratio is reached at around 55 years of age.

The impact of age is confirmed by [START_REF] Hamermesh | Ageing and productivity: Economists and others[END_REF]. This article also underlines that productivity depending on age evolves over time. These findings justify to focus on the John Bates Clark Medal, with a given threshold in age, and analyzing the pre and post threshold periods.

Comparison with the existing literature

Literature on awards is already well documented (for example [START_REF] Hansen | Towards a General Theory of Awards, or, Do Economists Need a Hall of Fame[END_REF] or more recently [START_REF] Frey | Knight Fever -Towards an Economics of Awards[END_REF]). Yet, the approach has, in the whole, rather been descriptive and not particularly focused on economics.

In a general perspective, using a principal-agent framework, Frey & Gallus (2014) underline under which conditions awards tend to raise performance.

From an empirical point of view, the potential impact of awards on productivity has already been tested in the realm of mathematics: Cf. [START_REF] Borjas | Prizes and Productivity: How Winning the Fields Medal Affects Scientific Output?[END_REF]. However, the present article differs from this in a number of ways:

 Among the variables of productivity, we consider not only the number of publications, but also their quality. As a robustness check, we use an indicator which is the sum of the following annual publication rankings for each economist2 :

o Fed New York ranking (attributing marks: 400, 200, etc. down to 25. We attribute the mark nil when a publication is not included in this ranking).

o RePEc ranking. [START_REF] Chan | Does the John Bates Clark Medal Boost Subsequent Productivity And Citation Success?[END_REF] also test the impact of the John Bates Clark Medal for economists using a diffin-diff methodology. They build a synthetic control group of non-recipient scholars with similar previous research performance. Yet, their analysis does not take into account the possible heterogeneity in the control group.

This heterogeneity inside the non-recipient scholars is taken into account in the fifth part, displaying an original iterative diff-in-diff methodology, that parts the initial treatment group [since the treatment group of the article corresponds to the idea of the control group of [START_REF] Chan | Does the John Bates Clark Medal Boost Subsequent Productivity And Citation Success?[END_REF]] with an additional criterion.

Results using a standard diff-in-diff methodology

In this first method, a treatment group is constructed, made of economists belonging to the Econometrics Society, fulfilling the conditions to compete for the John Bates Clark Medal. For the control group, members of the Econometrics Society are also taken into account, to guarantee a comparable "standard" in economics, except that they do not fulfil the conditions to compete for the John Bates Clark Medal, namely they are not American, and not in the USA before the age of 40 (if they had a position in the USA, but left before 35, they are also included). Thus, they cannot be disappointed for not receiving this reward. For the lists of these two initial treatment and control groups: Cf. Appendix.

Using these two groups, the performance in terms of average and median number of publications, depending on age, is exhibited on graphs 3 and 4, by year, and on graphs 5 and 6 on a rolling window of 5 years, to smooth the profiles, due to the time required for publications, which generates nonregular profiles.

The average and median numbers of working papers are also shown on graphs 7 and 8, using a rolling window of 5 years. The "initial" control group exhibits quite different characteristics from the "initial" treatment group, in terms of average level and evolutions of issuance of working papers (Cf. graphs 7 and 8), which tends to show that the American context is quite different from the non-American one, even for economist of a priori comparable standards.

As regards the average number of publications, by year (Cf. graph 3) and over a rolling window of 5 years (Cf. graph 5), the treatment and the control group respectively do not seem to be very different, and would even show a slight difference in favour of the control group around 40 (before 40, there is a rise in productivity for the control group and a stability for the treatment group, which was not expected).

Thus, when using this first method, results are not really conclusive: differences are minor as regards publications, and even in favour of the control group, and differences related to working papers are so big that they are probably connected to structural differences between American and non-American universities, rather than to differences of individual motivations.

Results using an iterative diff-in-diff methodology

Going back to medicine, where the diff-in-diff methodology was first used, suppose a drug is to be tested, to cure a disease that both affects adults and children, without knowing a priori that this drug is only efficient for children. A treatment group is built taking a representative sample of 1000 persons inside the population of a given country, the same for the control group. Suppose also that this country is a developed one with a very low proportion of children, say 10%.

If the results of the whole treatment group are compared with the ones of the whole control group, the positive results for children being diluted, they will be hardly significant.

In that case, the fraction of the treatment group with the results closest to what is expected (improvement of health considering the symptoms of the related disease) should be selected in contrast with either the rest of the treatment group or the initial control group. Then, a criterion should be found to distinguish this fraction with the rest of the initial treatment group, to finally find that age is the fundamental parameter. After this iteration, the "new" treatment group would be re-built applying this additional criterion (being a child) to the initial sample. When selecting individuals under a given age, the "new" treatment group may contain some children for which the medicine was not efficient and delete a few adults (especially "young" ones) for which the drug was efficient, but the overall effect should be significantly magnified compared to the initial treatment group.

This method is to be used when results of the initial treatment group are not significant, but it may also be implemented for cases in which treatment groups exhibit satisfactory results initially, which may even magnify them and set a more precise conclusion on the kind of population for which the treatment applies most efficiently.

In the case of this article, iterative diff-in-diff would consists in selecting, into an initial sample for the treatment group, the members of the sample that fit most the expected result, in the case of the present article, the decrease of their productivity. When comparing this sub-sample with the rest of the initial sample, one tries to find a criterion, using for instance data analysis, in line with the initial question raised (in the present case, the fact of attributing a high value to honors, proxied by the position of the honors section in their CV), which differentiates the two sub-samples.

One then applies this criteria to the initial sample (which selects mostly members of the sub-sample n°1 made of economists registering the biggest fall of productivity, it can lead of course to select also a few members of the sub-sample n°2) and re-performs the diff-in-diff methodology This method consists thus in building a more focused and up-to-the-point sample for the treatment group. By contrasting the results of this new sub-sample n°1 with the ones of the new sub-sample n°2, two utilizations are possible: 1/ compare the evolutions of these two groups with the ones of the initial control group, to have more tangible results or 2/ compare the evolutions between these two groups. In the case of the present article, this would lead to reformulate the initial question: "how does the productivity of disappointed economists evolve when missing a major prize, compared to economists of equivalent level (but potentially in a slightly different context, namely a non-American context)".

The new question would be: "inside a unified context, namely the American one, how does the productivity of potentially disappointed economists who pay most attention to honors evolve, compared with the productivity of economists who would also be potentially disappointed, but probably somewhat less since they should pay less attention to honors, when a major prize has not been awarded".

In the appendix, the ratio of the number of publications between 35 and 40 over the number of publications between 41 and 46 is calculated for each member of the sample and these members are ranked according to this criterion, which registers the fall of productivity beyond the threshold of 40.

An additional criterion is needed to discriminate between the members who are rather above or under the median. Using the information contained in the CV available online, it is found that the position of the "honors and awards" part of the CV is well correlated with the ranking linked to the ratio: over the first 44 members of the sample whose ratio is highest, 31 have an "honors and awards" part which is between the first page and the very beginning of the second page (indicated by the value "1,01" or "1,1" in the table of the appendix). To take a simple criterion, this limit of the beginning of the second page is retained as the frontier between the members of the treatment group who are most honors oriented (the "new" treatment group), and those who are least honors oriented. These "new" control group members are made of the sum of the members with an "honors and awards" part after the very beginning of the second page, and the members without such a category or without CV available online (or at least, for which their CV was not found), which is consistent with the fact that these members do not strongly intend to display their honors and awards.

When comparing the performance of the new treatment group with the ones of the new control group, the initial property that was tested needs thus to be re-formulated. In the case of this article, instead of reaching the conclusion that "the performance of the economists who contend to the John Bates Clark medal, disappointed by not receiving it, has abated X years after the age of 40, compared to noncontenders economists of equal standard", the conclusion would be "the performance of the economists who contend to the John Bates Clark medal, who attribute a high importance to honors, disappointed by not receiving it, has abated X years after the age of 40, compared to economists of equal standard who were also contenders, but attributing less importance to honors in general, and in particular to this prize." And indeed, when looking at the following results, it can be shown that the over-performance of the contenders paying a high importance to honors abates and that the difference of productivity between the "new" treatment and control groups, visible before 40, reverses after (the most spectacular result is for the mean number of working papers, for which the two curbs intersect at the age of 42: Cf. graph 13. The intersection appears later for the median number of working papers, but the change in the trend of the "new" treatment group appears around 40: Cf. graph 14).

Graph Being given that the number of publications or working papers does not take into account the quality dimension, two complementary calculations are made. Publications are weighted by their RePEc scores, which measure their influence (graphs 15 and 16, for the average and the median values respectively) and by their FED scores (using the scores attributed by the Fed for its ranking of reviews, from 400 to 0, if the review is not covered. Cf. graphs 17 and 18, for the average and the median values respectively).

All these graphs show that the performance of the contenders for the John Bates Clark Medal paying most attention to honors and awards ("New" treatment group) is higher around 40 (the over performance before 40 lasts for a few years after 40 due to the fact that some publications may take several years to be released) than the one of the contenders for the John Bates Clark Medal paying least attention to honors and awards ("New" control group). When summing up these two groups ("Old" treatment group), it can be observed that its average or median weighted performance is much higher than the one of the non-contenders for the John Bates Clark Medal ("Old" control group). This reinforces the interest of the new iterative method, which underlines the over performance before 40

for economists paying most interest to honors and awards, whereas the comparison between the "old"

Conclusion

This article tries to answer this question: do economists see their productivity decreasing if they are disappointed by not being awarded a major reward (namely the John Bates Clark Medal) they could hope for?

In that purpose, we use two different methods, the first one with a standard diff-in-diff approach which gives mixed results and the second one with an innovative method to build a treatment group, that is called iterative diff-in-diff, whose results are more in line with what is expected, with an increase in productivity, measured with different indicators, before the threshold of 40 for the John Bates Clark Medal.

More precisely, the most tangible results of our article, using this new methodology, are the following.

Economists potentially disappointed and attributing most attention to honors invest differently over time, with an overinvestment before the end of the period of award of the prize, and an underinvestment after, in particular as regards working papers. The initial control group exhibits quite different characteristics, in terms of average level of publications and number of working papers, which tends to show that the American context is quite different (in level, and potentially in evolution as regards for example working papers) from the non-American ones, even for economist of a priori comparable standards.

This original method may potentially have numerous applications, to get tangible results through diffin-diff methods that do not seem to work initially, with also more nuanced / complete conclusions. In the present case, it is found that the decrease of productivity and the concentration of efforts over time in relation with the timing of the John Bates Clark Medal is most tangible for economists which pay most attention to honors. It can also enable to build a control group (if the initial question is somewhat reformulated), starting from a group that should have been considered merely as a treatment group, when no obvious control group can be found. In other words, to go back to the medicine example, one would not contrast the results in terms of recovery for a treatment group of ill people receiving a medicine, with a control group of ill people receiving a placebo, but one would contrast the results in terms of recovery for a treatment group of ill people receiving a medicine, with a control group of ill people receiving the same treatment, but for which it is known from the start that the treatment will not be as efficient (if efficient): for example, giving a child medicine with a child posology both to children and to adults suffering the same illness. The method will be conclusive if there is a difference between the two groups.

If there was no effect in terms of disappointment and loss of productivity, no difference between these two groups would be found, which does not mean that if there is no difference between the two, there is no impact: perhaps the initial criteria (age, or importance given to honors and awards) was not the right one. It is perhaps the main shortcoming of this method but anyway, since it is applied when the diff-in-diff method does not work, it can only improve the results. But if there is a difference, then it is conclusive. Thus, starting from a treatment group, splitting the sample enables to construct a new treatment group and a new control group, on the condition to reformulate the initial question.

  Initial control group: members of the Econometric Society not fulfilling the conditions to compete for the John Bates Clark Medal (non-American economists, who have never worked in the USA, or have left it before 35 3 ) Sources: Who's Who in Economics, homepages, online CV 3 As regards European Economists, the ones who may have been influenced by the perspective of getting the Yrjö Jahnsson reward (best European economist before 45), first attributed in 1993, have been excluded.

  11: Mean number of published articles by year, calculated over a rolling window of five years,

	Graph 12: Median number of published articles by year, calculated over a rolling window of five Graph 14: Median number of published working papers by year, calculated over a rolling window of
	years, depending on age five years, depending on age
	12
	10
	8
	6
		new treatment group, most honors oriented
	4
		old control group
	2
		new control group, least honors oriented
	depending on age 0
		35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
	12
	10 Source: RePEc, author's calculations Source: RePEc, author's calculations
	8
	Graph 13: Mean number of published working papers by year, calculated over a rolling window of
	6
	new treatment group, most honors oriented five years, depending on age
	4
	20	old control group
	2 18	new control group, least honors oriented
	12 14 0 16	35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
	8 Source: RePEc, author's calculations 10
	6	new treatment group, most
	4	honors oriented old control group
	2
	0	new control group, least honors oriented
		35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
	Source: RePEc, author's calculations

  treatment and control group only exhibited a kind of structural over-performance of economists in the US compared to their non-US counterparts (due among other to the overall increased interest for the quality of publications from the US universities).

	Graph 16: Median total ranking (number of publications, weighted by their RePEc influence scores), Graph 18: Median total ranking (number of publications, weighted by their FED scores), calculated
	calculated over a rolling window of five years, depending on age over a rolling window of five years, depending on age
	14 12
	12 10									"New" treatment group
	10 Graph 15: Mean total ranking (number of publications, weighted by their RePEc influence scores), 8
	8 6 calculated over a rolling window of five years, depending on age "Old" treatment group
	4 6 14 4 12 2									"Old" control group new treatment group, most honors oriented old control group (non John Bates Clark "New" treatment group contenders)
	0 2 8 0 10	35	37	39	41	43	45	47	49	oriented "Old" treatment group new control group, least honors "New" control group
	6	35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 ans ans ans ans ans ans ans ans
	2 4 Source: RePEc, author's calculations Source: RePEc, author's calculations			"Old" control group (non John Bates Clark contenders)
										"New" control group
	0
	35 Graph 17: Mean total ranking (number of publications, weighted by their FED scores), calculated over 37 39 41 43 45 47 49
		ans	ans	ans	ans	ans	ans	ans	ans
	a rolling window of five years, depending on age
	Source: RePEc, author's calculations 2500
										"New" treatment
	2000								group
	1500								"Old" treatment group
	1000
										"Old" control group
										(non John Bates Clark
	500								contenders)
										"New" control group
		0
		35	37	39	41	43	45	47	49
		ans	ans	ans	ans	ans	ans	ans	ans
	Source: RePEc, author's calculations

As this indicator is the sum of the rankings for all publications in a given year for a given economist, a rise in the indicator does not necessarily mean that the average quality has increased. Even if the average quality diminishes, it may be compensated by an increased number of (ranked) publications.

No CV or "honors/awards" part Eugene F. Fama Sr. Yes No CV or "honors/awards" part Jonathan Gruber Yes No CV or "honors/awards" part Merton H. Miller Yes No CV or "honors/awards" part Douglass C. North Yes No CV or "honors/awards" part Paul Milgrom Yes No CV or "honors/awards" part Michael Rothschild Yes No CV or "honors/awards" part Charles I. Plosser Yes No CV or "honors/awards" part Barry Julian Eichengreen No No CV or "honors/awards" part David Easley No No CV or "honors/awards" part N. Gregory Mankiw No No CV or "honors/awards" part George J. Stigler No No CV or "honors/awards" part Andrew Theo Levin No No CV or "honors/awards" part George William Evans No No CV or "honors/awards" part Roger H. Gordon No No CV or "honors/awards" part David Dollar No No CV or "honors/awards" part Harry M. Markowitz No No CV or "honors/awards" part Robert W. Fogel No No CV or "honors/awards" part Robert E. Lucas Jr. No No CV or "honors/awards" part Stephen Ross No No CV or "honors/awards" part William Baumol No No CV or "honors/awards" part

Appendix: list of economist belonging to each group

Economists of the initial treatment group (members of the Econometric Society fulfilling the conditions to compete for the John Bates Clark Medal)