Luc Pronzato
email: pronzato@i3s.unice.fr

Anatoly Zhigljavsky
email: zhigljavskyaa@cardiff.ac.uk

BLUE against OLSE in the location model: energy minimization and asymptotic considerations

The main purpose of the paper is to uncover the connections between kriging, energy minimization and properties of the ordinary least squares and best linear unbiased estimators in the location model with correlated observations. We emphasize the special role of the constant function and illustrate our results by several examples.

Introduction

The paper makes connections between the following domains: (a) simple and ordinary kriging with kernel K for prediction of values of a random field indexed by a set X , (b) energy minimization for K, and (c) parameter estimation with the Ordinary Least Squares Estimator (OLSE) and the Best Linear Unbiased Estimator (BLUE) in the location model with observations whose correlation is defined by K. These three areas are well-studied in modern literature. For the theory and methodology of kriging, along with numerous references, see [START_REF] Santner | The Design and Analysis of Computer Experiments[END_REF]; the most advanced modern treatment of the theory of energy minimization for K is contained in the excellent book [START_REF] Borodachov | Discrete energy on rectifiable sets[END_REF], whereas [START_REF] Näther | Effective Observation of Random Fields[END_REF] contains a comprehensive survey on classical properties of the OLSE and the BLUE in regression models with correlated observations. For more recent results on properties of the OLSE and the BLUE, see [START_REF] Dette | The BLUE in continuoustime regression models with correlated errors[END_REF] and [START_REF] Zhigljavsky | A new approach to optimal design for linear models with correlated observations[END_REF]. Despite both kriging and designing for correlated observations are well-known and well-studied areas in the DOE ("design of experiments") community, the effect of substituting the OLSE for the BLUE as parameter estimator in kriging prediction models has not been adequately addressed in the DOE literature. Similarly, there is a wealth of research on energy minimization, but the implications of this well-studied area on the properties of the OLSE and the BLUE in regression models have not been sufficiently discussed in the DOE literature; see, however, [START_REF] Pronzato | Bayesian quadrature, energy minimization, and space-filling design[END_REF], a previous paper by the present authors, where some aspects of these implications have been mentioned. The present paper aims at closing these gaps by fully concentrating on the interplay between the above three areas.

The paper is organized as follows. In Section 2 we consider different versions of kriging, based on both the OLSE and BLUE. There are no new results in this section but the view-point towards different aspects of kriging is novel. Section 3 is auxiliary and summarizes some known properties of energies, potentials and minimum-energy measures [START_REF] Sriperumbudur | Hilbert space embeddings and metrics on probability measures[END_REF][START_REF] Sejdinovic | Equivalence of distance-based and RKHS-based statistics in hypothesis testing[END_REF][START_REF] Pronzato | Bayesian quadrature, energy minimization, and space-filling design[END_REF], these properties being used in Section 4. The new results of Section 3 are emphasized in Examples 1 and 2, which constitute a substantial part of this section. The most interesting discussions and important contributions are contained in Section 4, where various properties of the OLSE and BLUE are considered. In Subsection 4.1 we summarize and discuss properties of these estimators for fixed-size designs while in the rest of Section 4 we concentrate on their asymptotic properties. An important result, connecting the energy minimization problem with the asymptotic value of the sequence of variances of the BLUE, is contained in Corollary 4.1: lim n→∞ var( θ n BLUE ) > 0 if and only if the constant function on X belongs to the Reproducing Kernel Hilbert Space (RKHS) generated by the kernel K, and, moreover, this limit is equal to E * K = inf µ X 2 K(x, x )µ(dx)µ(dx ), where the infimum is taken over the set of all signed measures on X of total mass 1. Another important theoretical result of the paper is contained in subsection 4. [START_REF] Dette | Reproducing kernel Hilbert spaces, polynomials, and the classical moment problem[END_REF], where we extend some well-known results of Schoenberg [START_REF] Schoenberg | Metric spaces and positive definite functions[END_REF] to the general class of so-called reduced kernels.

Kriging with the BLUE and OLSE

Consider the location model y(x) = θ + ε(x), E{ε(x)} = 0, E{ε(x)ε(x )} = K(x, x ), x, x ∈ X ,

where X is some set and observations y(x j ) of y(x) are performed at distinct points x j of an n-point design X n = {x 1 , . . . , x n } ⊂ X . The kernel K is assumed to be strictly positive definite (SPD), which means that for any n ∈ N, the kernel matrix K n = (K(x i , x j )) n i,j=1 is SPD (as we assume that X n contains no repetitions). We denote

k n (•) = [K(x 1 , •), . . . , K(x n , •)] .
The construction of best linear predictors of the value of y(x 0 ) at a given point x 0 ∈ X is called "kriging". Linear predictors of y(x 0 ) have the form η n (x 0 ) = n i=1 w i y(x i ) = w n y n , where w n = (w 1 , . . . , w n ) is the vector of weights and y n = [y(x 1 ), . . . , y(x n )] is the vector of observations. The simpler case when θ is known (and hence can be assumed to be 0), corresponds to "simple kriging"; the general case of an unknown θ is called "ordinary kriging". In this paper we shall not consider the case of "universal kriging", where the mean of the random field y(x) is linearly parameterized, which corresponds to the situation when y(x) = β h(x) + ε(x) with h(x) = [h 1 (x), . . . , h p (x)] a vector of p known functions on X and β a vector of unknown parameters in R p .

Simple kriging

In this model, the Mean Squared Prediction Error (MSPE) of a general linear predictor η n (x 0 ) = w n y n is

ρ 2 n (x 0 , w n ) = E{[η n (x 0 ) -y(x 0 )] 2 } = K(x 0 , x 0 ) -2 w n k n (x 0 ) + w n K n w n .
The Best Linear Predictor (BLP) at x 0 minimizes ρ 2 n (x 0 , w n ) in the class of all linear predictors η n (x 0 ) = w n y n and is given by η

* n (x 0 ) = w * n y n with w * n = w * n (x 0 ) = K -1 n k n (x 0 ) .
Therefore, the BLP is

η * n (x 0 ) = k n (x 0 )K -1 n y n . (2)
Its MSPE equals

ρ * n 2 (x 0 ) = ρ 2 n (x 0 , w * n ) = K(x 0 , x 0 ) -k n (x 0 )K -1 n k n (x 0 ) . (3)

Ordinary kriging

BLUP. Consider the general model [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF]. The unbiasedness condition for the predictor η n (x 0 ) = w n y n is E{η n (x 0 )} = E{y(x 0 )} = θ for all θ, which implies w n 1 n = 1, where 1 n = (1, . . . , 1) ∈ R n . The Best Linear Unbiased Predictor (BLUP) at x 0 minimizes ρ 2 n (x 0 , w n ) with respect w n ∈ R n satisfying w n 1 n = 1. The Lagrangian of this constrained minimization problem is

L(w n , λ n ) = w n K n w n -2 w n k n (x 0 ) + 2 λ n (w n 1 n -1),
with λ n the Lagrange coefficient for the constraint, which gives the stationarity conditions

K n 1 n 1 n 0 w n λ n = k n (x 0 ) 1 . (4)
By direct calculation, the solution of ( 4) is

w n = w n (x 0 ) = K -1 n k n (x 0 ) + [1 -k n (x 0 )K -1 n 1 n ]K -1 n 1 n 1 n K -1 n 1 n . ( 5
)
The place of the BLUE of θ in the BLUP. From (5), we get the following explicit expression for the BLUP:

η n (x 0 ) = θ n BLUE + k n (x 0 )K -1 n (y n -θ n BLUE 1 n ) ( 6
)
where

θ n BLUE = 1 n K -1 n y n 1 n K -1 n 1 n
is the BLUE of θ in model [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF]. Indeed, the variance var( θ n ) of any linear estimator θ n = w n y n of θ equals w n K n w n , the unbiasedness constraint E{ θ n } = θ for any θ imposes 1 n w n = 1, so that θ n BLUE = w n,BLUE y n with

w n,BLUE = K -1 n 1 n 1 n K -1 n 1 n

= arg min

wn: 1 n wn=1

w n K n w n . (7)
The variance of θ n BLUE equals

var( θ n BLUE ) = w n,BLUE K n w n,BLUE = (1 n K -1 n 1 n ) -1 . (8)
In Section 4.1 we shall see that var( θ n BLUE ) is the energy for K of the minimumenergy signed measure µ * n of total mass 1, supported on X n , µ * n having weights w n,BLUE .

Equations ( 2) and ( 6) show that we can perceive the BLUP η n as the BLP η * n applied to model [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF] where the unknown θ is replaced by the BLUE θ n BLUE . Also, the BLP η * n (x 0 ) and the BLUP η n (x 0 ) are related by

η n (x 0 ) = η * n (x 0 ) + [1 -1 n w * n (x 0 )] θ n BLUE . (9)
Equation [START_REF] Knott | On the minimum efficiency of least squares[END_REF] shows that η n (x) corrects η * n (x) when 1 n w * n (x) = 1. The MSPE of η n (x 0 ), called the ordinary kriging variance, equals

ρ 2 n (x 0 ) = ρ * n 2 (x 0 ) + [1 -1 n w * n (x 0 )] 2 var( θ n BLUE ) = K(x 0 , x 0 ) -k n (x 0 )K -1 n k n (x 0 ) + (1 -k n (x 0 )K -1 n 1 n ) 2 1 n K -1 n 1 n . (10)
Therefore ρ 2 n (x 0 ) ≥ ρ * n 2 (x 0 ) given by (3); this is due to the presence of the unknown constant mean θ that needs to be estimated.

Use of a reduced kernel. Let us show that expressions of η n (x 0 ) and ρ 2 n (x 0 ) can be simplified by considering a formal modification of the model. Define the random field ε 0 (x) = ε(x) -ε(x 0 ); it is centered and has covariance function

R(x, x ) = E{ε 0 (x)ε 0 (x )} = K(x, x ) -K(x, x 0 ) -K(x , x 0 ) + K(x 0 , x 0 ) ,
sometimes called the reduction of K with respect to the delta measure δ x0 [START_REF] Schaback | Native Hilbert spaces for radial basis functions I[END_REF][START_REF] Gauthier | Convex relaxation for IMSE optimal design in random field models[END_REF], or the version of K centered at δ x0 [START_REF] Sejdinovic | Equivalence of distance-based and RKHS-based statistics in hypothesis testing[END_REF]. It satisfies R(x, x 0 ) = 0 for all x, and the kernel matrix R n with elements {R n } i,j = R(x i , x j ) satisfies

R n = [I n -1 n ] K n k n (x 0 ) k n (x 0 ) K(x 0 , x 0 ) I n -1 n ,
with I n the n × n identity matrix. The kernel matrix R n is strictly positive definite when the x i are all pairwise different and x 0 ∈ X n . Since we can write y(x) = θ + ε(x) = ϑ + ε 0 (x), with ϑ = θ + ε(x 0 ) defining a new unknown mean parameter, previous expressions for the predictor η n (x 0 ) and the kriging variance ρ 2 n (x 0 ) remain valid when we substitute R for K. Since R(x, x 0 ) = 0 for all x, we obtain

η n (x 0 ) = 1 n R -1 n y n 1 n R -1 n 1 n and ρ 2 n (x 0 ) = 1 1 n R -1 n 1 n .
Notice that η n (x 0 ) is the BLUE of ϑ and ρ 2 n (x 0 ) is the variance of this estimator. Reduction of K with respect to a general measure µ is considered in Section 4.5.

Prediction with the OLSE of θ. The OLSE θ n OLSE of θ in the model ( 1) is simply the empirical mean θ n OLSE = ȳn = w n,OLSE y n , where w n,OLSE = 1 n /n. As 1 n w n,OLSE = 1, θ n OLSE is unbiased; its variance equals

var( θ n OLSE ) = w n,OLSE K n w n,OLSE = 1 n 2 1 n K n 1 n . (11)
From the Cauchy-Schwarz inequality,

var( θ n OLSE ) var( θ n BLUE ) = 1 n 2 (1 n K n 1 n )(1 n K -1 n 1 n ) ≥ (1 n 1 n ) 2 n 2 = 1 . (12)
To predict with the OLSE θ n OLSE , we apply the simple kriging predictor (2) to the centered observations y n = y n -ȳn and then add the empirical mean ȳn to the predictor η * n . This yields a predictor having the same form as [START_REF] Dijk | Distribution Theory: Convolution, Fourier Transform, and Laplace Transform[END_REF], but with the OLSE ȳn = θ n OLSE substituted for the BLUE θ n BLUE :

η n,OLSE (x 0 ) = ȳn + k n (x 0 )K -1 n (y n -ȳn 1 n ) .
The MSPE of η n,OLSE (x 0 ) is

ρ 2 n,OLSE (x 0 ) = ρ * n 2 (x 0 ) + [1 -1 n w * n (x 0 )] 2 var( θ n OLSE ) = K(x 0 , x 0 ) -k n (x 0 )K -1 n k n (x 0 ) + (1 -k n (x 0 )K -1 n 1 n ) 2 1 n K n 1 n n 2 ,
and (12) implies ρ 2 n,OLSE (x 0 ) ≥ ρ 2 n (x), where ρ 2 n (x) = ρ 2 n,BLUE (x 0 ) is given by [START_REF] Näther | Effective Observation of Random Fields[END_REF]. Although var( θ n OLSE ) may be significantly larger than var( θ n BLUE ), in practice ρ 2 n,OLSE (x 0 ) is often only marginally larger than ρ 2 n (x) due to the factor

(1 -k n (x)K -1 n 1 n ) 2
, which tends to be small especially for large n. The right panel of Figure 4 provides an illustration.

The variance var( θ n OLSE ) corresponds in fact to the energy for K of the empirical measure µ n = (1/n) n i=1 δ xi on X n . The notion of energy of a measure for a kernel K is introduced in the next section.

3 Energy, potentials and minimum-energy measures

Energy and MMD

Let X be a compact set in a metric measurable space and K(x, x ) be a kernel; that is, a symmetric function on X × X → R. Unless otherwise stated, we assume that K is uniformly bounded; that is, there exists a constant C such that |K(x, x )| ≤ C < ∞ for all x and x in X . Let M and M (1) be respectively the sets of finite signed measures on X and of signed measures on X with total mass 1. Also, let M + and M + (1) be the sets of finite positive measures on X and of probability measures on X , respectively. Note that the set M + (1) is weakly compact whereas M (1) is not.

For any µ ∈ M , the energy of µ is defined as

E K (µ) = X 2 K(x, x )µ(dx)µ(dx ) ,
and

|E K (µ)| < ∞ as K is uniformly bounded. Definition 3.1. A kernel K is Integrally Positive Definite (IPD) when E K (ν) ≥ 0 for any ν ∈ M . It is Integrally Strictly Positive Definite (ISPD) if, in addition, E K (ν) = 0 implies ν = 0. Definition 3.2. A kernel K is Conditionally Integrally Positive Definite (CIPD) when E K (ν) ≥ 0 for all signed measures ν ∈ M such that ν(X ) = 0. It is Conditionally Strictly Integrally Positive Definite (CISPD) if, in addition, E K (ν) = 0 for a ν such that ν(X ) = 0 implies ν = 0.
If the kernel K is IPD, then E K (µ) ≥ 0 for any µ ∈ M . For µ, ν ∈ M , we also define the cross-energy by E K (µ, ν) = X 2 K(x, x )µ(dx)ν(dx ) . As the energies for both measures µ and ν are bounded, the Cauchy-Schwarz inequality implies that the cross-energy

E K (µ, ν) is also bounded. If K is IPD or more generally CIPD, then E K (•) is a convex functional on M (1); if K is CISPD, then E K (•) is strictly convex on M (1)
; see [START_REF] Pronzato | Bayesian quadrature, energy minimization, and space-filling design[END_REF]. These properties remain valid without the assumption that K is uniformly bounded (in particular for singular kernels such that K(x, x) = +∞), but we must then restrict our attention to measures with finite energy. In what follows, we assume that K is at least CIPD.

The Maximum Mean Discrepancy (MMD) between two measures µ and ν in M (1) is

MMD(µ, ν) = E K (µ -ν) . ( 13
)
As we have assumed that K is at least CIPD, MMD(µ, ν) is properly defined for all µ, ν ∈ M (1). If K is CISPD, then K is a characteristic kernel, and the MMD defines a metric on the space of probability measures M + (1); see [START_REF] Sriperumbudur | Hilbert space embeddings and metrics on probability measures[END_REF].

Another important notion is that of potential (kernel imbedding) of a (signed) measure µ ∈ M , defined by P µ (x) = X K(x, x )µ(dx ). The space of potentials is

P K = {P µ (•), µ ∈ M }; this set is dense in the RKHS H(K) associated with K [12, Prop. 2.1]. We also define P + K = {P µ (•), µ ∈ M + }
, the set of potentials associated with finite positive measures µ ∈ M + . Smoothness of the kernel K is naturally inherited by the potentials P µ (•) ∈ P K and subsequently by all functions of the RKHS H(K); see [START_REF] Saitoh | Theory of Reproducing Kernels and Applications[END_REF]Sect. 2.1.3] for a comprehensive exposition of the relation between the smoothness of K and that of the elements of H(K).

Minimum-energy measures: optimality conditions

We denote by µ * = arg min µ∈M (1) E K (µ) the minimum-energy signed measure of total mass 1 (when it exists), and by µ + = arg min µ∈M + (1) E K (µ) the minimum-energy probability measure, for a given kernel K. We also denote

E * K = inf µ∈M (1) E K (µ) and E + K = min µ∈M + (1) E K (µ) = E K (µ + ). Theorem 3.1. Assume that K is CIPD. (i) µ + is a minimum-energy probability measure if and only if P µ + (x) ≥ E + K for all x ∈ X . For any such measure µ + , P µ + (x) = E + K on the support of µ + .
(ii) µ * is a minimum-energy signed measure of total mass one if and only if

P µ * (x) = E * K for all x ∈ X .
(iii) There always exists a minimum-energy probability measure µ + , there may not exist a minimum-energy signed measure µ * of total mass one. If K is CISPD, then µ + is uniquely defined and µ * is uniquely defined when it exists.

For the proof of Theorem 3.1 see [14, Theorems 3.1 and 3.2].

A noticeable consequence of Theorem 3.1 is that minimum-energy measures for separable kernels can be deduced from their one-dimensional counterparts. Indeed, consider a separable (tensor product) kernel K(x, x ) =

d i=1 K i (x i , x i ) on X = X 1 × • • • × X d ,
where the K i are univariate (C)ISPD kernels. Then, the energy and potential of a product measure µ(dx)

= d i=1 µ i (dx i ) satisfy E K (µ) = d i=1 E Ki (µ i ) and P µ (x) = d i=1 P µi (x i
). Theorem 3.1 thus implies that the minimum-energy probability measure µ + K for K is the product of the univariate minimum-energy probability measures µ + Ki , i.e., µ + K (dx) = d i=1 µ + Ki (dx i ), and if there exists a minimum-energy signed measure µ * Ki for K i on X i for each i, the minimum-energy signed measure µ * K for K on X exists and equals µ * K (dx) = d i=1 µ * Ki (dx i ). Moreover, if, for each i, there exists a minimum-energy signed measure µ * Ki for K i on X i which coincides with µ + Ki , the minimum-energy probability measure for K i on X i , then µ * K for K on X exists and coincides with µ + K , the minimum-energy probability measure for K on X . Let 1 X denote constant function 1 on X (i.e., 1 X (x) = 1 for all x ∈ X ). Theorem 3.1 has the following corollary. (i) µ * exists if and only if 1

X ∈ P K , (ii) µ * = µ + if and only if 1 X ∈ P + K .
Corollary 3.1 raises the importance of the establishing conditions on K such that (a) 1 X ∈ P K and (b) 1 X ∈ P + K . The second condition will be considered in the next subsection and we will return to condition (a) and its consequences in Section 4.4; see Corollary 4.1.

Conditions guaranteeing that µ * exists and µ

* = µ + In view of Corollary 3.1-(ii ), 1 X ∈ P +
K is equivalent to the existence of a minimum-energy signed measure µ * of mass 1, with µ * being a probability measure. An easy condition guaranteeing this is given in the following property, which is a simple consequence of Theorem 3.1.

Corollary 3.2. Let µ + be a minimum-energy probability measure. If µ + is supported on the whole set X , then µ + is also a minimum-energy signed measure of total mass one.

The proof of the next theorem, provided in [START_REF] Pronzato | Bayesian quadrature, energy minimization, and space-filling design[END_REF], is based on the observation that under the conditions of the theorem, the potential P µ * is subharmonic outside the support of µ * . This theorem provides the most general sufficient condition known to us. Theorem 3.2 (see [START_REF] Pronzato | Bayesian quadrature, energy minimization, and space-filling design[END_REF]Th. 3.3]). Let K be an ISPD translation invariant kernel, with K(x, x ) = Ψ(x -x ) and Ψ continuous, twice differentiable except at the origin, with Laplacian ∆Ψ(x) =

d i=1 ∂ 2 Ψ(x)/∂x 2 i ≥ 0, ∀x = 0.
Then there exists a unique minimum-energy signed measure µ * of mass 1 and µ * is a probability measure.

Despite its generality, the conditions stated in Theorem 3.2 are rather disappointing. First, when d = 1 they correspond to Ψ(x) being convex for x > 0, a condition already mentioned in [START_REF] Hájek | Linear estimation of the mean value of a stationary random process with convex correlation function[END_REF]. Second, when d ≥ 2 and Ψ(x -x ) = ψ( xx ), ψ must have a singularity at 0 to have ∆Ψ(x) ≥ 0 for all x = 0 (this is the case of Riesz kernels with ψ(t) = t -s when s ∈ [d -2, d)). Finally, these conditions are not necessary. As the example below illustrates, it is indeed easy to construct translation invariant kernels for d = 1 such that Ψ(x) is not convex for x > 0 but the minimum-energy signed measure µ * of mass 1 is a probability measure. By modifying the relative weights of the exponential and Gaussian kernels, we can easily create situations where µ * exists but is not a probability measure. The right panel of Figure 1 For the exponential kernel alone, Ψ(x) = exp(-β|x|), β ∈ (0, ∞), the continuous component is the Lebesgue measure µ L = U [0,1] on X , and µ * = µ + = (δ 0 + δ 1 + β µ L )/(β + 2); see [START_REF] Pronzato | Bayesian quadrature, energy minimization, and space-filling design[END_REF]. The continuous BLUE is thus U [0,1] when Ψ(x) is bounded, nonnegative, non-constant, and non-increasing for x > 0: indeed, simple calculation shows that P µ L (0) < P µ L (1/2) so that P µ L is not constant on [0, 1]; see Theorem 3.1-(ii ).

θ ∞ BLUE = 1 β + 2 y(0) + y(1) + β 1 0 y(x)dx .
We formulate the following two conjectures for the case where X is a compact subset of R d with nonempty interior and K is translation invariant, i.e., K(x, x ) = Ψ(x -x ):

C1: For d > 1, µ * does not exist unless Ψ has a singularity at zero (see [START_REF] Pronzato | Minimum-energy measures for singular kernels[END_REF] for properties and regularization of singular kernels).

C2: If Ψ is differentiable at the origin, then µ * does not exist.

According to Corollary 3.1, to prove C2 it would be enough to show that 1 X ∈ P K when Ψ is differentiable at the origin. Below we provide a simple example that supports this conjecture.

Example 2. Consider the Matérn 3/2 kernel on X = [0, 1] with Ψ(x) = (1 + β|x|) exp(-β|x|). It defines an RKHS on R, which we denote by H R ; we denote by f H R the norm of a function f ∈ H R . The restriction of elements of H R to X defines another RKHS H X , for which the norm is defined by

∀f ∈ H X , f H X = min f0∈H R :f0(x)=f (x), ∀x∈X f 0 H R , see [1, Th. 6].
The inclusion 1 X ∈ P K means that there exists a signed measure µ ∈ M such that 1 = 1 X (x) = P µ (x) for all x ∈ X . Consider the function 1 X defined on R by

1 X (x) =    (1 -βx) exp(βx) for x ∈ (-∞, 0] , 1 for x ∈ [0, 1] , (1 + β[x -1]) exp[-β(x -1)] for x ∈ [1, ∞) .
It belongs to H R and coincides with 1 X on X (so that 1 X ∈ H X ). We then investigate the possibility that 1 X (x) = P µ (x) = Ψ(x -x )µ(dx ) for µ a measure supported on X . In the Fourier space, this gives 1

X (ω) = Ψ(ω) M (ω), with Ψ(ω) = 2 √ 2β 3 /[ √ π(β 2 + ω 2 ) 2
] the Fourier transform of Ψ and M the Fourier transform of µ. This yields after some calculation

M (ω) = i (β 2 -ω 2 ) [exp(-iω) -1] + 2 βω [exp(-iω) + 1] 4 βω ,
and then

µ = √ 2 π 4 β β 2 µ L + 2 β(δ 0 + δ 1 ) + δ (1) 0 -δ (1) 1
,

where R f (x)δ (1)
x dx = f (x) for any test function f , with f the derivative of f . Therefore, µ is not a measure, and the situation would be similar for any other extension 1 X of 1 X . The (generalized) continuous BLUE of Section 4.4 is

θ ∞ BLUE = 2 β [y(0) + y(1)] + β 2 1 0 y(x)dx + y (0) -y (1) β 2 + 2 β ,
which involves the first-order derivatives of y. We shall see in Section 4.4 (Theorem 4.2) that var(

θ ∞ BLUE ) = 1/ 1 X 2 H X , with here 1 X 2 H X = 1 X 2 H R = 1 + β/4; see Figure 3 for a plot of 1/ 1 X 2 H X as a function of β.
From Corollary 3.2, the non-existence of a minimum-energy signed measure of total mass 1 implies that the support of the minimum-energy probability measure µ + is strictly included in X . After some calculation, we obtain that for β large enough µ + is the mixture of delta measures with the uniform distribution on an interval [a * , 1 -a * ]: we have µ + = µ + (a * ), with µ + (a) = m 0 (δ 0 + δ 1 ) + m a (δ a + δ 1-a ) + α(1 -2 a)U [a,1-a] (a > 1/β), where m 0 = exp(βa)/D, m a = (βa-1)/D, α = β 2 a/D, and D = D(β, a) = 2 exp(βa)+2 (βa-1)+β 2 a(1-2 a); the potential P µ + (a) (•) is constant on [a, 1 -a] and a * is obtained by minimizing

E K(β) [µ + (a)] = 2[-2 βa + 6 β 2 a 2 + exp(2 βa) -1 -4 β 3 a 3 + 2 β 3 a 2 ] D 2 (β, a)
with respect to a or, equivalently, by solving the equation P µ + (a) (a) = P µ + (a) (0). The solution is a * = C/β, with C 1.572366, which gives

E K(β) (µ + ) = E K(β) [µ + (a * )] = 2 6C 2 + exp(2 C) + 2βC 2 -1 -2C -4C 3 [2 exp(C) + 2 (C -1) + C(β -2 C)] 2 . ( 14
)
When β → +∞, a * tends to 0, m 0 and m a tend to 0 and α tends to 1; that is, µ + converges in distribution to U [0,1] . This construction only makes sense when a * < 1/2, which corresponds to β > β 1 3.144731. For smaller values of β, µ + has no continuous component. When

β 1 ≥ β > β 0 2.512862, the minimum-energy probability measure is µ + (a) = m 0 (δ 0 + δ 1 ) + (1 -2 m 0 )δ 1/2 , with m 0 = m 0 (β) = (β + 2) exp(-β/2) -2 2 [2 (β + 2) exp(-β/2) -(β + 1) exp(-β) -3] < 1/2
which gives

E K(β) (µ + ) = (β 2 + 2 β + 2) exp(-β) -2 2 [2 (β + 2) exp(-β/2) -(β + 1) exp(-β) -3] . ( 15
)
When β ≤ β 0 , µ + = (δ 0 + δ 1 )/2, and E K(β) (µ + ) = [1 + (β + 1) exp(-β)]/2. In contrast with the Matérn 3/2 kernel considered above, when using the same approach for the exponential (Matérn 1/2) kernel for which Ψ(x) = exp(-β|x|) and Ψ(ω

) = √ 2β/[ √ π(β 2 + ω 2 )],
by taking 1 X (x) = exp(βx) on (-∞, 0], 1 on [0, 1], and exp[-β(x -1)] on [1, ∞), respectively, we get

M (ω) = {iβ [exp(-iω) -1] + ω[exp(-iω) + 1]}/(2 ω). This yields µ = √ 2 π(δ 0 + δ 1 + β µ L )/2,
which is indeed a signed measure (here positive); see also Example 1.

For smooth kernels, it is the non-existence of the signed measure µ * that causes the strongly oscillating behavior of the BLUE weights frequently observed in practice: the (generalized) continuous BLUE involves derivatives of y, the order of which is related to the degree of smoothness of Ψ; see the right panel of Figure 2. There, as Ψ(x) = exp(-βx 2 ) is infinitely differentiable at the origin, derivatives of all orders are involved. As 1 X ∈ H(K), var( θ n BLUE ) → 0 when the sequence of design points x 1 , x 2 . . . is dense in X ; see Section 4.4.

4 Properties of var( θ n OLSE ) and var( θ n BLUE )

Fixed-size designs

Let us return to the discussion in Section 2 and consider the variances var( θ n OLSE ) and var( θ n BLUE ) computed for the design X n = {x 1 , . . . , x n }, where the x i are assumed to be distinct and the kernel K is SPD. In view of [START_REF] Parzen | An approach to time series analysis[END_REF], var( θ n OLSE ) can be written as the energy

var( θ n OLSE ) = X 2 K(x, x )µ n (dx)µ n (dx ) = E K (µ n ) ( 16
)
for the empirical probability measure µ n assigning weights 1/n to the x i ∈ X n . From ( 7), the discrete BLUE θ n BLUE can be written as

θ n BLUE = X y(x)µ * n (dx) with µ * n = arg min νn X 2 K(x, x )ν n (dx)ν n (dx ),
where ν n belongs to the set of signed measures of total mass 1 supported on X n .

The n-point optimal measure µ * n concentrated on X n has weights w * n = w n,BLUE , and in view of [START_REF] Hájek | Linear estimation of the mean value of a stationary random process with convex correlation function[END_REF],

var( θ n BLUE ) = X 2 K(x, x )µ * n (dx)µ * n (dx ) = E K (µ * n ) .
The inequality [START_REF] Paulsen | An Introduction to the Theory of Reproducing Kernel Hilbert Spaces[END_REF] simply expresses the fact that E K (µ * n ) ≤ E K (µ n ), which obviously follows from the definition of µ * n . Let λ 1 and λ n respectively denote the minimum and maximum eigenvalues of the matrix K n associated with the design X n . We then have, for any u ∈ R n ,

(u u) 2 ≤ (u K n u) (u K -1 n u) ≤ 1 4 λ 1 λ n + λ n λ 1 2 (u u) 2 , ( 17
)
where we use the Cauchy-Schwarz inequality on the left and the Kantorovich inequality on the right. Since var( 8) and ( 11), taking u = 1 n in the left inequality, we obtain [START_REF] Paulsen | An Introduction to the Theory of Reproducing Kernel Hilbert Spaces[END_REF], where there is equality if and only if 1 n is an eigenvector of K n ; that is, K n 1 n = λ1 n for some λ > 0, which means that the row (and column) sums of K n are all identical. For other (equivalent) characterizations of the equality var( θ n BLUE ) = var( θ n OLSE ), see [START_REF] Puntanen | Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty[END_REF]Sect. 10.2]. One may notice that [START_REF] Pronzato | Performance analysis of greedy algorithms for minimising a maximum mean discrepancy[END_REF], where µ n and µ * n have respective weights w n,OLSE = 1 n /n and w n,BLUE given by [START_REF] Gauthier | Convex relaxation for IMSE optimal design in random field models[END_REF].

θ n BLUE ) = (1 n K -1 n 1 n ) -1 and var( θ n OLSE ) = 1 n K n 1 n /n 2 , see (
var( θ n OLSE ) -var( θ n BLUE ) = (w n,BLUE -w n,OLSE ) K n (w n,BLUE -w n,OLSE ) = MMD 2 (µ n , µ * n ) = E K (µ n -µ * n ) ≥ 0 , see (
The right inequality in [START_REF] Saitoh | Theory of Reproducing Kernels and Applications[END_REF] coincides with the celebrated upper bound on var( θ n OLSE )/var( θ n BLUE ) proved independently in [START_REF] Bloomfield | The inefficiency of least squares[END_REF] and [START_REF] Knott | On the minimum efficiency of least squares[END_REF]. However, this upper bound is often rather pessimistic, and a better bound can be obtained as follows. As E K (•) defines a convex functional on M (1), we have

E K (µ * n ) ≥ E K (µ n ) + F E K (µ n , µ * n ) ,
where, for any µ and ν in M (1), F E K (µ, ν) denotes the directional derivative of E K (•) at µ in the direction ν; that is,

F E K (µ, ν) = lim α→0 + E K [(1 -)µ + αν] -E K (µ) α .
Direct calculation gives

F E K (µ, ν) = 2 X [P µ (x) -E K (ν)]ν(dx)
, and thus

E K (µ * n ) ≥ E K (µ n ) + inf ν∈M (1)
F E K (µ n , ν) = 2 inf x∈X P µn (x) -E K (µ n ) .
We hence obtain the following bound.

var( θ n BLUE ) var( θ n OLSE ) ≥ 2 inf x∈X P µn (x) E K (µ n ) -1 ,
where µ n is the uniform probability measure on X n .

Designs with increasing n

We start with an illustrative example.

Example 3. The left panel of Figure 4 shows the typical behavior of the variances var( θ n OLSE ) and var( θ n BLUE ) as n increases. In this example, X = [0, 1], K is the exponential (Matérn 1/2) kernel K(x, x ) = exp(-5 |x -x |) and the sequence of designs

X n = {x (n) 1 , . . . , x (n)
n } consists of equidistant points x (n) i = (i -1)/(n -1), i = 1, . . . , n. We can see that var( θ n BLUE ) is monotonically decreasing with n, whereas the behavior of var( θ n OLSE ) is non-monotonic. The right panel demonstrates that the significantly larger variance of θ n OLSE compared to θ n BLUE has little consequence on the prediction errors: the figure shows that the difference between the integrated MSPEs for both estimators for the uniform measure, i.e., X ρ 2 n,OLSE (x)dx-X ρ 2 n,BLU E (x)dx, is negligible. Note that the situation might be different for d > 1 and a design X n less dense in X .

Assume that K is a continuous SPD kernel and that the sequence {µ n } of empirical measures µ n associated with the designs limiting probability measure µ ∞ (not necessarily uniform on X ). We assume that X is a compact subset of R d and that the support of µ ∞ coincides with X .

X n = {x (n) 1 , . . . , x (n) n } (consisting of distinct points x (n) i ∈ X ) weakly converges as n → ∞ to some
In the community of optimal design of experiments, the limiting probability measure µ ∞ would be considered as an "approximate design". As θ n OLSE = y(x)µ n (dx) for any X n , the OLSE for the approximate design µ ∞ is θ ∞ OLSE = y(x)µ ∞ (dx). We will call this OLSE the "continuous OLSE".

The inspection of Figure 4 suggests the following general questions.

We can give (at least partial) answers to all these questions. Let us start with question (i ). Since the probability measures µ n weakly converge to µ ∞ , the kernel K is continuous and bounded (as it is PD) and X is compact, [START_REF] Puntanen | Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty[END_REF] gives var(

θ n OLSE ) = E K (µ n ) → E K (µ ∞ ) = var( θ ∞ OLSE ) as n → ∞. As K is PD, E K (µ ∞ ) ≥ 0.
Families of kernels K and measures µ ∞ such that E K (µ ∞ ) = 0 are provided in Section 4.5; since var( θ n OLSE ) ≥ var( θ n BLUE ) for all n, these families provide examples where we have an affirmative answer to question (ii ).

The non-monotonic behavior of var( θ n OLSE ) queried in question (iii ) is related to "Smit's paradox" (see [10, p. 50]). This paradox is discussed in Section 4.3 below. Question (iv ) will be addressed in Section 4.4. Finally, concerning (v ), the conditions for [START_REF] Santner | The Design and Analysis of Computer Experiments[END_REF] to hold follow from those mentioned in Section 3.3: lim n→∞ var( θ n OLSE ) = lim n→∞ var( θ n BLUE ) if and only if µ ∞ = µ + = µ * : that is, the approximate design µ ∞ coincides with the minimum-energy probability measure µ + , which at the same time equals µ * , the minimum-energy signed measure of mass one.

Behaviour of var( θ n

OLSE ) as n increases

Smit's paradox refers to the non-monotonic behavior of var( θ n OLSE ) as shown in the left panel of Figure 4. It was firstly observed and investigated in [START_REF] Vilenkin | On the estimation of the mean in stationary processes[END_REF] and [START_REF] Smit | Estimation of the mean of a stationary stochastic process by equidistant observations[END_REF] for a particular class of stationary kernels K(x, x ) = ψ(x -x ) with 1 0 ψ(x)(1 -2x) dx > 0 in the case of designs formed by equidistant points in X = [0, 1] of the form (d) below for µ ∞ = U [0,1] . At first glance, such an increase of var( θ n OLSE ) with n does not look natural, but in fact, when µ ∞ = µ + = arg min µ∈M + (1) E K (µ) then any type of convergence of var( θ n OLSE ) towards E K (µ ∞ ) is possible, including the one depicted in Figure 4 (left). Example 4 will give a further illustration.

The following four classes of designs X n , defined for an arbitrary probability measure µ ∞ ∈ M + (1), are often used in practical considerations:

(a) X 1 = {x 1 } with arbitrary x 1 ∈ X and X n+1 = X n ∪ {x n+1 }, where

x n+1 = arg min x MMD 2 (µ n,x , µ ∞ ), where µ n,x is the discrete measure concentrated on the set of n + 1 points X n ∪ {x} and assigning the same weight 1/(n + 1) to all these points;

(b) X n = {x 1 , . . . , x n } are nested designs and the points {x 1 , x 2 . . . , } are chosen so that the sequence of empirical measures µ n corresponding to designs X n converges to a probability measure µ ∞ ;

(c) X = [0, 1] and

X n = {x (n) 1 , . . . , x (n)
n }, where x

(n) j

is the (j -0.5)/nquantile of a given probability measure µ ∞ ;

(d) X = [0, 1] and X n = {x (n) 1 , . . . , x (n) n }, where x (n) j
is the (j -1)/(n -1)quantile of a given probability measure µ ∞ .

In all four cases, if µ ∞ = µ + , then the empirical measures µ n of designs X n converge to µ + and therefore var(

θ n OLSE ) = E K (µ n ) → E + K .
In case (a), the sequence E K (µ n ) is monotonously decreasing (with rate at least O([log n]/n), see, e.g., [START_REF] Pronzato | Performance analysis of greedy algorithms for minimising a maximum mean discrepancy[END_REF]); in case (b) one should expect small fluctuations on the route to the limit. In cases (c) and (d) the behavior of var( θ n OLSE ) is often monotonic; however, for kernels such as K(x, x ) = cos 2 [βπ(x-x )] or K(x, x ) = max{0, 1-β|x-x |} with β > 1, a rather irregular behavior of var( θ n OLSE ) may be observed. Note finally that if θ is estimated by θ n OLSE , then the use of any other µ ∞ than µ + is not optimal, and hence unreasonable. From this view-point, Smit's paradox has little practical significance.

(x) = θf (x) + ε(x), E{ε(x)} = 0, E{ε(x)ε(x )} = K(x, x ),
which is a generalization of model ( 1) to an arbitrary measurable function f (we have f = 1 X in model ( 1)). As usual, assume that K is SPD so that all kernel matrices K n below are invertible and K generates an RKHS H(K).

Assume that x 1 , x 2 . . . is a dense sequence of distinct points in X and consider the sequence of nested designs X n = {x 1 , . . . , x n }, n = 1, 2 . . . From the definition of the discrete BLUE, we immediately conclude that the sequence of var( θ n BLUE ) is monotonically decreasing with n. Therefore, the limit lim n→∞ var( θ n BLUE ) exists and it is a non-negative number. The next theorem (which is a corollary of [11, Th. 6C]) gives the value of this limit in case f ∈ H(K).

var( θ n BLUE ) → 1/ f 2 H(K) as n → ∞. (19)
In our main case, when f = 1 X , the result ( 19) can be explained and specialized as follows. In view of ( 7), the n-point BLUE θ n BLUE can be written as θ n BLUE = y(x)µ * n (dx), where µ * n is the minimum-energy signed measure of mass 1 concentrated on X n and is defined by the vector of weights w n,BLUE . Let H n (K) be the RKHS induced by the kernel K on the set X n . The scalar product in H n (K) is defined by a, b Hn(K) = a K -1 n b for any a, b ∈ R n ; see e.g. [START_REF] Paulsen | An Introduction to the Theory of Reproducing Kernel Hilbert Spaces[END_REF]Sect. 2.3.3]. The restriction of the function 1 X to the set X n is the vector 1 n and therefore [START_REF] Hájek | Linear estimation of the mean value of a stationary random process with convex correlation function[END_REF] implies that the variance of θ n BLUE equals var(

θ n BLUE ) = 1/ 1 n 2
Hn(K) . Assume that the function 1 X belongs to H(K). Then, as this function is continuous and the sequence of points does not have a weak limit (as the set M (1) is not weakly compact, the sequence {µ * n } ⊂ M (1) does not necessarily have a convergent subsequence). In some cases, we can define a generalized continuous BLUE of θ and sometimes write it as θ ∞ BLUE = X y(x)φ(x)dx, where φ(x) is a generalized function or distribution (see e.g. [START_REF] Dijk | Distribution Theory: Convolution, Fourier Transform, and Laplace Transform[END_REF]). If φ(•) is a distribution rather than a function, then the expression θ ∞ BLUE = y(x)φ(x)dx involves mean-square derivatives of the random field y(x), which always exist if the kernel K is differentiable at the diagonal; see [START_REF] Dette | The BLUE in continuoustime regression models with correlated errors[END_REF] for details and references and Example 2 for an illustration. Theorem 4.2 in the case f = 1 X is also closely related to the minimization problem E K (µ) → min µ∈M (1) considered in Section 3.2. If K is IPD, then E * K = inf µ∈M + (1) ≥ 0 and the above arguments imply that E * K = lim n→∞ 1/var( θ n BLUE ) and 1 X ∈ H(K) if and only if E * K > 0. Summarizing, for the main case of interest here, with f = 1 X , we conclude the following (where we define 1 X H(K) = ∞ when 1 X / ∈ H(K)).

x 1 , x 2 . . . is dense in X , 1 n Hn(K) → 1 X H(K) as n → ∞, implying (19) for f = 1 X ; see [
Corollary 4.1.

(a) lim n→∞ var(

θ n BLUE ) = E * K = 1/ 1 X 2 H(K) .
(b) lim n→∞ var( θ n BLUE ) > 0 if and only if 1 X ∈ H(K).

Reduced kernels

This section shows that for any signed measure µ ∈ M (1), one can construct kernels that admit µ as minimum-energy signed measure of total mass 1. It also establishes links between parameter estimation and integration in the location model. Consider again model (1), with K an uniformly bounded kernel. Let µ ∈ M and define P as the orthogonal projection from L 2 (X , µ) onto the linear space Assume now that K is CISPD. Take any ξ ∈ M (1). From (21), we have E Kµ (ξ) = E K (ξ -µ) ≥ 0 with equality if and only if ξ = µ, which proves that µ is the unique minimum-energy signed measure in M (1) for K µ .

(v ) Take any n-point design X n ⊂ X and any z n = (z 1 , . . . , z n ) ∈ R n , z n = 0 n . If s n = 1 n z n = 0, denote ξ n = (1/s n ) n i=1 z i δ xi ; ξ n belongs to M (1), it has finite support and therefore cannot coincide with µ. This implies z n K µ n z n = s 2 n E Kµ (ξ n ) > 0. If s n = 0, denote ν n = n i=1 z i δ xi . From ( 21), we have z n K µ n z n = E Kµ (ν n ) = E K (ν n ), which is strictly positive since K is CISPD.

Example 5. The kernel K(x, x ) = -xx is CPD; for µ = δ 0 (the delta measure at the origin), the associated reduced kernel is K δ0 (x, x ) = x + xxx , i.e., the energy-distance kernel of [START_REF] Székely | Energy statistics: A class of statistics based on distances[END_REF], which is PD (but not SPD as K δ0 (0, 0) = 0). One may refer to [START_REF] Sejdinovic | Equivalence of distance-based and RKHS-based statistics in hypothesis testing[END_REF] for a thorough exposition on distance-induced kernels and their properties.

We finally make the following observation. Let K be a CI(S)PD kernel and assume that there exists a minimum-energy signed measure of total mass one µ * (µ * is uniquely defined from Theorem 3.1), with infinite support. From Corollary 3.2, this is the case in particular when the minimum-energy probability measure µ + (which always exists) has full support, since then µ * = µ + . The reduced kernel K µ * is thus (S)PD from Theorem 4.3-(v ). Since P K,µ * (x) = E K (µ * ) for all x ∈ X (see Theorem 3.1), we have, K µ * (x, x ) = K(x, x ) -E K (µ * ) for any x, x ∈ X , implying that K -C is (S)PD for any constant C ≤ E K (µ * ).

Conclusions

We have shown the existing connections between the following problems: kriging for prediction of values of a random field (Section 2), energy minimization (Section 3), and parameter estimation in the location model with correlated observations, using the OLSE or the BLUE. As shown in Section 4, the asymptotic variances of those estimators refer to different energy minimization problems. It appears that the constant function 1 X plays a very important role in our study. In particular, as outlined in Corollary 4.1, the limiting value of the variances of the BLUE for a dense sequence of designs is positive if and only if 1 X belongs to the RKHS generated by the kernel K. In Subsection 3.3, we have formulated two conjectures concerning the existence of µ * , the minimum energy signed measure of mass 1. In view of Corollary 3.1-(i), connecting the existence of µ * to the property 1 X ∈ P K , where P K is the space of potentials, these two conjectures can be reformulated in terms of the constant function 1 X , as written below (C1 and C2). We also add a third conjecture, C3, where the implication "if" has been established in [START_REF] Dette | Reproducing kernel Hilbert spaces, polynomials, and the classical moment problem[END_REF].

Assume that X is a compact subset of R d with nonempty interior and that the kernel K is translation invariant, i.e., K(x, x ) = Ψ(x -x ).

C1 : If Ψ(0) < ∞ and d > 1, then 1 X / ∈ P K .

Example 1 .

 1 Consider the following linear combination of exponential and Gaussian kernels: Ψ(x) = exp(-3 |x|/2) + exp(-x 2 ) with X = [0, 1]. Then, ∆Ψ(x) < 0 for x ∈ (a, b) with a 0.0978, b 0.3534, but µ * is a probability measure: µ * = α δ 0 + α δ 1 + (1 -2 α)ξ, with α 0.4331 and ξ having a density ϕ with respect to the Lebesgue measure on X ; see the left panel of Figure1for a plot of ϕ(x) (determined numerically).

  corresponds to Ψ(x) = exp(-3 |x|/2) + (3/2) exp(-x 2 ); there µ * = α δ 0 +α δ 1 +(1-2 α)ξ with α 0.4893 and ϕ(x) < 0 for x ∈ (c, d), c 0.2659, d 0.7341. Corollary 3.2 implies that the support of the minimum-energy probability measure µ + is strictly included in X (and indeed numerical computation shows that the density of the continuous component of µ + is zero in the central part of the interval [0, 1]).

Figure 1 :Figure 2 :

 12 Figure 1: Density ϕ(x) of the continuous component of the minimum-energy signed measure µ* on X = [0, 1]. Left: K(x, x ) = exp(-3 |x -x |/2) + exp(-|x -x | 2 ) (µ * is a probability measure). Right: K(x, x ) = exp(-3 |x -x |/2) + (3/2) exp(-|x -x | 2 ) (µ * is not a probability measure).The weights w n,BLUE defined by (7) are very different for the kernel Ψ(x) = exp(-3 |x|/2) + exp(-x 2 ) and for the Gaussian kernel Ψ(x) = exp(-x 2 ); Figure2gives an illustration (log-scale) when X n corresponds to n = 50 points equally spaced in [0, 1]. Whereas the measure that allocates w n,BLUE to X n forms a discrete approximation of µ * when Ψ(x) = exp(-3 |x|/2) + exp(-x 2 ), there is no minimum-energy signed measure of total mass one when Ψ(x) = exp(-x 2 ) and the (signed) components w i of w n,BLUE have very large magnitudes and strongly oscillate.In Example 1, when K(x, x ) = Ψ(x -x ) = exp(-β|x -x |) on X = [0, 1], the continuous component of µ * is uniform on X , but we cannot have µ * =

Figure 3 shows

 3 E K(β) (µ + ) as a function of β for β ∈ [1, 10].

Figure 3 : 2 H

 32 Figure 3: Energy of the minimum-energy probability measure µ + for Ψ(x) = (1 + β|x|) exp(-β|x|), x ∈ [0, 1]: E K(β) (µ + ) = [1 + (β + 1) exp(-β)]/2 for β ≤ β0, is given by (15) for β0 < β ≤ β1 and by (14) for β1 < β. The figure also shows var( θ ∞ BLUE ) = 1/ 1 X 2 H X as a function of β.

Theorem 4 . 1 .

 41 The variances of the BLUE and OLSE in model (1) satisfy

Figure 4 :

 4 Figure 4: Left: var( θ n OLSE ) and var( θ n BLUE ) as functions of n. Right: difference between the integrated MSPEs for θ n OLSE and θ n BLUE as function of n.

( i )

 i Does the limit lim n→∞ var( θ n OLSE ) exist and what is it? (ii ) Is it possible that lim n→∞ var( θ n OLSE ) = lim n→∞ var( θ n BLUE ) = 0? (iii ) Is the non-monotonic behavior of var( θ n OLSE ) similar to depicted in Figure 4 (left) typical? (iv ) Does the limit lim n→∞ var( θ n BLUE ) exist and what is it? (v ) Assuming lim n→∞ var( θ n BLUE ) > 0, what are the conditions guaranteeing lim n→∞ var( θ n OLSE ) = lim n→∞ var( θ n BLUE ) ?

Example 4 .

 4 We use again the Matérn 1/2 kernel, as in Example 3 but with a different correlation length. The behavior of var( θ n OLSE ) shown in the left panel of Figure 5 for the two designs (c) and (d) with µ ∞ = U [0,1] ( = µ + ) is now much different from that on the left panel of Figure 4. On the right panel the kernel is K(x, x ) = max{0, 1 -5 |x -x |}, and var( θ n OLSE ) oscillates during its convergence to E K (µ ∞ ) from below or from above.

Figure 5 :

 5 Figure 5: var( θ n OLSE ) as a function of n for two different designs in [0, 1]: X (1) n = {(i -1)/(n -1), i = 1, . . . , n} and X (2) n = {(i -1/2)/n, i = 1, . . . , n}. Left: K(x, x ) = exp(-|x -x |); Right: K(x, x ) = max{0, 1 -5 |x -x |}.

4. 4

 4 Computing lim n→∞ var( θ n BLUE ), continuous BLUE Consider the one-parameter model y

Theorem 4 . 2 .

 42 f ∈ H(K) if and only if

  17, Th. 2.8 & Remark 2.3]. Moreover, if 1 X ∈ P K and hence µ * , the minimum-energy signed measure of mass 1, exists (see Corollary 3.1-(i )), then the sequence of signed measures µ * n weakly converges to µ * . In this case, we can define the continuous BLUE of θ by θ ∞ BLUE = y(x)µ * (dx). If 1 X / ∈ P K , then µ * does not exist and the sequence of signed measures µ *

	n

Acknowledgments

The work of the first author was partly supported by project INDEX (INcremental Design of EXperiments) ANR-18-CE91-0007 of the French National Research Agency (ANR). The authors are grateful to Toni Karvonen (University of Helsinki) for useful advice and especially for proposing a family of translationinvariant kernels with non-convex functions Ψ and 1 X ∈ P K used in Example 1.

spanned by 1 X . We can write y(x) = θ + ε(x) = θ + Pε(x) + (Id L 2 -P)ε(x) = ϑ + ε µ (x) , where ϑ = θ + Pε(x) is a new parameter and ε µ (x) = (Id L 2 -P)ε(x) is a new random process with zero mean and covariance K µ given by the reduced kernel (the reduction of K with respect to µ)

In this model,

is its variance, and the continuous BLUE of ϑ always exists: ϑ ∞ BLUE = y(x)µ(dx); see [START_REF] Pronzato | Bayesian quadrature, energy minimization, and space-filling design[END_REF]. In the next theorem, we establish fundamental properties of reduced kernels showing their specificity. Property (ii) implies, in particular, that MMD 2 (µ, ν) = E K (µ -ν) = E Kµ (ν) for any µ ∈ M and ν ∈ M (1). A consequence of this is E Kµ (µ) = 0 for any kernel K and any signed measure µ. In particular, if K is an IPD kernel and µ ∈ M (1), then µ is the minimum-energy signed measure of mass 1 for K µ , with E * Kµ = E Kµ (µ) = 0 implying that the non-zero constant functions do not belong to the RKHS H(K µ ); see Corollary 4.1-(b). Moreover, for any dense sequence of design points [START_REF] Schoenberg | Metric spaces and positive definite functions[END_REF]Sect. 3] and [12, Sect. 9.1]) where µ is a delta measure. Properties (iv ) and (v ) are further extensions of this result. We recall that a kernel K is Conditionally Positive Definite (CPD) if and only if for any n ∈ N and any n-point design X n , z n K n z n ≥ 0 for any z n ∈ R n such that 1 n z n = 0. (i) For any µ ∈ M and ν ∈ M (1), we have

(ii) For any µ, ξ ∈ M , we have

(iii) Let µ ∈ M (1) be a discrete measure with finite support. Then

(iv) For any µ ∈ M (1), we have:

If, moreover, K is CISPD, then µ is the unique minimum-energy signed measure in M (1) for K µ .

(v) Let µ ∈ M (1) be a measure with infinite support and K be CISPD. Then K µ is SPD.

Proof. (i ) We have:

(ii ) Direct calculation using [START_REF] Schoenberg | Metric spaces and positive definite functions[END_REF] gives

where p K,n (µ) = [P K,µ (x 1 ), . . . , P K,µ (x n )] .

(

We have E K (µ) = u m K m u m and p K,n (µ) = K n,m u m , where K m is the kernel matrix for the support points s i of µ and {K n,m } i,j = K(x i , s j ); i = 1, . . . , n, j = 1, . . . , m.

For any z n ∈ R n , we have

where

Conversely, assume that K µ is PD. For any z n ∈ R n such that 1 n z n = 0, (23) implies z n K n z n = z n K µ n z n ≥ 0.

(iv ) Assume that K is CIPD. As µ ∈ M (1), for any ξ ∈ M the measure ξ -ξ(X )µ has total mass zero, and [START_REF] Sejdinovic | Equivalence of distance-based and RKHS-based statistics in hypothesis testing[END_REF] implies that E Kµ (ξ) ≥ 0. Conversely, assume that K µ is IPD. For any ξ ∈ M (0) we have E K (ξ) = E Kµ (ξ) ≥ 0.


MSc Fabienne Venet

PharmD Alexandre Pachot

PharmD Anne-Lise Debard

MD Julien Bohé

PhD Jacques Bienvenu

PhD Alain Lepape

MD Guillaume Monneret

Increased percentage of CD4؉CD25؉ regulatory T cells during septic shock is due to the decrease of CD4؉CD25؊ lymphocytes

S

epsis is still the leading cause of death in critically ill patients (1). It is now believed that those patients have features consistent with immunosuppression (1-2). The failure of the immune system may be defined by the impairment of monocytic functions, the dramatic apoptosisinduced loss of lymphocytes (B and T cells) and dendritic cells, and the anergy of T lymphocytes (1-2). We described an additional characteristic of sepsis-induced immunosuppression as we demonstrated an increased percentage of CD25ϩ T cells among CD4ϩ T lymphocytes in patients with septic shock (3). Although recently described in humans, CD4ϩCD25ϩ T lymphocytes (Treg) are already well defined as suppressor T cells. There is a growing body of evidence demonstrating that they are key cells in controlling different immune functions (4). In various animal models, bacterial infec-tions induce elevation of Treg, which play a role in avoiding a systemic inflammation (5). These findings support the preliminary observation of an elevated percentage of Treg in patients with septic shock in whom it may contribute to sepsis-induced immunosuppression.

The goal of the present study was to investigate whether the elevation of the percentage of Treg during shock was due to their differentiation. To this end, we determined the absolute count of those cells and we measured Foxp3 messenger RNA (mRNA) by quantitative real-time polymerase chain reaction (qRTPCR) since, at present, Foxp3 appears to be the gene best correlated with Treg (6).

METHODS

Patients and Study Design. The study group consisted of 54 patients with septic shock according to the diagnostic criteria of the American College of Chest Physicians/ Society of Critical Care Medicine (7). Septic shock was defined by an identifiable site of infection, hypotension persisting despite fluid resuscitation and requiring vasopressor therapy, and evidence of a systemic inflammatory response manifested by at least two of the following criteria: a) temperature Ͼ38°C or Ͻ36°C; b) heart rate Ͼ90 beats/min; c) respi-ratory rate Ͼ20 breaths/min; d) white blood cell count Ͼ12,000/mm 3 or Ͻ4000/mm 3 . Severity was assessed by the Simplified Acute Physiology Score II (8). Mortality was defined as death occurring within 28 days after the onset of shock. To provide a panel of values from healthy donors, we also included 30 individuals from laboratory staff of our hospital (aged 19 -58 yrs, 17 females, 13 males, without known comorbidity). Cell phenotyping was performed at days 1-2 and days 3-6 after the onset of shock on residual blood after completing routine follow-up performed in our intensive care unit (ICU), in accordance with the human experimentation guidelines for clinical research of our institute. For mRNA measurements, blood samples (2.5 mL) were drawn directly into PAXGene Blood RNA tubes (PreAnalytix, Hilden, Germany). All patients or relatives gave informed consent. Hence, missing values are due to patients who died (nonsurvivors) and to patients (survivors) who recovered and left the ICU before 6 days.

Flow Cytometry. Flow cytometric (EPICS XL, Beckman-Coulter, Hialeah, FL) expression of cell-surface markers was assessed on peripheral whole blood collected on EDTA anticoagulant tubes. Monocytes were characterized as previously described (3), and regulatory T cells were characterized by expression of CD3, CD4, CD25, and CD45RO and by the lack of expression of CD69 (3). A number of monoclonal antibodies and their respective isotype controls were used according to KEY WORDS: septic shock; regulatory T cells; CD4؉CD25؉; Foxp3; immunosuppression; sepsis the manufacturer's recommendations: ECDlabeled anti-CD3, PE-labeled anti-CD25, PC5-labeled anti-CD69, fluorescein isothiocyanate-labeled anti-CD4, ECD-labeled anti-CD45RO (Immunotech, Marseille, France). The red cells were lysed on the automated TQ Prep system (Beckman-Coulter). Absolute count was performed using Flow-Count Fluorosphere according to the manufacturer's recommendation (Beckman-Coulter). Results are expressed as percentages of CD25 positive cells out of CD4ϩ lymphocytes (CD3ϩ CD4ϩ) and as number of lymphocytes per microliter of whole blood.

Foxp3 mRNA Expression by qRT-PCR. Total RNA was extracted using PAXGene Blood RNA kit (PreAnalytix) and treated by DNAse (RNase-Free DNase set, Qiagen). Total RNA was reverse transcribed into complementary DNA using ThermoScript RT-PCR system (Invitrogen, Carlsbad, CA) according to the manufacturer's instructions. Foxp3 mRNA level was quantified using qRT-PCR and the Fast-Start DNA Master SYBR Green I real-time PCR kit (Roche Molecular Biochemicals). The complementary DNA calibrator for Foxp3 was prepared with the following primer combination: forward primer, 5'-TCACCTACGCCACGCT-CAT-3'; reverse primer, 5'-ACTCAGGTTGTG-GCGGATG-3' as previously described ( 9). Primer mixes and complementary DNA standard for the housekeeping gene PPIB (encoding for cyclophilin B) were obtained from Search-LC (Heidelberg, Germany). Relative quantification ratio between Foxp3 and the housekeeping gene PPIB was obtained with the LightCycler Relative Quantification Software.

Statistical Study. Due to the sample size, comparisons between groups were made using the nonparametric Mann-Whitney test. We considered p Ͻ .05 as significant without correction of the number of tests performed. Results are expressed as mean Ϯ SD.

RESULTS AND DISCUSSION

We included 54 patients exhibiting symptoms of septic shock. The demographic and clinical characteristics of the population are presented in Table 1. Mortality of the overall population was 24%. Thirty patients had no comorbidity. Main comorbidities were chronic obstructive pulmonary disease (17%), diabetes (9%), chronic cardiac failure (7%), and hemopathy (7%). Immunoparalysis was characterized by diminished expression of HLA-DR on monocytes (data not shown). Treg were distinguished from the recently activated T cells by the expression of CD45RO (marker of memory T cells, 86 Ϯ 15 %) and by the lack of expression of CD69 (marker of recently activated lymphocytes, 11 Ϯ 7%). We observed a significant increase in the percentage of CD4ϩCD25ϩ T lymphocytes in patients at days 1-2 (32 Ϯ 11) and days 3-6 (36 Ϯ 12) in comparison with healthy donors (20 Ϯ 6, p Ͻ .0001, Fig. 1A). Otherwise, we did not find any correlation between the percentage of Treg and Simplified Acute Physiology Score II at admission or with percentages of HLA-DR expressing monocytes (data not shown). Regarding cell count, as reported elsewhere (1), we observed a lymphopenia at days 1-2 (864 Ϯ 604/L) and days 3-6 (1239 Ϯ 949/ L) in comparison with healthy donors (3113 Ϯ 1956/L, p Ͻ .02). We next investigated the absolute count of circulating Treg. In healthy donors, we obtained similar values to those published elsewhere (10). We found that early after the onset of shock, both T CD4ϩ lineages (CD25ϩ and CD25Ϫ) were diminished. However, their magnitudes of decrease were quite different: We observed a marked diminution of CD4CD25Ϫ lymphocytes (70% below normal values) whereas that of Treg was less pronounced (43%, Fig. 1, B andC). Then, whereas Treg returned rapidly to normal values, CD4ϩCD25Ϫ T lymphocytes remained dramatically reduced. This suggested that the increased percentage of Treg was not due to an elevation of those cells but rather to a decrease of CD4ϩCD25-lymphocytes. To definitely rule out a lack of proliferation, we quantified Foxp3 mRNA in peripheral blood. Although Treg represent a small number of cells in whole blood, the reliability of measuring spe-cific mRNA from minority cell population has been recently validated for monocytes (11) and Th2 cells (12). Foxp3 is a transcriptional factor thought to be specifically linked to Treg. Its overexpression correlated with their differentiation (6). At the onset of shock, as expected, Foxp3 mRNA was not increased and was even found to be decreased in patients in comparison with healthy donors (p ϭ .015) and it returned to normal values at days 3-6 (Fig. 1D). Hence, we did not observe any increase in Foxp3 mRNA after septic shock, which agrees with results from absolute counting. In total, it suggests that Treg elevation is the consequence of a mechanism that does not imply Treg by themselves.

The elevated percentage of Treg has now been observed in a total of 69 patients with septic shock (including those of our preliminary study) (3). We suggest that this increase, independently of disease severity, may belong to (or be a consequence of) the homeostatic modulatory mechanisms rapidly occurring after the onset of shock, also known as compensatory anti-inflammatory response syndrome (13). One important mechanism of action proposed for Treg is cell-contact dependent. Hence, the relative abundance of those cells at the onset of shock may have a major impact on a patient's immune response. Experiments are currently under process in our laboratory to define a role for those cells during sepsisinduced immunoparalysis.

The main result of the present study is to demonstrate that the increased proportion of the CD25ϩ subset among CD4ϩ T cells is not due to a proliferation of Treg but rather to a selective depletion of CD4ϩCD25Ϫ T cells. Indeed, we have shown that, despite a moderate initial decrease, Treg tended to remain in normal range during septic shock, whereas CD4ϩCD25Ϫ were found to be dramatically decreased until day 6. Consequently, it seems that Treg are weakly affected by cell death reported during septic shock. As proposed in gastrointestinal malignancies (10), one may hypothesize that these cells are less sensitive to the massive sepsis-induced apoptotic processes than other lymphocytes. Accordingly, Treg have been demonstrated to be resistant to dexamethasone-induced apoptosis in mice ( 14) even though glucocorticoids are believed to be involved in sepsisinduced apoptosis (1). It has been also reported that Treg are resistant to Fasdependent apoptosis in vitro (15). This hypothesis remains to be further investigated. There has been growing recognition that apoptotic processes play a major role in septic shock pathophysiology. A better understanding of the interaction between apoptosis and immunosuppression may provide new targets for therapy of this hitherto deadly disease.

Objective:

  The elevation of the percentage of regulatory CD4؉CD25؉ T lymphocytes (Treg) has been recently described during septic shock. The objective of the present study was to investigate whether this increased percentage was due to Treg proliferation. Design: Observational study. Setting: Adult intensive care units in a university hospital. Subjects: Patients with septic shock (n ؍ 54) and healthy individuals (n ؍ 30). Interventions: None. Measurements and Main Results: In patients, we first confirmed the increased percentage of Treg among CD4؉ lymphocytes in comparison with healthy individuals. Surprisingly, regarding absolute counting, we demonstrated that both T CD4؉ lineages (CD25؉ and CD25؊) were diminished immediately after the onset of shock. Then, whereas Treg returned rapidly to healthy donors values, CD4؉CD25؊ T lymphocytes remained dramatically reduced. Finally, Foxp3 (Treg-related gene) messenger RNA quantification enabled us to definitively rule out a lack of proliferation since it was not increased during shock. Conclusion: The increased percentage of Treg after shock is due not to their proliferation but to a decrease in CD4؉CD25؊ T lymphocyte number. We hypothesize that it might be due to a resistance of Treg to apoptosis processes occurring during septic shock. (Crit Care Med 2004; 32:2329 -2331)

Figure 1 .

 1 Figure 1. Increased proportion of regulatory CD4ϩCD25ϩ T lymphocytes (Treg) during septic shock is due to the decrease of CD4ϩCD25Ϫ lymphocytes. A, CD4ϩ T cells were examined for their percentage of regulatory T cells (CD4ϩCD25ϩCD45ROϩCD69Ϫ). Circulating lymphocytes from 54 patients (open boxes) were phenotyped at days 1-2 (D1-D2, n ϭ 31) and days 3-6 (D3-D6, n ϭ 46) after the onset of shock. As control, values from 30 healthy donors (measured once) are represented (gray box). Data are shown as a box-plot representation. The nonparametric Mann-Whitney test was used to assess significance between groups (*p Ͻ .0001 vs. control values). Including results from our previous study, the percentage of Treg has now been determined in a total of 66 healthy individuals; results are 20.0 Ϯ 0.6 % (median, 18.6) on the overall normal population. B, absolute counting of CD4ϩCD25ϩ T lymphocytes was performed in whole blood from 30 patients with septic shock (open boxes) and 30 healthy individuals (gray box). Patients' lymphocytes were counted at days 1-2 (n ϭ 26) and days 3-6 (n ϭ 23) after the onset of shock; those of donors were measured once. Data are shown as a box-plot representation. The nonparametric Mann-Whitney test was used to assess significance between groups (*p ϭ .0001 vs. control values). C, absolute counting of CD4ϩCD25Ϫ T lymphocytes was performed in whole blood from 30 patients with septic shock (open boxes) and 30 healthy individuals (gray box). Patients' lymphocytes were counted at days 1-2 (n ϭ 26) and days 3-6 (n ϭ 23) after the onset of shock; those of donors were measured once. Data are shown as a box-plot representation. The nonparametric Mann-Whitney test was used to assess significance between groups (*p Ͻ .0001 vs. control values). D, Foxp3 gene expression was quantified in peripheral blood of 27 patients with septic shock at days 1-2 (n ϭ 22) and days 3-6 (n ϭ 14) after the onset of shock using real-time polymerase chain reaction (open boxes). Results are expressed as relative quantification ratio between Foxp3 and the housekeeping gene PPIB encoding for cyclophilin B. Normal expression levels were measured once in 16 healthy donors (gray box). Data are shown as a box-plot representation. The nonparametric Mann-Whitney test was used to assess significance between groups (*p Ͻ .02 vs. control values). mRNA, messenger RNA.

Table 1 .

 1 Demographic and clinical characteristics

		No. (%)
	Male	32 (59)
	Female	22 (41)
	Age, yrs, mean (95% CI)	63 (46-82)
	Main diagnosis category
	Medical	26 (48)
	Surgery	28 (52)
	Microbiologically documented	42 (78)
	diagnosis
	Bacilli Gram (Ϫ)	21 (39)
	Cocci Gram (ϩ)	26 (48)
	Fungi	15 (28)
	Others	8 (15)
	Type of infection
	Community acquired	29 (54)
	Nosocomial	25 (46)
	Site of infection
	Pulmonary	24 (44)
	Abdominal	22 (41)
	Others	8 (15)
	SAPS II at admission, mean	51 (35-68)
	(95% CI)
	CI, confidence interval; SAPS, Simplified
	Acute Physiology Score.

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ACKNOWLEDGMENTS

This work was conducted thanks to the logistic support (H. Thizy, F. Gueyffier) from the Centre d'Investigation Clinique (Clinical Research Centre) of INSERM and Hospices Civils de Lyon. We thank N. Voirin for the statistical part and the staff of the Flow Cytometry Unit (J. Baudot, C. Fernandez, MA. Guinand, MC Gutowski) at the Lyon Sud University Hospital.