2.3.1 The formal equivalence
We just discussed the two main approaches to estimate the jump distribution of a dis- cretely observed compound Poisson process. As mentioned at the start of each of the sec- tions, in essence, they focus on tackling two fundamentally different problems (although they implicitly tackle both): untangling the parameters, and rigorously inverting the iden- tities that relate them to the data, respectively. For this reason they lead to estimators for which different conclusions are obtained assuming distinct assumptions. However, as it may already be clear, the formal identities from where they are derived are formally equivalent and hence all the estimators are very similar to each other. Let us make this more precise for the Poisson case.
In view of expression (2.1.5) and recalling that %i =e−λ∆ (λ∆)i! i,i∈N, or equivalently due to (2.2.8), Γ(z) =eλ∆(z−1) and Γ0(z) := Γ(z)−%0=e−λ∆ eλ∆z−1, z∈C. Therefore, Γ−01(z) = 1 λ∆Log(1 +e λ∆z),
where Log should be the distinguished logarithm ifz=z(θ) is a continuous path inC\{0}
for the same reasons as those mentioned after (2.2.9). Inputting z= FdG0 into the last display, where we recall thatdG0 :=dG−%0δ0, (2.1.6) can be alternatively written as
FdF(u) = 1
λ∆Log(1 +e
λ∆FdG0(u)), u∈
R. (2.3.1)
The estimators of the direct approach were derived from a formal transformation of this expression, and so were those of the spectral approach in view of (2.2.10) and of its equivalent form (2.2.9). Therefore, this display shows that both approaches are formally equivalent and it explicitly highlights their distinction: the former differs from the latter in that, prior to taking Fourier inverse transforms, it insists on expanding the logarithm instead of working directly with it. This difference determines the conditions to be imposed and results in the superiority of the spectral approach: crucially, as it stands, the right hand side of the last display is well-defined for allu ∈R. This is because the argument of the logarithm equalseλ∆FdG(u) by the definition ofdG0, and due to the strictly positive lower bound forFdG(u) =ϕ∆in (1.4.2). However, the expansion of the logarithm is not always
justified and further assumptions are needed to embark on this route. This is the content of our discussion after (2.1.10) and the last display further clarifies it: takingt= ∆ and γ= 0 in (1.1.5) it follows that, if λ∆< π, the distinguished logarithm in the last display coincides with the principal branch of the complex logarithm1 because the argument does not wind about the origin. Furthermore, if λ∆ < log 2, |eλ∆FdG0(u)| ≤ 1 because the mass ofdG0 is 1−e−λ∆, and just in this case the logarithm can be rigorously expanded using Mercator’s complex series. For larger values ofλ∆ this expansion only makes sense if the Laplace transform L as defined in (2.1.13) is used assuming the support of dF, or equivalently ofdG0, is inR+and takingu >0 large enough so that|eλ∆LdG0(u)|is smaller
or equal to the radius of convergence of the series. Indeed, the precise condition is (2.1.13), which Buchmann and Gr¨ubel [2003], Hansen and Pitts [2006] and Bøgsted and Pitts [2010] assumed. Notice that, after this, the direct approach still has to invert the last display just as in the spectral approach. This justifies the fact that, at least in the low-frequency regime, the former has to make stronger assumptions than the latter and, thus, the spectral approach should be preferable in these problems. An alternative way of phrasing this is that, even though the approaches are formally equivalent, there is a fundamental difference between them: the direct approach insists on explicitly decompounding the sum in Z or, equivalently, untangling relationship (2.1.3), whilst in the spectral approach this is achieved at once by resorting to the frequency domain, hence requiring milder assumptions. Consequently, this raises a natural and interesting question: in the context of general compound distributions, is it possible to extend the result in (2.2.16) by working with Γ−01, as in the spectral approach, instead of with its expansion or with Λ in (2.1.8), as in the direct approach? Moreover, can (2.2.16) be extended to more general inverse problems in which the distribution of the observations G is related to that of interest F by FdG = Γ(FdF) for some Γ (the starting point of the last paragraph)? The only result in this direction of which we are aware is that of den Boer and Mandjes [2016]. They work with the Laplace transform assuming the supports ofdF and dG are inR+,
and show pointwise consistency of their estimator at rate logn/√n under much stronger assumptions than those used for (2.2.16). Comparing their results to (2.2.16), they seem far from optimal and, thus, the problems we just proposed remain very much open. We emphasise that some preliminary calculations suggest they should hold under appropriate conditions on Γ. We can further ask under what assumptions on Γ these hold for the multidimensional and/or noisy case. Although we do not include the proofs here, (2.2.16) does extend to the multidimensional and noisy compound Poisson case under the same (multidimensional-analogous) assumptions.
1This was missed by van Es et al. [2007], who mention this property holds forλ∆<log 2 and restrict
their simulations to the not-so-interesting case ofλ∆ = 0.3 to avoid the implementation of the distinguished logarithm. However, we remark that the practical construction of their estimators in Section 3 in van Es et al. [2007] is also valid forλ∆< π, at least in sets of probability approaching 1 asn→ ∞.
2.3.2 Similarities between existing estimators
With the comparison of the two approaches made above, we can explicitly compare the estimators derived from them. In view of (2.2.13), Coca [2015] exploits the fact that
F(x) = 1 λ∆
Z x
−∞1R
\{0}F−1[(Log(FdG))] (dy).
Furthermore, and as argued therein, the right hand side equals 1 λ∆ Z x −∞ F−1[(Log(FdG) +λ∆)] (dy)≈ 1 λ∆ Z x −∞ F−1[(Log(FdG)+λ∆)FKh](dy) = 1 λ∆ Z x −∞ F−1hLog 1+eλ∆FdG0 FKh i (dy),
where the last equality follows by the definition ofdG0. We remark that, unlike its series representation, the last expression is well-defined because (2.3.1) is, due to the spectral cut-off induced by FKh and to the fact that kKhk1 = kKk1 < ∞ in view of (2.2.5). Formally expanding the logarithm and rearranging terms, the last display can be written as Z x −∞ ∞ X i=1 (−1)i+1e iλ∆ iλ∆F −1 FdGi0FKh (dy) = Z x −∞ ∞ X i=1 (−1)i+1e iλ∆ iλ∆dG ∗i 0 ! ∗Kh(dy) = ∞ X i=1 (−1)i+1e iλ∆ iλ∆G ∗i 0 ! ∗Kh(x).
The last expression shows that our estimator can be interpreted as a kernel-regularised version of that in Buchmann and Gr¨ubel [2003] with the novelty that we can estimate a discrete jump component and that λ is substituted by an (efficient) estimator of it. On the left hand side, assumedG0= (1−e−λ∆)g0 for some densityg0 and takeFK =1[−1,1]. Then, in view of (2.1.16), this expression makes clear that our estimator is an integrated low-frequency version of that in Comte et al. [2014], but in which the estimator for the coefficients is a plug-in estimator using (an efficient) one forλ.
Let us comment on the estimator of van Es et al. [2007]. Under the density assumption, they exploit the fact that
f(x) = 1 λ∆F
−1hLog1+(eλ∆−1)Fg0i(x)≈ 1 λ∆F
−1hLog1+(eλ∆−1)Fg0FKhi(x), where we again remark that, unlike its series representation, the last expression is also well-defined by for the same reasons the second to last display was. Formally arguing as
before, the last display can be written as ∞ X i=1 (−1)i+1 e λ∆−1i iλ∆ (g0∗Kh) ∗i = ∞ X i=1 (−1)i+1e iλ∆ iλ∆(dG0∗Kh) ∗i .
In view of the right hand side, the estimator of van Es et al. [2007] can be interpreted as the density version of the estimator of Buchmann and Gr¨ubel [2003], but in which the kernel density estimator is plugged-in in place of the empirical distribution function. Furthermore, in view of the left hand side and by the discussion right after expression (2.1.16), the estimator of Comte et al. [2014] is the high-frequency version of that in van Es et al. [2007] withFK=1[−1,1] assuming no knowledge ofλ. With this interpretation, the estimator in Duval [2013a] is the wavelet density version of them in the high-frequency regime. Since using a kernel or wavelet estimator does not affect the asymptotic results, we conclude that, formally, all the estimators are very close relatives of each other. When λ∆ < log 2 they behave similarly in practice. As we point out in Section 2.6, the true distinction between them lies in whether they estimate λ or not (regardless of whether knowledge of it is available or not), as their asymptotic properties do depend on this.
Recall that at the end of Sections 2.1.2 and 2.2.2 we also briefly mentioned the con- struction of the estimators of Buchmann and Gr¨ubel [2003] and Coca [2015] for the jump mass function. In particular, assuming dF only has a discrete component, we remarked that the cumulative sum of the former coincides withFenwhen an estimate ofλis plugged-
in, and that of the latter is, up to negligible terms, equal toFbn. By the arguments of the
previous paragraph, they are thus formally equal up to convolution with the kernel intro- duced in Section 2.2.2. Indeed, as we mention in Section 2.6, the covariance of the limiting processes in their central limit theorems is the same and, furthermore, it coincides with the information lower bound of the estimation problem when no restriction on the support of F is made. Buchmann and Gr¨ubel [2003] restricted the support to R+ and Trabs [2015a]
assumed the L´evy measure is absolutely continuous with respect to Lebesgue’s measure, so efficiency cannot be concluded but, if such changes in the model do not change the information bounds, both will be efficient.
We remark that, with this unified view, all the estimators can be interpreted as arising from an infinite sum of terms with alternating signs. This corresponds to their attempt to invert the compounding operation and justifies the common feature that the resulting estimates of the distribution are not monotonic. In other words, this feature does not de- pend on the type of estimator used such as a kernel one and, a priori, seems unavoidable without any further transformation such as the ones Buchmann and Gr¨ubel [2004] made in the purely atomic case mentioned at the end of Section 2.1.2. Due to the close relation- ship between the decompounding problem and the estimation of discretely observed L´evy processes, this also justifies the appearance of this feature in the estimators of functionals of the L´evy measure.
Even though the estimators of Nickl and Reiß [2012] and Nickl et al. [2016] were derived for more general L´evy processes, a natural question is how they compare to that in Coca [2015] when the underlying process is compound Poisson. These also turn out to be the same, except for exponentially negligible quantities, although we postpone the comparison until Section 2.5 for reasons that will become apparent there. Lastly, we remark that a third approach has recently been introduced to nonparametrically estimate the density of a discretely observed compound Poisson process. Gugushvili et al. [2015] and Gugushvili et al. [2016] use a Bayesian approach and are able to tackle the multidimensional case in the low- and the high-frequency regimes, respectively. Their methodology considerably differs from the one we deal with here and therefore refer the reader to their works for more details.