J.-F Bercher
email: jean-francois.bercher@univ-paris-est.fr

Keywords: numbers: 02.50.-r, 05.90.+m, 89.70.-a AMS classication scheme numbers: 28D20, 94A17, 62B10, 39B62

In the present work, we show how the generalized Cramér-Rao inequality for the estimation of a parameter, presented in a recent paper, can be extended to the mutidimensional case with general norms on R n , and to a wider context. As a particular case, we obtain a new multidimensional Cramér-Rao inequality which is saturated by generalized q-Gaussian distributions. We also give another related Cramér-Rao inequality, for a general norm, which is saturated as well by these distributions. Finally, we derive uncertainty relations from these Cramér-Rao inequalities. These uncertainty relations involve moments computed with respect to escort distributions, and we show that some of these relations are saturated by generalized q-Gaussian distributions. These results introduce extended versions of Fisher information, new Cramér-Rao inequalities, and new characterizations of generalized q-Gaussian distributions which are important in several areas of physics and mathematics.

Introduction

It is well known that the Gaussian distribution is a central distribution with respect to classical information measures and inequalities. In particular, the Gaussian distribution is both a maximum entropy and a minimum Fisher information distribution over all distributions with the same variance. We will show that the same kind of result holds for the family of generalized q-Gaussians, for Rényi or Tsallis entropy and a suitable extension of the Fisher information.

These generalized q-Gaussians appear in statistical physics, where they are the maximum entropy distributions of the nonextensive thermostatistics [START_REF] Tsallis | Introduction to Nonextensive Statistical Mechanics[END_REF].

The generalized q-Gaussian distributions dene a versatile family that can describe problems with compact support as well as problems with heavy tailed distributions. They are also analytical solutions of actual physical problems, see [START_REF] Lutz | Anomalous diusion and Tsallis statistics in an optical lattice[END_REF], [START_REF] Schwämmle | q-Gaussians in the porous-medium equation: stability and time evolution[END_REF] [4], [START_REF] Ohara | Information geometry of q-Gaussian densities and behaviors of solutions to related diusion equations[END_REF], and are sometimes known as Barenblatt-Pattle functions, following their identication by [START_REF] Barenblatt | On some unsteady motions of a liquid and a gas in a porous medium[END_REF], [START_REF] Pattle | Diusion from an instantaneous point source with concentration dependent coecient[END_REF]. We shall also mention that the Generalized q-Gaussian distributions appear in other elds, namely as the solution of non-linear diusion equations, or as the distributions that saturate some sharp inequalities in functional analysis [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diusions[END_REF], [START_REF] Pino | The optimal euclidean Lp-Sobolev logarithmic inequality[END_REF], [START_REF] Cordero-Erausquin | A mass-transportation approach to sharp sobolev and Gagliardo-Nirenberg inequalities[END_REF], [START_REF] Agueh | Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations[END_REF].

In the literature, and in particular within nonextensive thermostatistics, several extensions of the Fisher information and of the Cramér-Rao inequality have been proposed, e.g. [START_REF] Pennini | Semiclassical information from deformed and escort information measures[END_REF], [START_REF] Furuichi | On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics[END_REF], [START_REF] Furuichi | On generalized Fisher informations and Cramér-Rao type inequalities[END_REF], [START_REF] Naudts | Generalised exponential families and associated entropy functions[END_REF], [START_REF] Naudts | The q-exponential family in statistical physics[END_REF]. In information theory, the remarkable work by Lutwak et al. [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF], [START_REF] Lutwak | Extensions of Fisher information and Stam's inequality. Information Theory[END_REF] also denes an extended Fisher information and a Cramér-Rao inequality saturated by q-Gaussian distributions.

However the Fisher information is originally dened in a broader context as the information about a parameter of a parametric family of distributions. It is only in the special case of a location parameter that it reduces to the Fisher information of the distribution. The Fisher information is especially important for the formulation of the Cramér-Rao inequality. This well-known inequality appears in the context of estimation theory, where it denes a lower bound on the variance of any estimator of a parameter.

In our recent work [START_REF] Bercher | On generalized Cramér-Rao inequalities, generalized Fisher information and characterizations of generalized q-Gaussian distributions[END_REF], we have thrown a bridge between concepts in estimation theory and tools of nonextensive thermostatistics.

Using the notion of escort distribution, we have established an extended version of the Cramér-Rao inequality for the estimation of a parameter.

This new Cramér-Rao inequality includes the standard one, as well as Barankin-Vajda versions [20, Corollary 5.1], [START_REF] Vajda | χ α -divergence and generalized Fisher information[END_REF] as particular cases. Furthermore, in the case of a location parameter, we have obtained extended versions of the standard Cramér-Rao inequality, which are saturated by the generalized q-Gaussians. This means that among all distributions with a given moment, the generalized q-Gaussians are also the minimizers of extended versions of the Fisher information, just as the standard Gaussian minimizes Fisher information over all distributions with a given variance.

This result yields a new information-theoretic characterization of these generalized Gaussian distributions.

However, a quite frustrating point is that these results were limited to the univariate case, while the multidimensional case is obviously of high importance. This restriction is overcome in the present paper where we show that previous results can be extended to the multidimensional case. More than that, we consider an even wider context where moments of the error are computed with respect to two dierent probability distributions. In addition, giving our results for general norms will be hardly more dicult than for Euclidean norms, so we consider this general case from the beginning. Finally, we derive new uncertainty relations from the multidimensional Cramér-Rao inequalities. Let us now give a brief overview of the results, together with the organization of the paper.

Let θ ∈ Θ ⊆ R n be a multidimensional parameter that we wish to estimate using data x. We show that for θ(x) an estimator of θ, if f (x; θ) and g(x; θ) are two probability densities, and if α and β are Hölder conjugates of each other, then

E θ(x) -θ α 1 α I β [f |g; θ] 1 β ≥ |n + ∇ θ . B f (θ)| , (1)
where . is a general norm on R n , ∇ θ . B f (θ) represents the divergence of the bias between θ(x) and θ, and I β [f |g; θ] stands for a generalized Fisher information that measures the information in f about θ, and is taken with respect to g. This general result is established in section 3. Then we discuss in subsection 3.2 some special cases of this general inequality. In particular, if f and g is a pair of q-escort distributions, we obtain

E θ(x) -θ α 1 α I β,q [f |g; θ] 1 β ≥ n + ∇ θ .E q θ(x) -θ (2)
E q θ(x) -θ α 1 α I β,q [f |g; θ] 1 β ≥ n + ∇ θ .E θ(x) -θ , (3)
where I β,q [f |g; θ] is the generalized (β, q)-Fisher information, and E q [.] denotes the q-expectation which is used in nonextensive statistics. These results are the mutidimensional extensions, with an arbitrary norm, of our previous q-Cramér-Rao inequalities [START_REF] Bercher | On generalized Cramér-Rao inequalities, generalized Fisher information and characterizations of generalized q-Gaussian distributions[END_REF]. In the monodimensional case and q = 1, these inequalities reduce to the Barankin-Vajda Cramér-Rao inequality, and to the standard Cramér-Rao inequality for α = β = 2. In addition, in the case of a location parameter, we show that

E [ x α ] 1 α I β,q [g] 1 β ≥ n (4)
which reduces again to our previous results in the univariate case. Examining carefully the cases for equality in (4), we exhibit that the lower bound is attained by generalized q-Gaussian distributions, and we prove that these generalized Gaussian are the unique extremal functions, provided that the dual norm is strictly convex. For a random vector x in R n , these generalized q-Gaussian have the probability density

G γ (x) = 1 Z(γ) (1 -(q -1) γ x α ) 1 q-1 + for q = 1 1 Z(γ) exp (-γ x α ) if q = 1 (5)
for α ∈ (0, ∞), γ a real positive parameter and q > (n -α)/n, where we use the notation (x) + = max {x, 0}, and where Z(γ) is the partition function §. For q > 1, the density has a compact support, while for q ≤ 1 it is dened on the whole R n and behaves as a power distribution for x → ∞. A shorthand notation for the expression of the generalized q-Gaussian density is

G γ (x) = 1 Z(γ) exp q * (-γ x α ) , (6)
with q * = 2 -q, and where the so-called q-exponential function is dened by

exp q (x) := (1 + (1 -q)x) 1 1-q

+

, for q = 1 and exp q=1 (x) := exp(x).

(7) § In the case of the Euclidean norm, the general expressions of the main information measures attached to the generalized Gaussians are derived in Appendix A of [START_REF] Bercher | On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions[END_REF]. Similar expressions can readily be obtained in the case of a general norm, using the change of variable in polar coordinates x = r u, with u = x/||x|| and the representation of the Lebesgue measure dx = r n-1 dr dσ(u), c.f. [23, p. 87], where dσ(u) denotes the surface element on the unit sphere. By this remark, the expressions in [22, Appendix A] are valid, with the proviso that ωn will denote the volume of the n-dimensional unit ball B = {x ∈ R n : x ≤ 1}.

In section 4, we present another Cramér-Rao type inequality which is also saturated by the generalized q-Gaussian. This inequality has been originally established by [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF], and extended to the multidimensional case in [START_REF] Lutwak | Extensions of Fisher information and Stam's inequality. Information Theory[END_REF] and independently in [START_REF] Bercher | On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions[END_REF] in the case of an Euclidean norm. We show here that this last inequality can readily be stated and proved in the case of an arbitrary norm.

Finally, in section 5, we derive some new multidimensional uncertainty relations from the generalized q-Cramér-Rao inequalities. These uncertainty relations involve escort mean values and are saturated by generalized Gaussians. In particular, we obtain an inequality of the form

E k 2 [ x γ 2 ] 1 γ E ξ θ 2 1 θλ > 1 M k 2 [|ψ| 2 ] 1 kλ E k 2 [ x γ 2 ] 1 γ E ξ θ 2 1 θλ ≥ K, (8)
where x and ξ are two Fourier dual variables, γ ≥ 2, θ ≥ 2, and E k 2

[.] denotes an expectation computed with respect to an escort distribution of order k 2 . The lower bound K is xed and attained when the underlying wave function is a q-Gaussian. For γ = θ = 2, q = 1, [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diusions[END_REF] gives the multidimensional version of the well-known Weyl-Heisenberg uncertainty principle.

In order to derive these dierent inequalities, we will need some preliminary results, in particular concerning some properties of general norms on R n . This is the objective of section 2 where we rst dene the notion of dual norms and prove a result on the gradient of a general norm. Then, we establish, together with its equality conditions, a general Hölder-type inequality. This inequality will be an essential ingredient in the derivation of the general Cramér-Rao inequality (1).

Preliminary results

As mentioned above, we will consider here general norms on R n . Let us rst simply recall that a norm on R n is a function . : R n → R + such that for any x, y ∈ R n and γ ∈ R, then (a) γx = |γ| x , (b) x + y ≤ x + y , and (c) x = 0 i x = 0. . We shall mention that it is possible to use weighted versions of the norms above, e.g. x w,p = (

n i=1 w i |x| p i ) 1
p , with w i > 0, and that any injective linear transformation A leads to a new norm, such as x A = Ax . Finally, it is also possible to construct new norms by combining dierent norms dened on subvectors of x.

A related important notion is the notion of dual-norm. Let E = (R n , . ) a n-dimensional normed space, where . is an arbitrary norm, and let us denote E * = (R n , . * ) its dual space. For Y ∈ E * , the dual norm . * is dened by

Y * = sup X ≤1
X.Y, [START_REF] Cordero-Erausquin | A mass-transportation approach to sharp sobolev and Gagliardo-Nirenberg inequalities[END_REF] where X.Y is the standard scalar product X.Y = n i=1 X i Y i . In particular, it is well known that if . is a L p -norm, then . * is the L q -norm, where p and q are Hölder conjugates of each other, i.e.

p -1 + q -1 = 1, see e.g. [24, chapter 5]. By a direct consequence of the denition of the dual norm, we always have X.Y ≤ X Y * . [START_REF] Agueh | Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations[END_REF] Note that when the dot product X.Y is negative, we can always take the minus of one of the elements to get |X.Y | = X. (-Y ) ≤ X -Y * = X Y * . Hence, we see that we actually have an extension of Hölder's inequality for vectors:

|X.Y | ≤ X Y * . (12)
Obviously, we recover here the Cauchy-Schwarz inequality if . = . 2 and the standard Hölder inequality for vectors if . = . p , and thus . * = . q .

In the following, we will need several facts on the gradient of a norm. These facts are stated in the next Lemma.

Lemma 1. Let . be dierentiable at x ∈ E, and denote x * = ∇ x . (x) ∈ E * the gradient of the norm at x. The gradient of x satises (a) x.x * = x and (b) x * * = 1. Furthermore, when the dual norm . * is strictly convex, then the gradient x * is the unique vector that satises (a) and (b).

Proof. We begin by equality (a). Let x and v two vectors of E and λ a real parameter. By the triangle inequality, we have x

+ λv ≤ x + λ v , so that lim λ→0 x + λv -x λ ≤ v . (13)
In the other hand, the chain rule for derivation

d Z dλ = dZ dλ .∇ Z Z , with Z = x + λv, gives d x + λv dλ λ=0 = v.∇ x x ≤ v , (14)
where the right inequality follows from [START_REF] Furuichi | On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics[END_REF]. Of course, taking v = x in [START_REF] Furuichi | On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics[END_REF] gives the equality sign, and ( 14) becomes x.∇ x x = x , that is (a).

By [START_REF] Furuichi | On generalized Fisher informations and Cramér-Rao type inequalities[END_REF], we also get that

∇ x x . v v ≤ 1, (15)
with equality if v = x. Therefore, in the denition of the dual norm ∇ x x * = sup w ≤1 w.∇ x x , the supremum is equal to one and is attained for w = x/ x . This proves (b).

In functional analysis, the existence of a solution to a system analogue to (a),(b) is granted by a consequence of the Hahn-Banach theorem, see e.g. the background material in [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF]. In this context, the uniqueness of extension in the Hahn-Banach theorem, therefore the uniqueness of x * , is guaranteed if the primal space E is smooth, which in turn is equivalent to the strict convexity of the dual norm [START_REF] Limaye | Functional Analysis[END_REF]Chapter 2]. In our setting, it is easy to check that we have uniqueness of the solution to the system (a),(b) provided that the dual norm is strictly convex. Indeed, if x * 1 and x * 2 are two solutions to (a),(b), we have by (a)

x x . (x * 1 + x * 2 ) = 2. Accordingly, the dual norm x * 1 + x * 2 * = sup w ≤1 w. (x * 1 + x * 2 ) is necessarily greater than 2: x * 1 + x * 2 * ≥ 2.
In the other hand, if the dual norm is strictly convex and using (b), we have x

* 1 + x * 2 * ≤ x * 1 * + x * 2 * = 2, with equality if and only if x * 1 = x * 2 .
Combining the two inequalities, we see that the two solutions are necessarily equal. Finally, since we have already identied that the gradient of the norm satises (a),(b), we get the last item in the Lemma.

The standard Hölder's inequality works for functions and relates the L 1 norm of the product of two functions to the product of their L p and L q norms: f g 1 ≤ f p g q , with 1 ≤ p, q ≤ ∞ and 1/p + 1/q = 1. For vectors and an arbitrary norm . , the inequality [START_REF] Pennini | Semiclassical information from deformed and escort information measures[END_REF] gives another kind of Hölder inequality (actually, this inequality is also true in a broader context, see e.g. [START_REF] Morrison | Functional Analysis: An Introduction to Banach Space Theory[END_REF]).

By combining these two inequalities, we obtain another Hölder-type inequality for vector-valued functions, which involves arbitrary norms. This inequality will be a key in the derivation of the new multidimensional Cramér-Rao inequality. It is given, with its equality condition, in the following Lemma.

Lemma 2. Let E = (R n , . ) be a n-dimensional normed space and E * = (R n , . * ) its dual space. If X(t) and Y (t) are two functions taking values respectively in E and E * , and if w(t) is a weight function, then

ˆ X(t) α w(t)dt 1 α ˆ Y (t) β * w(t)dt 1 β ≥ ˆ|X(t).Y (t)| w(t)dt, (16)
≥ ˆX(t).Y (t) w(t)dt [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] with α and β Hölder conjugates of each other, i.e.

α -1 + β -1 = 1, α ≥ 1.
The equality is obtained if

Y (t) = K X(t) α-1 ∇ X(t) X(t) , (18)
with K ∈ R for inequality [START_REF] Naudts | The q-exponential family in statistical physics[END_REF], and with K ∈ R + for the lower bound [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF]. If the dual norm is strictly convex, then the function Y (t) in ( 18) above is the unique function which saturates the inequalities [START_REF] Naudts | The q-exponential family in statistical physics[END_REF][START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF].

Proof.

Obviously, we always have

ˆX(t).Y (t) w(t)dt ≤ ˆ|X(t).Y (t)| w(t)dt, (20)
with equality if X(t).Y (t) ≥ 0 everywhere. Then, it only remains to apply the standard Hölder inequality to right hand side of ( 19):

ˆ X(t) Y (t) * w(t)dt ≤ ˆ X(t) α w(t)dt 1 α ˆ Y (t) β * w(t)dt 1 β (21)
to obtain [START_REF] Naudts | The q-exponential family in statistical physics[END_REF]. The inequality [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] then follows by [START_REF] Barankin | Locally best unbiased estimates[END_REF].

As far as the cases of equality are concerned, we know that in the Hölder inequality [START_REF] Vajda | χ α -divergence and generalized Fisher information[END_REF], the equality is obtained if and only if Y (t) β * = K X(t) α , with K a positive constant. Using the fact that α/β = α -1, the condition becomes Y (t) * = K X(t) α-1 . This condition implies that Y (t) must be of the form

Y (t) = K X(t) α-1 u, ( 22
)
where u is a vector of E * with unit norm: u * = 1. By inequality [START_REF] Pennini | Semiclassical information from deformed and escort information measures[END_REF] we see that the integrand in the left side of ( 19) is always less or equal the integrand on the right. Therefore, we will only get equality in [START_REF] Bercher | On generalized Cramér-Rao inequalities, generalized Fisher information and characterizations of generalized q-Gaussian distributions[END_REF] if the integrands are equal. Then if we plug [START_REF] Bercher | On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions[END_REF] in the inequality [START_REF] Pennini | Semiclassical information from deformed and escort information measures[END_REF], we obtain the condition X(t) α-1 |X(t).u| = X(t) α , that is nally |X(t).u| = X(t) . Since we know by Lemma 1 that v = ∇ X(t) X(t) is a unit vector that satises X(t).v = X(t) , and is unique if the dual norm is strictly convex, we see that u = ±v = ±∇ X(t) X(t) and this concludes the proof of the rst inequality. For equality to hold in the lower bound [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF], the integrand must be positive, which in turn implies that X(t).u = X(t) and u = ∇ X(t) X(t) .

The generalized Cramér-Rao inequality

In this section, we rst derive a main Cramér-Rao inequality for the estimation of a multidimensional parameter and introduce a generalized version of Fisher information.

Next, we examine the particular case of a pair of escort distributions, and then the case of a location parameter. So doing, we obtain multidimensional versions of the q-Cramér-Rao inequality and a Cramér-Rao inequality characterizing generalized q-Gaussian distributions.

The main Cramér-Rao inequality for the estimation of a parameter

The problem of estimation is to determine a function θ(x) in order to estimate an unknown parameter θ. Let f (x; θ) and g(x; θ) be two probability density functions, with x ∈ X ⊆ R k and θ a parameter of these densities, θ ∈ R n . An underlying idea in the statement of the new Cramér-Rao inequality is that it is possible to evaluate the moments of the error with respect to dierent probability distributions. For instance, in the estimation setting the estimation error is θ(x) -θ. The bias can be evaluated with respect to f according to

B f (θ) = ˆX θ(x) -θ f (x; θ) dx = E f θ(x) -θ (23)
while a general moment of a norm of the error can be computed with respect to another distribution, g(x; θ), as in

E g θ(x) -θ β = ˆX θ(x) -θ β g(x; θ) dx (24)
The distributions f (x; θ) and g(x, θ) can be chosen arbitrarily and are not necessarily directly related. However, g(x; θ) can be designed as a transformation of f (x; θ) that highlights, or perhaps scores out, some characteristics of f (x; θ). Typically, g(x; θ) can be a weighted version of f (x; θ), i.e. g(x; θ) = h(x; θ)f (x; θ). The distribution g(x; θ) can also be a quantized version of f (x; θ), such as g(x; θ) = [f (x; θ)] , where [.] denotes the integer part. Another important special case is when g(x; θ) is dened as the escort distribution of order q of f (x; θ), where q plays the role of a tuning parameter. We will see that this special case, which is particularly important in the context of nonextensive statistical physics, will lead to generalized q-Gaussians as the extremal functions. We are now in position to state and prove an extended version of the Cramér-Rao inequality.

Theorem 1. Let f (x; θ) be a multivariate probability density function dened over a subset X⊆ R n , and θ ∈ Θ ⊆ R k a parameter of the density. The set Θ is equipped with a norm . , and the corresponding dual norm is denoted . * . Let g(x; θ) denote another probability density function also dened on (X; Θ). Assume that f (x; θ) is a jointly measurable function of x and θ, is integrable with respect to x, is absolutely continuous with respect to θ, and that the derivatives with respect to each component of θ are locally integrable. For any estimator θ(x) of θ, we have

E g θ(x) -θ α 1 α I β [f |g; θ] 1 β ≥ |n + ∇ θ . B f (θ)| (25)
with α and β Hölder conjugates of each other, i.e. α -1 +β -1 = 1, α ≥ 1, and where the (β, g)-Fisher information

I β [f |g; θ] = ˆX ∇ θ f (x; θ) g(x; θ) β * g(x; θ) dx (26)
is the generalized Fisher information of order β on the parameter θ contained in the distribution f and taken with respect to g. The equality case is obtained if

∇ θ f (x; θ) g(x; θ) = K θ(x) -θ α-1 ∇ θ(x)-θ θ(x) -θ , (27)
with K > 0.

Proof. The bias in ( 23) is a n-dimensional vector. Let us consider its divergence with respect to variations of θ:

div B f (θ) = ∇ θ . B f (θ). (28)
The regularity conditions in the statement of the theorem enable to interchange integration with respect to x and dierentiation with respect to θ, and

∇ θ . B f (θ) = ˆX ∇ θ . θ(x) -θ f (x; θ) dx + ˆX ∇ θ f (x; θ). θ(x) -θ dx. ( 29
)
In the rst term on the right, we have ∇ θ .θ = n, and the integral reduces to -n ´X f (x; θ) dx = -n, since f (x; θ) is a probability density on X. The second term can be rearranged so as to obtain an integration with respect to the density g(x; θ), assuming that the derivatives with respect to each component of θ are absolutely continuous with respect to g(x; θ), i.e. g(x; θ)

∇ θ f (x; θ). This gives n + ∇ θ . B f (θ) = ˆX ∇ θ f (x; θ) g(x; θ) . θ(x) -θ g(x; θ) dx. (30)
Now, it only remains to apply the generalized Hölder-type inequality [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF] in Lemma 2 to the integral on the right side, with X(x) = θ(x) -θ, Y (x) = ∇ θ f (x;θ) g(x;θ) , and w(x) = g(x; θ). This yields in all generality

ˆX θ(x) -θ α g(x; θ) dx 1 α ˆX ∇ θ f (x; θ) g(x; θ) β * g(x; θ) dx 1 β ≥ |n + ∇ θ . B f (θ)| (31)
which is [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF].

By Lemma 2 again, we know that the case of equality occurs if Y (t) = K X(t) α-1 ∇ X(t) X(t) , K > 0, which gives [START_REF] Morrison | Functional Analysis: An Introduction to Banach Space Theory[END_REF].

3.2. Main consequences of the general result 3.2.1. Case of a q-escort distribution Let f (x; θ) and g(x; θ) be a pair of of q-escort distributions linked by

f (x; θ) = g(x; θ) q M q [g; θ] and g(x; θ) = f (x; θ) q M q [f ; θ] , (32)
with q > 0, q = 1/q, and the information generating function M q [g; θ] dened by

M q [g; θ] = ˆX g(x; θ) dx. ( 33
)
As usual, we will denote by E q [.] the q-expectation, which is the expectation taken with respect to an escort distribution of order q. Here we see that the expectation with respect to f (x; θ) is also the q-expectation with respect to g(x; θ). Let us also recall that the inverse function of the deformed q-exponential (7), the so-called q-logarithm, is dened by

ln q (x) := x 1-q -1 1 -q . ( 34
)
With these notations, we have the following corollary of the general Cramér-Rao inequality.

Corollary 1. For the pair of escort distributions [START_REF] Angulo | Uncertainty relationships in many-body systems[END_REF], the equivalent Cramér-Rao inequalities

E θ(x) -θ α 1 α I β,q [f |g; θ] 1 β ≥ n + ∇ θ .E q θ(x) -θ (35)
E q θ(x) -θ α 1 α I β,q [f |g; θ] 1 β ≥ n + ∇ θ .E θ(x) -θ , (36)
hold, where the generalized (β, q)-Fisher information is given by

I β,q [f |g; θ] = 1 M q [g; θ] β E g(x; θ) β(q-1) ∇ θ ln g(x; θ) q M q [g; θ] β * (37) = M q [f ; θ] β E q f (x; θ) β(1-q) ∇ θ ln f (x; θ) β * , (38)
and where equality occurs if

∇ θ ln q f (x; θ) = K θ(x) -θ α-1 ∇ θ(x)-θ θ(x) -θ , (39)
with K > 0.

Proof. The Cramér-Rao inequalities [START_REF] Bialynicki-Birula | Formulation of the uncertainty relations in terms of the Rényi entropies[END_REF] and [START_REF] Zozor | On classes of non-gaussian asymptotic minimizers in entropic uncertainty principles[END_REF] directly follow from the general Cramér-Rao inequality [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF], the relations (32) between f (x; θ) and g(x; θ), and the notation of q-expectations.

The expressions of the generalized (β, q)-Fisher information also follow by direct calculation. Finally, the equality condition yields

f (x; θ) (1-q) ∇ θ ln f (x; θ) = K θ(x) -θ α-1 ∇ θ(x)-θ θ(x) -θ . (40)
Noticing that the term on the left is nothing but the gradient of the deformed q-logarithm, f (x; θ) (1-q) ∇ θ ln f (x; θ) = ∇ θ ln q f (x; θ), we immediately obtain (39).

Case of a translation family

In the particular case of a translation parameter, the generalized Cramér-Rao inequality induces a new class of inequalities.

Let θ ∈ R n be a location parameter, x ∈ X ⊆ R n , and dene by f (x; θ) the family of density f (x; θ) = f (x -θ). In this case, we have ∇ θ f (x; θ) = -∇ x f (x -θ), provided that f is dierentiable at x -θ, and the Fisher information becomes a characteristic of the information in the distribution. If X is a bounded subset, we will assume that f (x) vanishes and is dierentiable on the boundary ∂X (otherwise the Fisher information dened for the function extended to R n is not dened).

Let us denote by µ f the mean of f (x). We immediately get that the mean of f (x; θ) is (µ f + θ), so that an unbiased estimator of θ could be θ(x) = x -µ f . If we choose θ(x) = x, the estimator will be biased, B f (θ) =E q θ(x) -θ = µ f , but independent of θ, so that the gradient of the bias with respect to θ is zero. In these conditions, the generalized Cramér-Rao inequality becomes ˆ

x -θ α g(x; θ) dx

1 α ˆ ∇ x f (x -θ) g(x; θ) β * g(x; θ) dx 1 β ≥ n. (41)
Furthermore, we can also choose θ = 0, and obtain, as a corollary, the following interesting functional inequality.

Corollary 2. Let f (x) and g(x) be two multivariate probability density functions dened over a subset X of R n . Assume that f (x) is a measurable dierentiable function of x, which vanishes and is dierentiable on the boundary ∂X, that ∇ x f (x) is absolutely continuous with respect to g(x), and nally that the involved integrals exist and are nite. Then, the following inequality holds

ˆX x α g(x) dx 1 α ˆX ∇ x f (x) g(x) β * g(x) dx 1 β ≥ n, (42)
with equality if (and only if when the dual norm is strictly convex)

∇ x f (x) = -K g(x) x α-1 ∇ x x , (43)
with K > 0.

As an elementary application, let us consider the univariate case, with X = [0, 1]. Let us take for g(x) the uniform distribution on the interval. Finally, let us choose for f (x) a β-

distribution: f (x) = x a-1 (1 -x) b-1 /B(a, b), with B(a, b) the β-function. Firstly, we obviously have ´1 0 |x| α dx = 1. Secondly, f (x) = (a -1)x a-2 (1 -x) b-1 + (b -1)x a-1 (1 -x) b-2 , so that the inequality is ˆ1 0 (a -1)x a-2 (1 -x) b-1 + (b -1)x a-1 (1 -x) b-2 β dx 1 β ≥ B(a, b). (44)
Taking now β = 1 and a > 1, b > 1, we obtain the following inequality for β-functions:

(a -1)B(a -1, b) + (b -1)B(a, b -1) ≥ B(a, b) (45)
3.2.3. Case of a location parameter within a pair of escort distributions By combining the two aspects presented above, namely the case of a pair of escort distributions and the case of a location parameter, we will obtain a new Cramér-Rao inequality saturated by multivariate generalized q-Gaussians. This provides a new information theoretic characterization of generalized q-Gaussian and extend our previous results to the multivariate case and arbitrary norms. As in Corollary 1, we use a pair of of q-escort distributions:

f (x) = g(x) q M q [g] and g(x) = f (x) q M q [f ] , (46)
with q = 1/q, and we denote by E [.] the standard expectation with respect to g(x), and by E q [.] the q-expectation with respect to f (x), which is simply the standard expectation taken with respect to the escort f (x) q /M q [f ] . In the statement of the following corollary, we will use the deformed exponential and logarithm dened in ( 7), [START_REF] Hirschman | A note on entropy[END_REF]. We will also use the notation q * = 2 -q that changes the quantities (1 -q * ) into (q -1).

Corollary 3. Let g(x) be a multivariate probability density function dened over a subset X ⊆ R n . Assume that g(x) is a measurable dierentiable function of x, which vanishes and is dierentiable on the boundary ∂X, and nally that the involved integrals exist and are nite. Then, for the pair of escort distributions (46), the following q-Cramér-Rao inequality holds

m α [g] 1 α I β,q [g] 1 β ≥ n (47) with          m α [g] = E [ x α ] I β,q [g] = (q/M q [g]) β E g(x) β(q-1) ∇ x ln g(x) β * = (q/M q [g]) β E ∇ x ln q * g(x) β * , (48)
where α and β are Hölder conjugates of each other, i.e. α -1 + β -1 = 1, α ≥ 1, and where I β,q [g] denotes the generalized (β, q)-Fisher information.

In terms of q-expectations with respect to f (x), it can also be written

m α,q [f ] 1 α Īβ,q [f ] 1 β ≥ n (49) with          m α,q [f ] = E q [ x α ] Īβ,q [f ] = M q [f ] β E q f (x) β(1-q) ∇ x ln f (x) β * = M q [f ] β E q ∇ x ln q f (x) β * . ( 50
)
In both cases, equality occurs if

f (x) ∝ exp q (-γ x α )
, or equivalently g(x) ∝ exp q * (-γ x α ) , with γ > 0.

(

) 51
If the dual norm is strictly convex, then this generalized q-Gaussian is the unique probability density function that achieves the equality in the extended Cramér-Rao inequalities.

Proof. As indicated above, the result is a direct consequence of Theorem 1 in the case of a pair of escort distributions and of the estimation of a location parameter, with θ(x) = x and θ = 0. Using (46), the condition for equality (43) becomes

g(x) q-1 ∇ x g(x) = -K x α-1 ∇ x x g(x). ( 52
)
From this equation, we see that g(x) will only be a function of the norm of x, and therefore will be radially symmetric. Furthermore, we see that the gradient of g(x) behaves as the negative of the gradient of x . This means that g(x), which is a function of x , is non increasing with x .

For g(x) = 0, the equality condition can be written

g(x) q-2 ∇ x g(x) = 1 q -1 ∇ x g(x) q-1 = - K α ∇ x x α , (53)
which, after integration of the two sides, gives

g(x) q-1 = - K α (q -1) x α + C, ( 54
)
where C is a constant of integration. Since g(x) is a probability density function, the solution is restricted to the domain where the right hand side is non negative, and g(x) = 0 elsewhere. In particular, when C is negative, we see that g(x) vanishes around the origin and presents a singularity at x = Cα/K(q -1). Since we assumed g(x) dierentiable everywhere, this solution must be discarded.

Therefore, the constant of integration C must be positive, and

g(x) ∝ 1 - K Cα (q -1) x α 1 q-1 ∝ exp q * (-γ x α ) (55)
which is (51). The expression of f (x) simply follows from the fact that f (x) is the escort distribution of g(x).

Another Cramér-Rao inequality saturated by generalized Gaussians

We nish this paper with another Cramér-Rao type inequality, which involves a variant φ β,q [g] of the generalized Fisher information I β,q [g] above, and which is also saturated by generalized q-Gaussian distributions. However, this inequality is less directly related to estimation results than the inequality (47) which is just a special case of the general Cramér-Rao inequality. The monodimensional version of this inequality has been established by [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF], and extended to the multidimensional case in [START_REF] Lutwak | Extensions of Fisher information and Stam's inequality. Information Theory[END_REF] and in [START_REF] Bercher | On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions[END_REF] in the case of an Euclidean norm. Actually, the inequality can readily be stated and proved for a general norm.

Theorem 2. For n ≥ 1, β and α Hölder conjugates of each other, α > 1, q > max {(n -1)/n, n/(n + α)} then for any probability density g on R n , supposed continuously dierentiable and such that the involved information measures are nite,

m α [g] 1 α φ β,q [g] 1 βλ ≥ m α [G] 1 α φ β,q [G] 1 βλ (56)
with λ = n(q -1) + 1 and where the general Fisher information is given by

φ β,q [g] = (M q [g] /q) β I β,q [g] = E g(x) β(q-1) ∇ x ln g(x) β * , (57)
and where the equality holds i g is a generalized Gaussian g = G γ .

For the proof of this inequality, we will use two general inequalities relating the moment m α [g], the generalized Fisher information φ β,q [g] and the information generating function M q [g] = ´g(x) q dx. We will also use the notation N q [g] = M q [g] 1 1-q ,which is known as the Rényi entropy power. In the theorem above as well as in the lemma below, φ β,q [G], m α [G] and N q [G] are the values taken by the Fisher information, the moment of order α and the entropy power when the probability density g is the generalized q-Gaussian G. The exact expressions of these quantities are given in the Appendix A of [START_REF] Bercher | On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions[END_REF], taking into account the footnote page 3. Lemma 3. For n ≥ 1, α ∈ (0, ∞), q > n/(n + α), and if g is a probability density on random vectors of

R n with m α [g] = E[ x α ] < ∞, N q [g] < ∞, then m α [g] 1 α N q [g] 1 n ≥ m α [G] 1 α N q [G] 1 n , (58)
with equality if and only if g is a generalized Gaussian.

Proof. The inequality (58) has been stated and proved in [START_REF] Lutwak | Moment-Entropy inequalities for a random vector[END_REF] in the case of an Euclidean norm.

We simply note here that the proof in [START_REF] Lutwak | Moment-Entropy inequalities for a random vector[END_REF] works as well in the case of a general norm.

We will also use a generalized Stam inequality derived from a general sharp Gagliardo-Nirenberg inequality proved in the remarkable paper of Cordero et al. [START_REF] Cordero-Erausquin | A mass-transportation approach to sharp sobolev and Gagliardo-Nirenberg inequalities[END_REF].

Lemma 4. For n ≥ 1, β and α Hölder conjugates of each other, α > 1, and q > max {(n -1)/n, n/(n + α)}, then for any probability density on R n , supposed continuously dierentiable, the following generalized Stam inequality holds

φ β,q [g] 1 βλ N q [g] 1 n ≥ φ β,q [G] 1 βλ N q [G] 1 n , (59)
with λ = n(q -1) + 1 and with equality if and only if g is any generalized q-Gaussian (6).

Proof. In our notations, the sharp Gagliardo-Nirenberg inequality [10, Eq. (34) p. 320], with a > 1

and ∇u β = ´ ∇u β * dx 1 β
, is

∇u β u 1 θ -1 a(β-1)+1 u 1 θ aβ ≥ K (60)
where K is a sharp constant which is attained if and only if u is a generalized Gaussian with exponent 1/(1 -a), and where θ is given by θ = n(a -1)/a(nβ -(aβ + 1 -a)(n -β)). The idea is to take u = g t , g being a probability density function, with aβt = 1, and to note q = [a(β -1) + 1] t.

With these notations, we get that βt = β(q -1) + 1, and (60) becomes

φ β,q [g] 1 β N q [g] (θ-1)(1-q)t θq ≥ K. (61)
Simplifying the expression of θ and the exponent in (61), we nally obtain, with q < 1, the generalized Stam inequality (59), with equality if and only if g is any generalized q-Gaussian (6). Actually, this generalized Stam inequality is also valid in the case q > 1, as it can be checked from [START_REF] Cordero-Erausquin | A mass-transportation approach to sharp sobolev and Gagliardo-Nirenberg inequalities[END_REF]'s results using similar steps as above. The conditions on q simply ensure the existence of the information measures for the generalized Gaussian.

We end with the proof of theorem 2, which is now an easy task.

Proof. The Cramér-Rao inequality (47) can also be written

m α [g] 1 α φ β,q [g] 1 β n q M q [g] ≥ 1. (62)
Eliminating M q [g] between this inequality and the moment-entropy inequality (58) with q > 1, we arrive at (56). Similarly, in the case q < 1, the elimination of M q [g] between the extended q-Cramér-Rao inequality for a location parameter (62) and the generalized Stam inequality (59) also leads to (56). The case of equality directly follows from the cases of equality in the initial inequalities. Alternatively, we can observe that (56) also follows at once by the combination of ( 58) and (59).

5. Uncertainty relations associated with the q-Cramér-Rao inequalities It is known that the Weyl-Heisenberg uncertainty principle in statistical physics corresponds to the standard Cramér-Rao inequality for the location parameter, see e.g. [START_REF] Stam | Some inequalities satised by the quantities of information of Fisher and Shannon[END_REF]. Following this idea, we derive new uncertainty relations from the extended Cramér-Rao inequalities. These uncertainty relations involve escort mean values and are saturated by generalized Gaussians. These uncertainty relations thus give a way to measure the uncertainty not only with respect to the original wave function, but also with respect to an escort-deformed version of the associated probability density.

Of course, the standard Weyl-Heisenberg uncertainty inequality is recovered in the case q = 1.

Let us recall that the uncertainty principle originates in Heisenberg's work and is stated in Weyl's book [30, page 77], who credits Pauli. It indicates that if x and ξ are dual Fourier variables (e.g. position and momentum), then the less the uncertainty in x, the greater in ξ and conversely:

E[|x| 2 ]E[|ξ| 2 ] ≥ 1 16π 2 , ( 63
)
with equality if and only if the probability density of x is a Gaussian density. Many improvements and variations on this inequality have been given, see for instance the review [START_REF] Folland | The uncertainty principle: A mathematical survey[END_REF]. Interesting, but not well-known, moment inequalities have been given in [START_REF] Angulo | Uncertainty relationships in many-body systems[END_REF][START_REF] Angulo | Information entropy and uncertainty in d-dimensional many-body systems[END_REF]. Similar formulations are obtained below. Other entropic uncertainty relations have been stated by Hirshman [START_REF] Hirschman | A note on entropy[END_REF] and improved in [START_REF] Bialynicki-Birula | Formulation of the uncertainty relations in terms of the Rényi entropies[END_REF][START_REF] Zozor | On classes of non-gaussian asymptotic minimizers in entropic uncertainty principles[END_REF]. It is worth mentioning that all the uncertainty inequalities are extremely useful in the analysis of complex systems, see the recent book [START_REF] Sen | Statistical Complexity: Applications in Electronic Structure[END_REF], particularly the chapter [START_REF] Sen | Statistical Complexity: Applications in Electronic Structure[END_REF], and the paper [START_REF] Zozor | Position-momentum uncertainty relations based on moments of arbitrary order[END_REF].

In the following, we consider a complex amplitude wave function ψ(x), x ∈ R n , with unit Euclidean norm 2, and denote ψ(ξ) its Fourier transform, ξ ∈ R We begin by a simple change of function which enables to express the generalized Fisher information as a Dirichlet energy. This leads to the following result. Proposition 1. For k = β/ (β(q -1) + 1) , λ = n(q -1) + 1, q > max {(n -1)/n, n/(n + α)}, we have

1 M kq 2 [|ψ| 2 ] ˆRn x α |ψ(x)| k dx 1 α ˆRn ∇ x |ψ(x)| β * dx 1 β ≥ n kq (64) 1 M k 2 [|ψ| 2 ] 1 kλ E k 2 [ x α ] 1 α ˆRn ∇ x |ψ(x)| β * dx 1 βλ ≥ 1 k 1 λ m α [G] 1 α φ β,q [G] 1 βλ , (65)
where E q [.] denote the q-expectation, the expectation computed with respect to the escort distribution of order q, and M q [.] the information generating function of order q. In these inequalities, the case of equality is obtained if and only if

|ψ(x)| k /M k 2 [|ψ| 2 ] is a generalized q-Gaussian. Proof. Let f (x) = |ψ(x)| 2 k 2 ´Rn (|ψ(x)| 2 ) k 2 dx = |ψ(x)| k M k 2 [|ψ| 2 ] , (66)
with k = β/ (β(q -1) + 1) . Note that we also have the conjugation relation 1 α + 1 k = q. The change of function ( 66) is chosen in order to reduce φ β,q [f ] to the simple form (67) given below. In the other hand, the moment of order α with respect to f (x) is nothing but the generalized k 2 -escort moment (68) of order α computed with respect to |ψ(x)| 2 . Finally, the information generating function M q [f ] can also be expressed in term of |ψ(x)| 2 :

φ β,q [f ] = ˆRn f (x) β(q-1)+1 ∇ x ln f (x) β * dx = |k| β M k 2 [|ψ| 2 ] β k ˆRn ∇ x |ψ(x, t)| β * dx. ( 67
) m α [f ] = ˆRn x α |ψ(x, t)| 2 k 2 ´Rn (|ψ(x, t)| 2 ) k 2 dx dx = E k 2 [ x α ] . ( 68
) M q [f ] = M kq 2 [|ψ| 2 ] M k 2 [|ψ| 2 ] q . (69)
With these notations, the relation between φ β,q [f ] and I β,q [f ], i.e. I β,q [f ]

1 β = φ β,q [f ] 1 β 1 q Mq[f ]
, and the conjugation relation 1 α + 1 k = q, the generalized Cramér-Rao (47) becomes (64). Similarly, the second Cramér-Rao inequality (56) gives (65).

Beginning with the relations (64), (65) and using a relation between the norm of a function and the norm of its Fourier transform, we obtain a pair of general uncertainty relations, for any exponent, and that involve the expectations computed with respect to escort distributions. This result is stated as follows:

Theorem 3. For 2 ≥ α ≥ 1, β its Hölder conjugate, q > max {(n -1)/n, n/(n + α)} , k = β/ (β(q -1) + 1) , and λ = n(q -1) + 1, then we have

M α 2 [| ψ| 2 ] 1 α M k 2 [|ψ| 2 ] 1 α M kq 2 [|ψ| 2 ] E k 2 [ x α α ] 1 α E α 2 ξ β β 1 β ≥ n 2πkq β 1 β α 1 α -n 2β . (70) M α 2 [| ψ| 2 ] 1 αλ M k 2 [|ψ| 2 ] 1 kλ E k 2 [ x α α ] 1 α E α 2 ξ β β 1 βλ ≥ 1 (2πk) 1 λ β 1 β α 1 α -n 2βλ m α [G] 1 α φ β,q [G] 1 βλ (71)
Proof. We rst observe that for any complex valued function, we have

∂|ψ(x)| ∂x i 2 = ∂ψ(x) ∂x i ∂ψ(x) * ∂x i -|ψ(x)| 2 ∂ arg ψ(x) ∂x i 2 . ( 72
= n i=1 |∂ i ψ(x)| β .
At this point, we can invoke the sharp version of the Hausdor-Young inequality due to Babenko and Beckner [START_REF] Beckner | Inequalities in Fourier analysis on R n[END_REF], which states that for a pair of Fourier transforms g and ĝ, then for 1 ≤ α ≤ 2

g β ≤ β 1 β α 1 α n 2 ĝ α , (74)
with equality if and only if g is a Gaussian function or if α = β = 2 (Parseval's identity). Thus, it

comes ˆRn ∇ x |ψ(x, t)| β β dx = n i=1 ˆRn |∂ i ψ(x)| β dx ≤ β 1 β α 1 α n 2 (2π) β n i=1 ˆRn |ξ i ψ(ξ)| α dξ β α (75)
Inserting this inequality in (64), and taking account of the fact that x

α = x α α = n i=1 |x i | α , we obtain 1 M kq 2 [|ψ| 2 ] n i=1 ˆRn |x i | α |ψ(x, t)| k dx 1 α n i=1 ˆRn |ξ i | α | ψ(ξ)| α dξ β α 1 β ≥ n 2πkq β 1 β α 1 α -n 2β (76)
Now, we can observe that by the Hölder inequality

ˆRn |ξ i | α | ψ(ξ)| α dξ ≤ ˆRn |ξ i | β | ψ(ξ)| α dξ α β ˆRn | ψ(ξ)| α dξ 1-α β . (77)
Plugging in this inequality in (76), we see that the exponent β/α simplies, and that the inequality can be written in a weaker but more symmetric form:

M α 2 [| ψ| 2 ] 1 α -1 β M kq 2 [|ψ| 2 ] ˆRn x α α |ψ(x, t)| k dx 1 α ˆRn ξ β β | ψ(ξ)| α dξ 1 β ≥ n 2πkq β 1 β α 1 α -n 2β , (78)
which can also be written in terms of escort expectations as (70).

For the second Cramér-Rao inequality, we follow the very same steps, beginning with (65). By

(75) it comes 1 M k 2 [|ψ| 2 ] 1 kλ E k 2 [ x α α ] 1 α n i=1 ˆRn |ξ i ψ(ξ)| α dξ β α 1 βλ ≥ 1 (2πk) 1 λ m α [G] 1 α φ β,q [G] 1 βλ β 1 β α 1 α -n 2βλ , (79)
which, by the Hölder inequality (77), gives (71).

In the general case, the inequalities (70) and (71) are not sharp, because they follow from the Babenko-Beckner inequality (74) and the Hölder inequality (77), where the conditions for equality are not met simultaneously. However, we can still get a sharp uncertainty relation, saturated by generalized q-Gaussians, in the case α = β = 2. Of course, for q = 1 (which, with β = 2 gives k = 2), we obtain a multidimensional version of Heisenberg inequality, which reduces to (63) in the scalar case.

Corollary 4. For k = β/ (β(q -1) + 1) , λ = n(q -1) + 1, q > max {(n -1)/n, n/(n + α)} and γ ≥ 2, θ ≥ 2, the following uncertainty relations hold:

(2πk)

1 λ m 2 [G] 1 2 φ 2,q [G] 1 2λ (85)
In both inequalities, the lower bound is attained if f

(x) = |ψ(x)| k /M k 2 [|ψ| 2
] is a generalized q-Gaussian (which means that |ψ(x)| 2 is also a generalized q-Gaussian with a dierent entropic index: q = 1 + k(q -1)/2). By Jensen's inequality, we know that for b ≥ a, we always have

E[|X| b ] a b ≥ E[|X| a ]
. Therefore, applying this inequality with γ ≥ 2, θ ≥ 2, we obtain (80) and (81). Finally, by the general power mean inequality, we know that M a [f ]

Conclusions

This paper improves and build on our previous ndings presented in [START_REF] Bercher | On generalized Cramér-Rao inequalities, generalized Fisher information and characterizations of generalized q-Gaussian distributions[END_REF]. We connect concepts in estimation theory to tools used in nonextensive thermostatistics and establish general Cramér-Rao type inequalities valid for estimation purposes. These results are given in the mutidimensional case, and a feature of our approach is that it works for arbitrary norms on R n . As a direct consequence, we obtain multidimensional versions of our q-Cramér-Rao inequalities, which include the Barankin-Vajda as well as the standard Cramér-Rao inequality as particular cases. Furthermore, in the case of a translation family, we have shown that the corresponding Cramér-Rao type inequality is saturated by multidimensional q-Gaussian distributions. We have also presented a related general Cramér-Rao inequality which is saturated by the same q-Gaussian distributions. These results imply in particular that the generalized q-Gaussians are the minimizers of an extended version of the Fisher information, among all distributions with a given moment, just as the standard Gaussian minimizes Fisher information over all distributions with a given variance. Since these generalized Gaussian are already known to be the maximum entropy distributions for Rényi or Tsallis entropies, this yields a new, complementary, information theoretic characterization of these distributions.

Finally, we have derived new multidimensional uncertainty relations from the extended Cramér-Rao inequalities. These uncertainty relations involve generalized expectations computed with respect to escort distributions, and we have shown that some of these uncertainty relations are saturated by generalized q-Gaussian distributions, thus generalizing the fact that the standard Heisenberg uncertainty relation is saturated by a standard Gaussian. Some of the presented uncertainty inequalities are not sharp, and it would certainly be of interest to try to improve them, by looking to the extremal functions or to optimal constants. It would also be interesting to try to document the properties of the two generalized Fisher information.

Finally, in their recent work [START_REF] Lutwak | Extensions of Fisher information and Stam's inequality. Information Theory[END_REF], Lutwak et al. have introduced an abstract, implicit, notion of Fisher information matrix attached to a probability density. It would be of interest to examine whether this notion could be extended and interpreted in the estimation theory framework.

1 p

 1 of norms is the class of L p -norms, p ≥ 1, given by x p = ( . As important particular cases, we have the L 1 -norm, x 1 = n i=1 |x| i , the max-norm or L ∞ -norm x ∞ = max (|x 1 |, . . . |x n |) , and of course the Euclidean L 2 -norm x 2 =

  n . By the Parseval equality ´Rn |ψ(x)| 2 dx = ´Rn | ψ(ξ)| 2 dξ = 1 and both |ψ(x)| 2 and | ψ(ξ)| 2 are probability density functions.

1 a 1 bk 2 > M 1 = 1 ,

 11211 ≥ M b [f ] for a ≥ b. Therefore, when k > 2, that is 3 2 -1β > q, then M and the inequality (81) yields (82).

  = c, where c is any constant. Therefore, we see that we always have ∇ x |ψ(x, t)| * ≤ |∇ x ψ(x, t)| * . In the following, we will take for . * a β-norm and for . its dual α-norm. So doing, we get |∇ x ψ(x, t)|

	This relation implies that
	∂|ψ(x)| ∂x i	≤	∂ψ(x) ∂x i	,	(73)
	with equality if and only if arg ψ(x) β
	β
				)

Acknowledgments

The author thanks the anonymous referees for their suggestions that helped improve the article.

The author is also very grateful to Deane Yang (Polytechnic Institute, NY), who suggested to look at general norms, and to Matthieu Fradelizi (LAMA, Univ. Paris-Est), for helpful discussions related to this work. Thanks are extended to Lodie Garbell for her friendly proofreading of the manuscript.