Adaptive Laguerre density estimation for mixed Poisson models

. In this paper, we consider the observation of n i.i.d. mixed Poisson processes with random intensity having an unknown density f on R + . For ﬁxed observation time T , we propose a nonparametric adaptive strategy to estimate f . We use an appropriate Laguerre basis to build adaptive projection estimators. Non-asymptotic upper bounds of the L 2 -integrated risk are obtained and a lower bound is provided, which proves the optimality of the estimator. For large T , the variance of the previous method increases, therefore we propose another adaptive strategy. The procedures are illustrated on simulated data. March 13


Introduction
Consider n independent Poisson processes (N j (t), j = 1, . . ., n) with unit intensity and n i.i.d.positive random variables (C j , j = 1, . . ., n).Assume that the processes (N j (t), j = 1, . . ., n) and the sequence (C j , j = 1, . . ., n) are independent.Under these assumptions, the random time changed processes (X j (t) = N j (C j t), t ≥ 0) are i.i.d. and such that the conditional distribution of X j given C j = c is the distribution of a time-homogeneous Poisson process with intensity c.The process X j is known as a mixed Poisson process (see e.g.Grandell (1997), Mikosch (2009)).Such processes are of common use in non-life insurance mathematics as well as in numerous other areas of applications (see Fabio et al. and references therein).
In this paper, we assume that the random variables C j have an unknown density f on (0, +∞) and our concern is the nonparametric estimation of f from the observation of a nsample (X j (T ), j = 1, . . ., n) for a given value T .We investigate this subject for large n and both for fixed T and large T with two different methods.The fixed T method performs well for small T (e.g.T = 1) and deteriorates as T increases while the large T method performs better and better as T increases.Thus, the two methods are complementary.
In Section 2, we consider the case T = 1.The distribution of X j (1) = N j (C j ) is given by: (1) which can be estimated by: (2) The problem of estimating f from the discrete observations (N j (C j ), j = 1, . . ., n) is thus an inverse problem, the problem of estimating a mixing density in a Poisson mixture.Several authors have considered this topic whether by kernel or projection methods, see Simar (1976), Karr (1984), Zhang (1995), Loh andZhang (1996, 1997), Hengartner (1997).These authors are mainly interested in estimating f on a compact subset of (0, +∞).We discuss with more details the links between the present results and the previous references in subsection 2. 4.
In this paper, we assume that (H) f ∈ L 2 ((0, +∞)) and propose a solution without any constraint on the support of the unknown function.We study the L 2 ((0, +∞))-risk and prove upper and lower bounds on an adequate function space.Our approach is a penalized projection method (see Massart (1997)) which provides a concrete adaptive estimator of f easily implementable.It is based on the following idea.By relations (1), α ℓ (f ) is the L 2 scalar product of f and the function c → e −c c ℓ /ℓ!.Choosing an orthonormal basis (ϕ k ) of L 2 ((0, +∞)), (1) can be written as: k are respectively the k-th component of f and e −c c ℓ /ℓ! on the basis.The problem is to choose a basis such that the mapping (θ k (f ), k ≥ 0) → (α ℓ (f ), ℓ ≥ 0) can be simply and explicitly inverted.Then, by plugging the estimators αℓ in the inverse mapping, we get estimators of the coefficients θ k (f ) and deduce estimators of f .An appropriate choice of (ϕ k ) is thus a key tool: we consider the Laguerre bases defined by (( √ aL k (at)e −at/2 , k ≥ 0)) where (L k (t)) are the Laguerre polynomials.Here, the choice a = 2 is especially relevant.Indeed, with k = 0 for all k > ℓ and the matrix Ω ℓ = (Ω (i) k ) 0≤i,k≤ℓ is lower triangular and explicitly invertible (Propositions 2.1 and 2.2).Therefore, the inverse problem has a solution: the linear mapping on R ℓ+1 (4) α ℓ = (α k (f ), k = 0, . . ., ℓ) ′ → θ ℓ = (θ k (f ), k = 0, . . ., ℓ) ′ = Ω −1 ℓ α ℓ .Moreover, a crucial consistency property holds: the first ℓ − 1 coordinates of α ℓ and θ ℓ are equal to those of α ℓ−1 and θ ℓ−1 .Note that, in Comte et al. (2013), another type of inverse problem involving functions of L 2 ((0, +∞)), has been solved also using a Laguerre basis.
So, we define a collection of estimators of f by fℓ = ℓ k=0 θk ϕ k , where ( θk ) are defined using (2) and (4).We study their L 2 -risk (Proposition 2.3).For this, we introduce appropriate regularity subspaces of L 2 ((0, +∞)), the Sobolev-Laguerre spaces with index s > 0. These spaces are defined in Shen (2000) and Bongioanni and Torrea (2009).We precise (see Section 7) the rate of decay of the coefficients of a function f developed in a Laguerre basis when f belongs to a Sobolev-Laguerre space with index s.This allows to evaluate the order of the bias term f − f ℓ 2 where .denotes the L 2 ((0, +∞))-norm.Using these regularity spaces, we discuss the possible rates of convergence of the L 2 -risk of fℓ .Functions belonging to a Sobolev-Laguerre ball with index s yield rates of order O((log n) −s ).This rate is optimal, as we prove a lower-bound result.Afterwards, we propose a data-driven choice l of the dimension ℓ and study the L 2 -risk of the resulting adaptive estimator (Theorem 2.2).We interpret the results in the case where the observation is (N j (C j T ), j = 1, . . ., n).This amounts to a change of scale which multiplies the variance term of the risk by a factor T and implies a deterioration of the estimator as T increases.
Section 3 is devoted to the estimation of f for large T .Our method relies on the property that for each j, C j,T = N j (C j T )/T is a consistent estimator of the random variable C j as T tends to infinity.Then, we use the i.i.d.sample ( C j,T ) 1≤j≤n to build estimators of f .We propose projection estimators on the Laguerre basis (3) using other estimators of the coefficients θ k (f ) together with an adaptive choice of the space dimension (Proposition 3.1, Theorem 3.1).The criterion for the model selection is non standard: it involves a penalization which is the sum of two terms, one depending on n, ℓ and the other on T, ℓ.
Section 4 gives numerical simulation results and some concluding remarks are stated in Section 5. Proofs are gathered in Section 6.In Section 7, regularity spaces associated with Laguerre bases are discussed and useful inequality is recalled in Section 8 .

2.
Estimation of the mixing density for T = 1.
Proposition 2.1.The coefficients Ω (ℓ) k defined by (7) are equal to Define the vectors The matrix Ω ℓ is therefore invertible and its inverse is explicitly computed in the following proposition.
Proposition 2.3.The estimator fℓ of f defined by ( 2)-( 8)-( 9)-( 10) satisfies Proposition 2.3 states a squared-bias/variance decomposition, and we need now to specify the bias order on adequate functional spaces, in order to evaluate optimal rates.

2.2.
Rates and rate optimality.As it is always the case in nonparametric estimation 1 , we must link the bias term f − f ℓ 2 with regularity properties of function f .In our context, these should be expressed in relation with the rate of decay of the coefficients (θ k (f )) k≥0 .The Laguerre-Sobolev spaces described in Section 7 provide an adequate solution.
For any h ∈ W s 2 ((0, +∞), K), we have h − h ℓ2 = ∞ k=ℓ+1 θ 2 k (h) ≤ K/ℓ s where h ℓ is the orthogonal projection of h on S ℓ .Proposition 2.4.Let for 0 < ǫ < 1, We have and Note that lǫ does not depend on s and is thus adaptive.With ℓ ⋆ , the bias and variance terms have the same order (log(n)) −s , which is better.In addition, the constant is improved.Nevertheless, this choice depends on s.
Proof.For f ∈ W s 2 (0, +∞), K), the risk bound in Proposition 2.3 writes The variance term has exponential order 2 4ℓ with respect to ℓ.Thus, we can not make the classical bias variance compromise.First we can choose ℓ such that the bias term dominates: this is obtained by choosing ℓ = ℓ ǫ .Second, a more precise tuning of both terms is obtained with ℓ = ℓ ⋆ .In both cases, the rate is of order O([log(n)] −s ).
We now prove that, for densities lying in Laguerre-Sobolev balls W s 2 ((0, +∞), K), the rate (log n) −s is optimal. 2 Theorem 2.1.Assume that s is a positive integer and let K ≥ 1.There exists a constant c > 0 such that liminf where inf fn denotes the infimum over all estimators of f based on (N j (C j )) 1≤j≤n .
The proof uses several lemmas established in Zhang (1995) and Loh and Zhang (1996).

Model selection.
Model selection is justified as the bias may have much smaller order.For instance, it can be null if f admits a finite development in the Laguerre basis.Exponential distributions also provide examples of smaller bias.Indeed, consider f an exponential density E(θ).Then . Choosing .
The rate depends on θ and can be O(n −β ) for any β < 1.For instance if θ = 5/3 the rate is O(n −1/2 ), for θ = 1/2, the rate is O(n −0.44 ) (see Section 4) and it tends to O(n −1 ) (the parametric rate) when θ tends to 1, which is coherent with the fact that the bias is null for θ = 1.This kind of result can be generalized to the case of a distribution f defined as a mixture of exponential distributions and to Gamma distributions Γ(p, θ), with p an integer.More precisely, if f p is the density Γ(p, θ), This term can be computed explicitly and we get, for Note that the bias is null for θ = 1 and ℓ > p − 1, which is expected since f p ∈ S p−1 .Moreover, the bias order depends on θ, which can be seen in simulations.Now we have to define an automatic selection rule of the adequate dimension ℓ.We make the selection among the following set: where [x] denotes the integer part of the real number x.For κ a numerical constant, we define (13) l = arg min ℓ∈Mn − fℓ 2 + pen(ℓ) , with pen(ℓ) = κ ℓ2 4ℓ n .
We can prove the following result Theorem 2.2.Consider the estimator fl defined by ( 10) and (13).For any κ ≥ 8, we have The infimum in the right-hand-side of the inequality above shows that the estimator is indeed adaptive.Note that the penalty is, up to a constant, equal to the variance multiplied by ℓ.This implies a possible negligible loss in the rate of the adaptive estimator w.r.t. the expected optimal rate.
Remark.Let us now assume that the observation is (N j (C j T ), j = 1, . . ., n).The previous method applies directly to estimate the density f T of C j T i.e. f T (t) = (1/T )f (t/T ).We can deduce the results for f (c) = T f T (T c).The function f is developed on the basis (ϕ ), k ≥ 0) and the following relation holds θ the orthogonal projection of f on the space S (T ) ℓ spanned by (ϕ where fT,ℓ is the estimator built for f T using (N j (C j T ), j = 1, . . ., n).The estimator Moreover, with l defined in ( 13), there exists κ > 0 such that The variance term in the L 2 -risk is multiplied by a factor T .This explains that the method may be worse when T increases.Actually, this was clear on simulated data.
2.4.Related works.In Simar (1976), it is proved that the cumulative distribution function F (x) of C j can be consistently estimated using (α ℓ ).The method is theoretical and concrete implementation is not easy.Noting that α 0 (f ) is simply the Laplace transform of f , Karr (1984) studies the properties of α0 to estimate α 0 (f ) in the more general context of mixed point Poisson processes.
For comparison purposes, we detail some of the results of Zhang (1995), Hengartner (1995) and Loh andZhang (1996, 1997) in the case of Poisson mixtures.In the case where f has compact support [0, θ ⋆ ], Zhang (1995) gives a kernel estimator of f (a) and studies pointwise quadratic risk on Hölder classes with index r (i.e.functions f admitting ⌊r⌋ derivatives such that f (⌊r⌋)3 is r − ⌊r⌋-Hölder).The estimator has a MSE of order [log(n)/ log log(n)] −2r which does not correspond to his lower bound which is [log(n)] −2r .In the case of non compact support for f , the kernel estimator MSE has order (log(n)) −r/2 , with no associated lower bound.Loh and Zhang (1996) generalize the results of Zhang (1995) by studying a weighted-L p -risk.
Hengartner (1997) considers the case where f has a compact support.He builds projection estimators using orthogonal polynomials on the support.The upper bound of MISE has order [log(n)/ log log(n)] −2r on the same class as above and on Sobolev classes with index r.On the latter classes, he proves a lower bound of order [log(n)/ log log(n)] −2r .
Loh and Zhang (1997), in the case of non compact support for f , use Laguerre polynomials and build projection estimators.Thus, the function is estimated by a polynomial; they study a weighted L 2 -risk.The upper bound is O([log(m)] −m/2 ) on the class of functions such that j≥m j m τ 2 j (f ) < M where τ j (f ) is the coefficient of f on the development with respect to the Laguerre polynomials.Their lower bound is O([log(n)] −m ), which does not correspond to the upper bound.
In all cases, the number of coefficients in the projection estimators does not depend on the regularity space.In this sense, the above methods are adaptive.
Let us now clarify our contribution.First, we use a L 2 ((0, +∞))-basis and a usual MISE, which is more fitted to the problem.Second, we clarify the functional spaces associated to the context of Laguerre bases on (0, +∞) and provide explicit links between regularity and coefficients of a development on these spaces.Upper and lower bounds match globally and without weights.
Here, the proof of our lower bound is inspired of Loh and Zhang's constructions.Therefore, our results synthesize and improves all these previous works.
Lastly, when the function under estimation has stronger regularity properties than considered in lower bounds, we show that the rate can be improved (polynomial instead of logarithmic).This justifies the proposal of an adaptive procedure, see Theorem 2.2, which is moreover non asymptotic.

Estimation for large
Conditionally to C j = c, we know that C j,T converges almost surely to c as T tends to infinity.Consequently, C j,T converges almost surely to C j .We now use the i.i.d.sample ( C j,T ) 1≤j≤n to build projection estimators of f , where the coefficients θ k (f ) are now estimated as follows.
( 14) Note that S ℓ has the norm-connection property: as can be seen from Lemma 6.1.We obtain the following risk bound.
Proposition 3.1.Recall that f ℓ is the orthogonal projection of f on S ℓ = span(ϕ 0 , . . ., ϕ ℓ ).Then The bound contains the usual decomposition into a squared-bias term f −f ℓ 2 and a variance term.The latter term is the sum of two components: the first one 2(ℓ + 1)/n is classical and no more exponential in ℓ, the second one is due to the approximation of the C j 's by the C j,T 's and gets small when T increases.To define a penalization procedure, we must estimate s 2 .Let ( 16) and ( 17) The following holds.
the estimator defined by ( 14) and ( 17).Then there exist numerical constants κ1 , κ2 such that where C is a numerical constant and C ′ a positive constant.
Thus, the estimator f (T ) l is adaptive and its risk automatically reaches the order of the biasvariance compromise.

Numerical simulations
In this paragraph, we illustrate on simulated data the two adaptive projection methods using the Laguerre basis: method 1 corresponds to Section 2 when T = 1, method 2 corresponds to section 3 for large T .We consider different distributions for the C j 's: (1) a Gamma Γ(p, θ) a mixed Gamma density 0.3Γ(3, 0.25) + 0.7Γ(10, 0.6). ( a Weibull density f (p,θ) (x) = θp −θ x θ−1 e −(x/p) θ 1 x>0 for p = 3 and θ = 2.Note that, as θ = 1, the density (1) has only three nonzero coefficients θ 0 , θ 1 , θ 2 in its exact development in the Laguerre basis.For density (3), we know that the rate of the L 2 risk depends on the value of θ (n −0.44 for θ = 1/2, see Section 2).In Figures 1-5, we illustrate the first method for T = 1 and n = 10000, n = 100000 and the second for sample sizes n = 1000 and T = 10, and n = 4000, T = 40, for the five densities defined above.We plot 25 consecutive estimates on the same picture together with the unknown density to recover, to show variability bands and illustrate the stability of the procedures.
• Comments on method 1.The method is easy to implement.As it is standard for penalized methods, the theoretical constant is too large and in practice, is calibrated by preliminary simulations.We have selected the constant κ = 0.001 in the penalty.This prevents from possible explosion of the variance, which has exponential order.The adaptive estimator performs reasonably well for large values of n (n ≥ 10000) but is very sensitive to the parameter values for distributions Gamma or exponential, as expected.The mixture density and the Pareto and Weibull densities, which do not admit finite developments in the basis, are correctly estimated.Increasing n improves significantly the estimation.We choose to select ℓ in {0, 1, . . ., 2⌊log(n)⌋ − 1}.
• Comments on method 2. The method is also easy to implement.We have selected the constants κ1 = 1.5, κ2 = 10 −5 .The very small value of κ2 simply kills the effect of the second term in the penalty in order to allow not too large values of T .This second method gives better results than the first method, as soon as T ≥ 10 (even T ≥ 5 provides good estimators).The number of observations need not be very large.We kept the same set of possible values for ℓ in the selection algorithm; here again, the selected values l are in {0, 1, . . ., 4}.

Concluding remarks
In this paper, we study the nonparametric density estimation of a positive random variable C from the observation of (N j (C j T ), j = 1, . . ., n), where (N j ) are i.i.d.Poisson processes with unit intensity, (C j ) are i.i.d.random variables distributed as C, and (N j ) and (C j ) are independent.Under the assumption that the unknown density f of the unobserved variables (C j ) is in L 2 ((0, +∞)) and for a fixed value T , we express the nonparametric problem as an inverse problem, which can be solved by using a Laguerre basis of L 2 ((0, +∞)).Explicit estimators of the coefficients of f on the basis are proposed and used to define a collection of projection estimators.The space dimension is then selected by a data driven criterion.For functions belonging to Sobolev-Laguerre spaces described in Section 2, f is estimated at a rate O((log(n)) −s ).So, an interesting question is to know whether there exist other functions than those of these spaces estimated at the same rate.This problem amounts to finding maximal functional classes for which a given rate of convergence of the estimators can be achieved.
For large T , estimators C j,T of the C j 's are used to build adaptive projection estimators in the Laguerre basis.In this approach, a moment condition on C j is required.
The numerical simulation results show that the Laguerre basis is indeed appropriate, to obtain estimators with no boundary effects at 0.
Possible developments of this work are the following.We may use specific kernel estimators on R + , as in Comte and Genon-Catalot Finally, ( 18) where we know that Ω (ℓ) k is a polynomial of degree k which is equal to 0 for ℓ = 0, 1, . . ., k − 1.Hence, we have Hence the result.
6.2.Proof of proposition 2.2.Denote by R ℓ [X] the space of polynomials with real coefficients and degree less than or equal to ℓ.The transpose of the matrix √ 2Ω ℓ represents the linear application of R ℓ [X], P (X) → P 1−X 2 , in the canonical basis (1, X, . . ., X ℓ ).The inverse linear mapping is Q(X) → Q (1 − 2X).Hence the result.We have ). Next, we write the variance term as follows: where cov(α i , αj ) = (α i δ j i − α i α j )/n and δ j i is the Kronecker symbol.Thus, for M symmetric and nonnegative, where D α = diag(α 0 , . . ., α ℓ ).Here, we get Note that Tr( t Ω −1 ℓ Ω −1 ℓ ) is known as the squared Frobenius norm of the matrix Ω −1 ℓ .It follows from Proposition 2.2 that ( 21) Noting that we get ( 22) As a consequence, we obtain the risk decomposition announced in Proposition 2.3.

Now, we define
Then, w 1n is chosen such that f 1n = 1 which yields: Finally, δ 0 is chosen by Proof.We first study f 0n .By construction, On the space of polynomials of degree 2s + 1 on [c 0 , c 1 ], is a norm and all norms are equivalent.Therefore, there exists C such that By Lemma 3 of Loh and Zhang (1996), |γ We deduce We have and . Therefore, provided that ε is small enough, the first term of f 0n (χ 0 + χ 2 ) is lower bounded as c 0 > 0 and the second term is O(u (s/2)+(1/4) /n) = o(1).Thus, we can choose ε small enough to have f 0n (( We have We check that f 1n ≥ 0 in the same way as for f 0n . • Step 2. For j = 0, 1, f jn ∈ W s 2 ((0, +∞), K) for all K ≥ 1.
Proof.This part is specific to our context as we do not have the same function spaces as Loh andZhang (1996,1997).
• Step 4.There exists C > 0 such that Proof.This part is also specific to our study: we only use two functions instead of three and our bound is global and not local.We have where w 1n /w 0n = o(1/n) and | cos | ≤ 1.Therefore, it is enough to bound from below: And as we only look at First T 1 = O( √ u) because, by Lemma 3 of Loh and Zhang (1996, p.573), Next, we write and as above by the choice of δ 0 .Therefore T ′ 2 = O( √ u/n 2 ).Consequently It follows that This concludes step 3 as u = δ 0 log(n).
The result of Proposition 6.1 inserted in Inequality (24), shows that for κ ≥ 8, we obtain 4p(ℓ, l) ≤ pen( l) + pen(ℓ) and Proof of Proposition 6.1.We apply the Talagrand Inequality recalled in Lemma 8.1 of Section 8. First note that Let us define M 2 = Tr( t M M ) and ρ 2 (M ) the largest eigenvalue of t M M .We consider the centered empirical process given by where Recall that ℓ * = ℓ ∨ ℓ ′ and define the unit ball for the maximization by B ℓ * = {t ∈ S ℓ * , |t| = 1}.
Next since β i,L has only one nonzero coordinate, equal to 1, we have to bound ψ t ( x) = t, Ω −1 L x for x = e j vector of the canonical basis of R L+1 , with j ≤ ℓ * and t ∈ B ℓ * .For such vectors x, We take ǫ 2 = δℓ * and for δ to be chosen afterwards, we get /C 3 ) and ℓ * ≥ 1 gives the result.
The proof of Proposition 6.2 is given in Section 6.8.Now, the definitions of p 1 , p 2 and pen(.)imply that 8p 1 (ℓ, ℓ ′ ) + 8p 2 (ℓ, ℓ ′ ) ≤ pen(ℓ) + pen(ℓ ′ ) for κ1 ≥ 32 and κ2 ≥ 64, ∀ℓ, ℓ ′ ∈ M n,T .Therefore, we obtain by Lemma 6.1.Moreover, using (15), on B ℓ∨ℓ ′ , t ∞ ≤ 2(ℓ ∨ ℓ ′ + 1) := M .Next, to find v, we split in two parts: sup where We write that where we apply the Taylor Formula and ξ T ∈ (C 1 , C 1,T ).Using Lemma 6.1 again, we get To conclude we use that E . Now the Talagrand Inequality implies that there exist constants A i , i = 1, 2, 3 such that √ n so that as which is the announced bound.Now we study R(t).Let D = sup t∈B l∨ℓ |R(t)| 2 − p 2 (ℓ, l) By Inequality (28), the first rhs term is zero.To deal with the second term, let Using the definition of M n,T , we get since ( 1 2 s 2 − s 2 ) + 1 Ω = 0.By the Markov inequality, we have ) and we use the Rosenthal Inequality (see Hall and Heyde (1980, p.23)) to get ) where m 4 is the fourth centered moment of X j = 3 C 2 j,T − 2 C j,T /T and m 2 2 the variance of X j .We write After some elementary computations using the centered moments of a Poisson distribution, we obtain that, if E(C When ρ ≡ 1, we denote this space as usual by L For any orthonormal basis We are especially interested in the weight functions (30) ρ(x) = x α e −x = w α (x), α ≥ 0 and the associated orthonormal bases of L 2 (R + , w α ), namely the Laguerre polynomials.Consider the second order differential equation: The solution is g(x) = L α k (x) the Laguerre polynomial with index α and order k.The function L α k is a polynomial of degree k, and the sequence (L α k ) is orthogonal with respect to the weight function w α .The orthogonality relations are equivalent to: We have The following holds, for all integer k and α ≥ 0 : , the sequence (φ α k ), k ≥ 0) constitutes an orthonormal basis of the space L 2 ((0, +∞), w α ).In particular, φ 0 k (x) = L 0 k (x) = L k (x), k ≥ 0 constitute an orthonormal basis of L 2 ((0, +∞), w), with w(x) = w 0 (x) = e −x .Noting that x α+1 e −x ′ = x α e −x (α + 1 − x), we obtain, using (31) and ( 33 For these formulas, see Abramowitz and Stegun (1964).
We can now prove the following result.

Figure 2 .
Figure 2. Estimation of the mixed Gamma density with method 1 (top left n = 10000 and top right n = 100000, for T = 1) and method 2 (bottom left, n = 1000, T = 10 and bottom right n = 4000, T = 40): true -thick (blue) line and 25 estimated (dashed (red) lines).The selected ℓ is 3 except for the bottom right plot where it is 4.
(2012), to compare them with projection Laguerre estimators.As in Fabio et al. (2012), we may enrich the data by considering several observation times.Another relevant extension is to study mixed compound Poisson processes, e.g. using the approach of Comte et al. (2014), or more general mixed Lévy processes.

Figure 4 .Proofs 6 . 1 .
Figure 4. Estimation of the Pareto density with projection method 1 (top left n = 10000 and top right n = 100000, for T = 1) and method 2 (bottom left, n = 1000, T = 10 and bottom right n = 4000, T = 40): true -thick (blue) line and 25 estimated (dashed (red) lines).Most of the time l = 2 for the top pictures and 0 for the bottom ones.
Proof of Proposition 6.2.First we study νn (t) and apply the Talagrand Inequality.To do this, we evaluate the bounds H 2 , M, v as defined in Lemma 8.1.Clearly 8 j ) < +∞, then there exist constants c 1 , c 2 such that m44 ≤ c 1 and m 2 2 ≤ c 2 .Laguerre polynomials and associated regularity spaces: General properties.For ρ : R +