Stat Comput DOI 10.1007/s11222-017-9748-4
Nonparametric estimation for compound Poisson process via variational analysis on measures
Alexey Lindo1 · Sergei Zuyev2 · Serik Sagitov2
Received: 2 February 2016 / Accepted: 17 April 2017 © The Author(s) 2017. This article is an open access publication
Abstract The paper develops new methods of nonpara- 1 Introduction metric estimation of a compound Poisson process. Our key estimator for the compounding (jump) measure is based on The paper develops new methods of nonparametric esti- series decomposition of functionals of a measure and relies mation of the distribution of compound Poisson data. A on the steepest descent technique. Our simulation studies for compound Poisson process (Wt )t≥0 isaMarkovjumppro- various examples of such measures demonstrate flexibility of cess with W0 = 0 characterised by a finite compounding our methods. They are particularly suited for discrete jump measure Λ defined on the real line R = (−∞, +∞) such distributions, not necessarily concentrated on a grid nor on that the positive or negative semi-axis. Our estimators also appli- cable for continuous jump distributions with an additional Λ({0}) = 0, Λ:=Λ(R) ∈ (0, ∞). (1) smoothing step. The jumps of this process occur at the constant rate Λ, Keywords Compound Poisson distribution · Decompound- and the jump sizes are independent random variables with a ing · Measure optimisation · Gradient methods · Steepest common distribution Λ(dx)/Λ. In a more general context, descent algorithms the compound Poisson process is a particular case of a Lévy process with Λ being the corresponding integrable Lévy mea- Mathematics Subject Classification Primary: 62G05; sure. Inference problems for such processes naturally arises Secondary: 62M05 · 65C60 in financial mathematics (Cont and Tankov 2003), queueing theory (Asmussen 2008), insurance (Mikosch 2009) and in many other situations modelled by compound Poisson and Lévy processes. Suppose the compound Poisson process is observed at reg- ularly spaced times (Wh, W2h,...,Wnh) for some time step > = − B h 0. The consecutive increments Xi Wih W(i−1)h then Sergei Zuyev ( ,... ) [email protected] form a vector X1 Xn of independent random variables having a common compound Poisson distribution with the Alexey Lindo [email protected] characteristic function Serik Sagitov θ ψ(θ) θ [email protected] ϕ(θ) = Eei Wh = eh ,ψ(θ)= (ei x − 1)Λ(dx). 1 School of Mathematics and Statistics, University of Glasgow, (2) Glasgow, UK 2 Department of Mathematical Sciences, Chalmers University R of Technology and University of Gothenburg, Gothenburg, Here and below the integrals are taken over the whole Sweden unless specified otherwise. Estimation of the measure Λ in 123 Stat Comput termsofasample(X1,...Xn) is usually called decompound- ing which is the main object of study in this paper. 8 We propose a combination of two nonparametric meth- ods which we call characteristic function fitting (ChF) and 6 convolution fitting (CoF). ChF may deal with a more gen-
eral class of Lévy processes, while CoF explicitly targets the Distance compound Poisson processes. 4 The ChF estimator for the jump measure Λ is obtained by minimisation of the loss functional 2 5 hψ(θ) 2 4 LChF(Λ) = |e −ˆϕn(θ)| ω(θ)dθ, (3) 0 3 −4 2 where ψ(θ) ≡ ψ(θ,Λ) is given by (2), −2 lambda 0 1 x 2 n 4 1 θ ϕˆ (θ) = ei Xk . n n k=1 Fig. 1 Illustration of intrinsic difficulties faced by any characteristic function fitting procedure. Plotted is the integrated squared modulus of is the empirical characteristic function and ω(θ) is a weight the difference between two characteristic functions with measures Λ = δ Λ = λδ ∈[− , ],λ∈ ( , ) function. It was shown in Neumann and Reiss (2009)ina 1 and x , x 5 5 0 5 . Clearly, any algorithm based on closeness of characteristic functions, like (3), would have difficulties more general Lévy process setting that minimising (3) leads converging to the global minimum attained at point x = 1,λ= 1even to a consistent estimator of the Lévy triplet. Typically, ω(θ) in this simple two-parameter model is a positive constant for θ ∈[θ1,θ2] and zero otherwise, but it can also be chosen to grow as θ → 0, this would lead to boosting an agreement of the moments of a fitted jump distribution with the empirical moments. for such a two-step procedure will be denoted by Λ˜ in the We compute explicitly the derivative of the loss func- k sequel. tional (3) with respect to the measure Λ,formula(18)in To give an early impression of our approach, let us demon- “Appendix”, and perform the steepest descent directly on the strate the performance of our methods on the famous data by cone of non-negative measures to a local minimiser, further Ladislaus Bortkiewicz who collected the numbers of Prus- developing the approach by Molchanov and Zuyev (2002). sian soldiers killed by a horse kick in 10 cavalry corps over It must be noted that, as a simple example reveal, the func- a 20-year period (Bortkiewicz 1898). The counts 0, 1, 2, 3 tionals based on the empirical characteristic function usually and 4 were observed 109, 65, 22, 3 and 1 times, with 0.6100 have a very irregular structure, see Fig. 1. As a result, the deaths per year per cavalry unit. The author argues that the steepest descent often fails to attend the global optimal solu- data are Poisson distributed which corresponds to the mea- tion, unless the starting point of the optimisation procedure sure Λ = λδ concentrated on the point {1} (only jumps of is carefully chosen. 1 size 1) and the mass λ being the parameter of the Poisson The CoF estimation method uses the fact that the convo- distribution estimated by the sample mean to be 0.61. Fig- lution of F(x) = P(W ≤ x), h ure 2 on its top panel presents the estimated Lévy measures for the cut-off values k = 1, 2, 3 when using CoF method. ∗2( ) = ( ) ( − ) , F x F y F x y dy For the values of k = 1, 2, the result is a measure having many atoms. This is explained by the fact that the accuracy as a functional of Λ has an explicit form of an infinite Taylor of the convolution approximation is not enough for these ˆ series involving direct products of measures Λ, see Theo- data, but k = 3 already results in a measure Λ3 essentially rem 2 in Sect. 4. After truncating it to only the first k terms, concentrated at {1}, thus supporting the Poisson model with (k) Λˆ = . we build a loss function LCoF by comparing two estimates parameter 3 0 6098. In Sect. 4, we return to this exam- of F∗2: the one based on the truncated series and the other ple and explain why the choice of k = 3 is reasonable here. 2∗ being the empirical convolution Fn . CoF is able to produce Caused by a possibly very irregular behaviour of the score ˆ nearly optimal estimates Λk when large values of k are taken, function LChF demonstrated above, we practically observed but at the expense a drastically increased computation time. that the convergence of the ChF method depends critically on A practical combination of these methods recommended the choice of the initial measure, especially on its total mass. ˆ by this paper is to find Λk usingCoFwithalowvalueofk and However, the proposed combination of CoF followed by ChF ˆ then apply ChF with Λk as the starting value. The estimate demonstrates (the bottom plot) that this two-step (faster) pro- 123 Stat Comput
^ ^ for estimating the unknown density of the measure Λ.In Λ1 ||Λ1|| = 0.4589 ^ ^ contrast, we do not distinguish between discrete and contin- Λ2 ||Λ2|| = 0.5881 ^ ^ uous Λ in that our algorithms, based on direct optimisation Λ3 ||Λ3|| = 0.6099 of functionals of a measure, work for both situations on a dis- cretised phase space of Λ. However, if one sees many small atoms appearing in the solution, which fill a thin grid, this may indicate that the true measure is absolutely continuous and some kind of smoothing should yield its density. In this paper, we do not address estimation of more general Lévy processes allowing for Λ(−1, 1) =∞. In the Lévy pro- cess setting, the most straightforward approach for estimating the distribution F(x) = P(Wh ≤ x) is the moments fitting, see Feuerverger and McDunnough (1981b) and Carrasco and Florens (2000). Estimates of Λ can be obtained by maximis- ing the likelihood ratio (see e.g. Quin and Lawless 1994)orby minimising some measure of proximity between F and the 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ˆ 1 n empirical distribution function Fn(x) = = 1I {X ≤x}, 0246810 n k 1 k where the dependence on Λ comes through F via the inver- x sion formula of the characteristic function: Λ^ Λ^ 1 y 3 || 3|| = 0.6099 F(x) − F(x − 0) = lim exp{hψ(θ) − iθx}dθ. Λ~ Λ~ →∞ 1 || 1|| = 0.6109 2π y −y
For the estimation, the characteristic function in the integral above is replaced by the empirical characteristic function. Parametric inference procedures based on the empiri- cal characteristic function have been known for some time, see Feuerverger and McDunnough (1981a) and Sueishi and Nishiyama (2005), and the references therein. Algorithms based on the inversion of the empirical characteristic func- tion and on the relation between its derivatives were proposed in Watteel and Kulperger (2003). Note that the inversion of the empirical characteristic function, in contrast to the inversion of its theoretical counterpart, generally leads to a complex valued measure which needs to be dealt with.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 One of the reviewers has drawn our attention to the recent 0246810preprint Coca (2015) which promises to be useful for testing x the presence of discrete and/or continuous jump distribution components as well as for obtaining approximation accuracy Fig. 2 The analysis of Bortkiewicz horse kick data. Top panel com- bounds based on the central limit theorem. Practical imple- = , , parison of CoF estimates for k 1 2 3. Bottom panel comparison of mentations of these theoretical results are yet to be explored. the estimate by CoF with k = 3 and a combination of CoF with k = 1 followed by ChF The rest of the paper has the following structure. Sec- tion 2 introduces the theoretical basis of our approach—a constraint optimisation technique in the space of measures. ˜ cedure results in the estimate Λ1, which is as good as more Section 3 provides an algorithmic implementation of the cor- ˆ computationally demanding Λ3. responding steepest descent method in R-language. Section 4 Previously developed methods include discrete decom- develops the necessary ingredients for the CoF method based pounding approach based on the inversion of Panjer recur- on the main analytical result of the paper, Theorem 2. Sec- sions as proposed in Buchmann and Grübel (2003). van Es tion 5 contains a broad range of simulation results illustrating et al. (2007) and, lately, Duval (2013), Comte et al. (2014) performance of our algorithms on simulated data with var- studied the continuous decompounding problem when the ious compounding measures, both discrete and continuous. measure Λ is assumed to have a density. They apply Fourier Section 6 presents an application of our approach to real inversion in combination with kernel smoothing techniques currency exchange data. Section 7 summarises our approach 123 Stat Comput and gives some practical recommendations. We conclude by Taking into account condition (1), we aim to find a solution “Appendix” with proofs and explicit formulas for the gradi- to the following constraint minimisation problem: ents of the two loss functions used in our steepest descent algorithm. Λ = arg min{L(η) : η ∈ M+,η({0}) = 0}, (6)
where L : M+ → R is strongly differentiable functional of 2 Optimisation of functionals of a measure a measure. The following necessary condition of a minimum is proven in Appendix. In this section, we briefly present the main ingredients of the constrained optimisation of functionals of a measure. Theorem 1 Suppose that a Λ solves (6), and the functional Theorem 1 gives necessary conditions for a local minimum L possesses a gradient function ∇L(x; Λ) at this Λ. Then of a strongly differentiable functional. This theorem justi- fies a major step in our optimisation algorithm described in ∇L(x; Λ) ≥ 0 for all x ∈ R \{0}, Sect. 3. Further details of the underlying theory can be found (7) ∇L(x; Λ) = 0 Λ − almost everywhere. in Molchanov and Zuyev (2000a, b). Denote by M and M+ the class of signed and, respectively, non-negative measures with a finite total variation. The set Remark 1 It can be shown similarly that the necessary con- M then becomes a Banach space with sum and multiplication dition (7) also holds for optimisation over the class of Lévy measures satisfying, in addition to (1), the integrability con- by real numbers defined set-wise: (η1 + η2)(B) := η1(B) + dition min{1, x2} Λ(dx)<∞. η2(B) and (tη)(B) := tη(B) for any Borel set B and any real t.ThesetM+ is a pointed cone in M meaning that the zero measure is in M+, μ1 + μ2 ∈ M+, and tμ ∈ M+ as long as μ1,μ2,μ∈ M+ and t ≥ 0. 3 Steepest descent algorithm on the cone of A functional G : M → R is called Fréchet (or strongly) positive measures differentiable at η ∈ M if there exists a bounded linear oper- ator (a differential) DG(η)[·] : M → R such that There is an extensive number of algorithms realising a parametric optimisation over a finite number of continuous G(η + ν) − G(η) = DG(η)[ν]+o(ν), ν→0, (4) variables, but optimisation algorithms over the cone of mea- sures have been proposed only recently in Molchanov and where ν is the total variation of a signed measure ν ∈ Zuyev (2002) for the case of measures with a fixed total M. If for a given η ∈ M there exists a bounded function mass. The variation analysis of functionals of a measure out- ∇G( ·;η) : R → R such that lined in the previous section allows us to develop a steepest descent type algorithm for minimisation of functionals of a compounding measure which we describe next. This algo- (η)[ν]= ∇ ( ; η) ν( ) ν ∈ M, DG G x dx for all rithm has been used to obtain the simulation results presented in Sect. 5. then ∇G( ·;η) is called the gradient function for G at η. Recall that the principal optimisation problem has the Typically in applications, and it is indeed the case for the form (6), where the functional L(Λ) is minimised over the functionals of a measure considered in this paper, the gradient measures Λ subject to the constraint (1). For computational functions exist so that the differentials indeed have an integral purposes, a measure Λ ∈ M+ is replaced by its discrete form. approximation which has a form of a linear combination Λ = l λ δ of Dirac measures on a finite regular grid As a simple illustration, consider an integral of a bounded i=1 i xi ,..., ∈ R = + Δ function G(η) = f (x)η(dx). Since this is already a x1 xl , xi+1 xi 2 . Specifically, for a given bounded linear functional of η, we get ∇G(x; η) = f (x) measure Λ, the atoms of Λ are given by for any η. More generally, for a composition G(η) = u( f (x)η(dx)), the gradient function can be obtained by λ1 := Λ((−∞, x1 + Δ)), the Chain rule: λi := Λ([xi − Δ, xi + Δ)), for i = 2,...,l − 1, (8) λ := Λ([ − Δ, ∞)). l xl ∇G(x; η) = u f (y)η(dy) f (x). (5)
Clearly, the larger is l and the finer is the grid {x1,...,xl } The functional G in this example is strongly differentiable if the better is the approximation, however, at a higher compu- both functions u and f are bounded. tational cost. 123 Stat Comput
Respectively, the discretised version of the gradient func- l (ΛΛ + νν) − (ΛΛΛ) = ν + (|ν|). tion ∇L(x; Λ) is the vector L L gi i o i=1 g = (g1,...,gl ), gi := ∇L(xi ;ΛΛ), i = 1,...,l. (9) This minimisation is subject to the following linear con- Our main optimisation algorithm has the following struc- straints ture: In the master algorithm description above, the line 3 l |νi |≤ε, νi ≥−λi , i = 1,...,l. Steepest descent algorithm i=1 Input: initial vector Λ GoSteep Λ 1: function ( ) The just described linear programming task has a straight- 2: initialise the discretised gradient 3: g ← (∇L (x1;ΛΛ), . . . , ∇L(xl ;ΛΛ)) forward solution given below. < −τ >τ ≥ ··· ≥ 4: while mini gi 2 or max{i : λi >τ1} gi 2 do For simplicity, we assume that g1 gl . Note that 5: choose a favourable step size ε depending on L and Λ this ordering can always be achieved by a permutation of Λ ← MakeStep ε Λ g 6: compute new vector ( , , g) the components of the vector g and respectively, Λ. Assume 7: compute gradient at the new Λ: Λ 8: g ← (∇L(x1;ΛΛ), . . . , ∇L(xl ;ΛΛ)) also that the total mass of is bigger than the given positive 9: end while stepsize ε. Define two indices 10: return Λ 11: end function i−1 ig = max{i : gi ≥|gl |}, iε = max{i : λ j <ε}. = uses the necessary condition (7) as a test condition for the j 1 main cycle. In the computer realisations, we usually want to ≤ ν ∗ discard the atoms of a negligible size: for this purpose, we If iε ig, then the coordinates of are given by τ use a zero-value threshold parameter 1. Another threshold ⎧ τ −λ ≤ , parameter 2 decides when the coordinates of the gradient ⎨ i for i iε ν∗ := iε−1 λ − ε = + , vector are sufficiently small to be discarded. For the exam- i ⎩ j=1 j for i iε 1 ples considered in the next section, we typically used the 0fori ≥ iε + 2, −2 −6 following values: ω ≡ 1, τ1 = 10 and τ2 = 10 .The MakeStep key subroutine, mentioned on line 6, is described and if iε > ig, then below. It calculates the admissible steepest direction ν ∗ of ∗ ∗ ⎧ size ν ≤ε and returns an updated vector Λ ← Λ + ν . ⎨⎪ −λi , for i ≤ ig, ν∗ := 0fori < i < l, i ⎪ g Algorithm for a steepest descent move ⎩ ig ε − = λ j for i = l. Input: maximal step size ε, current variable value Λ and current gra- j 1 dient value g 1: function MakeStep(ε, Λ, g) The presented algorithm is realised in the statistical com- ν∗ ← 2: initialise the optimal step 0 putation environment R Core Team (2015) in the form of a 3: initialise the running coordinate i ← 0 4: initialise the total mass available E ← ε library mesop which is freely downloadable from one of the 5: while ((E > 0) and (i ≤ l)) do authors’ webpage.1 6: if gi > |gl | then ν∗ ← (−λ , − ) 7: i max i E ← − ν∗ 8: E E i 9: else 4 Description of the CoF method ν∗ ← 10: l E ← 11: E 0 As it was alluded in Introduction, the CoF method uses a 12: end if 13: i ← i + 1 representation of the convolution as a function of the com- 14: end while pounding measure Λ. We now formulate the main theoretical ∗ 15: return Λ + ν result of the paper on which the CoF method is based. The 16: end function proof is given in Appendix. We will need the following notation. For a function F, MakeStep ν ∗ denote Ux F(y) = F(y − x) − F(x) and The subroutine looks for a vector which l ν minimises the linear form i=1 gi i appearing in the Taylor expansion 1 http://www.math.chalmers.se/~sergei/download.html. 123 Stat Comput
U ,..., F(y) := U (U ,..., F(y)) the values R (0.61) = 0.58, 0.21, 0.06 for k = 1, 2, 3. x1 xn xnx1 xn−1 k n−|J| This indicates that k = 3 is rather adequate cut-off for the = (−1) F(y − j∈J x j ), data. J⊆{1,2,...,n} If the expected number of jumps, hΛ, in the time inter- [ , ] where the sum is taken over all the subsets J of {1, 2,...,n} val 0 h , is large, the sample values Xi , in the case of a finite variances, would have approximately normal distri- including the empty set. Denote 0(F,Λ,y) = F(y), and bution. Since the normal distribution is determined by the first two moments only and not by the entire compound- ( ,Λ, ) = 1 ( )Λ( )...Λ( ), i F y Ux1,...,xi F y dx1 dxi Λ/Λ i! Ri ing distribution, an effective estimation of is hardly i ≥ 1. possible, see Duval (2014) for a related discussion. Indeed, to get the upper bound close to 0.2 given hΛ=8, one = Theorem 2 Let (Wt )t≥0 be a compound Poisson process would need to take k 41 which is hardly computationally characterised by (2), and F(y) = Fh(y) be the cumulative possible. distribution function of Wh for a given positive h. Then for To summarise, if one has a control on the choice of h, each real y, one has it should be taken so that the estimated value of hΛ is close to 1. For large values of this parameter, the central limit ∞ theorem prevents an effective estimation of λ, while the small ∗2 i F (y) = h i (F,Λ,y). (10) values would result in almost always single jumps and the i=0 optimisation procesure giving basically the sample measure as a solution. Similarly to the problem of choice of a kernel Recall that the empirical convolution of a sample estimator or the histogram’s bin width, a compromise should (X ,...,X ), 1 n be sought. A practical approach would be to try various values of h, as we demonstrate below in Sect. 6 on the real FX data. ˆ ∗2( ) := 1 { + ≤ }. Fn y n 1I Xi X j y (11) 2 1≤i< j≤n 5 Simulation results is an unbiased and consistent estimator of F∗2(x),seeFrees (1986). The CoF method looks for a finite measure Λ that To illustrate the performance of our estimation methods, we minimises the following loss function generated samples of size n = 1000 for compound Poisson processes driven by various kinds of measure Λ. In Sects. 5.1, k 2 5.2 and 5.3, we considered examples of discrete jump size (k) (Λ) = i ( ˆ ,Λ, ) − ˆ ∗2( ) ω( ) . LCoF h i Fn y Fn y y dy distributions. Note that lattice distributions with both positive = i 0 and negative jumps are particularly challenging because of (12) possible cancellations of jumps, the case barely considered in the literature so far. k The infinite sum in (10) is truncated to terms in (12)for In Sects. 5.4 and 5.5, we present simulation results for two computational reasons. The error introduced by the trunca- cases of continuously distributed jumps: non-negative and tion can be accurately estimated by bounding the remainder general. The continuous measures are replaced in the sim- term in the finite expansion formula (16) in the proof. Alter- ulations by their discretised versions given by (8). The grid natively, turning to (10) and using 0 ≤ F(y) ≤ 1, we obtain Δ = . − size in these examples was 0 25. Note that no special sup |U ,..., F(y)|≤2i 1, yielding a uniform bound y∈R x1 xi account is given to the fact that the measure is continuous and the algorithms work the same way as with genuine discrete ∞ i measures. However, the presence of atoms filling the consec- sup h i (F,Λ,y) ≤ Rk(hΛ), where ∈R utive grid ranges should indicate that the true compounding y i=k+1 measure is probably continuous. A separate analysis could ∞ ( )n 1 2x be tried to formally check this hypothesis, for instance, by the Rk(x) = . (13) 2 n! n=k+1 methods proposed in Coca (2015). If confirmed, some kind of kernel smoothing could be used to produce an estimated Thus, to have a good estimate with this method, the upper density curve or specific estimation methods for continu- bound Rk(hΛ) should be small, which could be achieved ously distributed jumps employed, like the ones mentioned by reducing the time step h or/and increasing k. For instance, in Introduction. for the horse kick data considered in Introduction, we have For all the considered examples, we applied three versions h = 1 and the estimated value of Λ is 0.61, giving of the CoF with h = 1, k = 1, 2, 3 and ω ≡ 1. We also apply 123 Stat Comput
= ChF using the estimate of CoF with k 1. Observe that CoF ^ Λ Λ−Λ1 || || = 1 with k = 1 can be made particularly fast because here we ^ ^ Λ−Λ2 ||Λ1|| = 0.6153 have a non-negative least squares optimisation problem. If ^ ^ Λ−Λ3 ||Λ2|| = 0.8605 the computation time is no concern, one can also implement ^ ||Λ3|| = 0.927 CoF with higher values of k to use the resulting measure as a starting point to ChF. Given a complicated nature of the loss function, this may or may not lead to a better fit. In all these examples Λ=1, which explains our particular choice of h = 1, see the discussion above in connection to the error estimate (13).
5.1 Degenerate jump measure
Consider first the simplest measure Λ(dx) = δ1(dx) cor- responding to a standard Poisson process with rate 1. Since all the jumps are integer valued and non-negative, it would be logical to take the non-negative integer grid for possi- −0.2 0.0 0.2 0.4 0.6 −2−1012345 ble atom positions of the discretised Λ.Thisisthewaywe x have done it for the horse kick data analysis in Introduction. However, to test the robustness of our methods, we took the ^ Λ grid {0, ±1/4, ±2/4,...}. As a result, the estimated mea- Λ−Λ3 || || = 1 Λ−Λ~ ^ sures might place some mass on non-integer points or even 1 ||Λ3|| = 0.927 Λ~ on negative values of x to compensate for inaccurately fitted || 1|| = 0.9514 positive jumps. We have chosen to show on the graphs the discrepancies between the estimated and the true measure. An important indicator of the effectiveness of an estimation ˆ is the closeness of the total masses Λ and Λ.ForΛ = δ1, the probability to have more than 3 jumps is approximately 0.02; therefore, with the CoF method we expect that k = 3 would give an adequate estimate for these data. Indeed, the top panel of Fig. 3 demonstrates that the CoF with k = 3is much more effective in detecting the jumps of the Poisson process compared to k = 2 and, especially, to k = 1. The lat- ter methods generate large discrepancies both in atom sizes and in the total mass of the obtained measure. Observe also the presence of artifactual small atoms at large x and even at −0.02 0.00 0.02 0.04 0.06 0.08 some non-integer locations. −2−1012345 The bottom panel shows that a good alternative to a rather x computationally demanding CoF method with k = 3isa ˆ Λ = δ much faster combined CoF–ChF method when Λ measure Fig. 3 Simulation results for 1. Top panel the differences between 1 Λ({ }) Λˆ ({ }) = , , is used as the initial measure in the ChF algorithm. The result- x and their estimates k x obtained by CoF with k 1 2 3. Zero values of the differences are not plotted. Bottom panel comparison ing measure Λ˜ is almost identical to Λˆ , but also has the ˆ ˜ ˆ 1 3 of Λ3 with Λ1 obtained by ChF initiated at Λ1. Notice a drastic change total mass closer to the target value 1. The total variation in the vertical axis scale as we go from the top to the bottom panel ˆ distances between the estimated measures Λk and the theo- retical measure Λ are 0.435, 0.084 and 0.053 for k = 1, 2, 3, respectively. The best fit provides the combined CoF–ChF Λ=1 and jumps of sizes −1, 1, 2 having respective ˜ method which produces a measure Λ1 within the distance of probabilities 0.2, 0.2 and 0.6. Figure 4 presents the results 0.043 from Λ. of our simulations. The presence of negative jumps can- celling positive jumps creates an additional difficulty for the 5.2 Discrete positive and negative jumps estimation task. This phenomenon explains why the approx- imation obtained with k = 2 is worse than with k = 1 and Consider now a jump measure Λ = 0.2δ−1 + 0.2δ1 + 0.6δ2. k = 3: two jumps of sizes +1 and −1 sometimes cancel each This gives rise to a compound Poisson process with rate other, which is indistinguishable from no jumps case, see the 123 Stat Comput
^ Λ ^ Λ Λ−Λ1 || || = 1 Λ−Λ1 || || = 1 ^ ^ ^ ^ Λ−Λ2 ||Λ1|| = 0.6122 Λ−Λ2 ||Λ1|| = 0.6406 ^ ^ ^ ^ Λ−Λ3 ||Λ2|| = 0.9019 Λ−Λ3 ||Λ2|| = 1.2047 ^ ^ ||Λ3|| = 0.9041 ||Λ3|| = 0.9539 −0.4 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 0.4
−2 −1 0 1 2 3 4 5 −2−1012345 x x
Λ−Λ^ ||Λ|| = 1 Λ−Λ^ ||Λ|| = 1 ~3 3 Λ−Λ Λ^ Λ−Λ~ ^ 1 || 3|| = 0.9041 1 ||Λ3|| = 0.9539 Λ~ ~ || 1|| = 1.038 ||Λ1|| = 1.0279 0.00 0.05 0.10 −0.05 0.00 0.05 0.10
−2 −1 0 1 2 3 4 5 −2−1012345 x x
Fig. 4 Simulation results for Λ = 0.2δ−1 + 0.2δ1 + 0.6δ2. Top panel Fig. 5 Simulation results for a shifted Poisson distribution Λ({x}) = ˆ −1 the differences between Λ({x}) and their estimates Λk ({x}) obtained e /(x − 1)! for x = 1, 2,.... Top panel the differences between ˆ ˜ ˆ by CoF with k = 1, 2, 3. Bottom panel comparison of Λ3 with Λ1 Λ({x}) and their estimates Λk ({x}) obtained by CoF with k = 1, 2, 3. ˆ ˜ Bottom panel comparison of Λ3 with Λ1 obtained by ChF initiated at ˆ Λ1 top panel of Fig. 4. Moreover, −1 and 2 added together are the same as having a single size 1 jump. The phenomenon tom plot, it is again a clear winner in this case too. It is also still persists when we increased the sample size: k = 1 and much faster. k = 3 still perform better. Notice that going from k = 1 through k = 2uptok = 3 improves the performance of CoF, 5.3 Unbounded compounding distribution although the computing time increases dramatically. The cor- ˆ responding total variation distances of Λk to the theoretical As an example of a measure Λ with unbounded support, distribution are 0.3669, 0.6268 and 0.1558. The combined we take a shifted Poisson distribution with parameter 1. method gives the distance 0.0975, and according to the bot- Figure 5 presents our simulation results for this case; for 123 Stat Comput
∈[−, ] computation purposes, we took the interval x 2 5 as Λ ||Λ|| = 1 the support range for the estimated measure. In practice, the ^ ^ Λ1 ||Λ1|| = 0.6192 support range should be enlarged if atoms start appearing ^ ^ Λ2 ||Λ2|| = 1.0362 ^ on the boundaries of the chosen interval indicating a wider Λ ||Λ^ || = 0.9018 ~3 ~3 support of the estimated measure, see also Buchmann and Λ1 ||Λ1|| = 0.9047 Grübel (2003) for a related discussion. As the top panel reveals, also in this case the CoF method with k = 3gives a better approximation than those with k = 1ork = 2 (the total variation distance to the theoretical distribution is 0.1150 compared to 0.3256 and 0.9235, respectively) and the combined (faster) method gives an even better estimate with ˜ dTV(Λ1,Λ) = 0.0386. Interestingly, the case of k = 2was the worst in terms of the total variation distance to the original measure. We suspect that the ’pairing effect’ may be respon- sible: the jumps are better fitted with a single integer- valued variable rather than with the sum of two. The algorithm may 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 also got stuck in a local minimum producing small atoms at 012345 non-integer positions. x
5.4 Continuous non-negative compounding distribution
Consider a compound Poisson process of rate 1 with the compounding distribution being exponential with parame- ter 1. The top plot of Fig. 6 shows that, as expected, the approximation accuracy increases with k. Observe that the ˆ total variation distance dTV(Λ3,ΛΛΛ) = 0.0985 is comparable with the discretisation error: dTV(Λ,ΛΛΛ) = 0.075. A Gaus- ˆ sian kernel smoothed version of Λ3 is presented at the bottom plot of Fig. 6. The visible discrepancy for small values of x is explained by the fact that there were no sufficiently many small jumps in the simulated sample for the algorithm to put more mass around 0. Optimisation in the space of measures usually tends to produce atomic measures since these are boundary ˜ points of typical constraint sets in M. Indeed, Λ1 has smaller number of atoms than Λ does and still it approx- 0.0 0.2 0.4 0.6 0.8 imates better the empirical characteristic function of the 012345 sample. Fig. 6 Simulation results for a compound Poisson process with jump intensity 1 and jump sizes having an exponential distribution with 5.5 Continuous compounding distribution parameter 1. Top plot obtained measures for various algorithms, the bottom plot the theoretical exponential density and the smoothed ver- ˜ Finally, Fig. 7 takes up the important example of com- sion of Λ1 measure with a Gaussian kernel with the standard deviation pound Poisson processes with normally distributed jumps 0.4 having both positive and negative values. Once again, the ˆ estimates Λk improve as k increases, and the combined Similarly to a general smoothing procedure, optimisation of ˆ method CoF–ChF gives an estimate similar to Λ3. Notice the kernel width requires additional criteria to be employed, an inflection around 0 caused by the restraint on the esti- like information criteria. It is also possible to add a specific mated measure which imposes the origin to have a zero mass. term into the score function of the optimisation procedure This shows as a dip in the curve produced by the kernel depending on the kernel function which is responsible for the smoother. goodness of fit of the smoothed curve to the empirical data. In the presented examples with continuous compounding We do not address, however, these issues here considering distribution, when choosing the kernel smoother width, we it a separate problem from a nonparametric measure fitting, were guided by a visual smoothness of the resulting curve. see also Discussion section below. 123 Stat Comput
GBP to EUR FX log−return increments from 2014−01−01 to 2014−10−10 Λ ||Λ|| = 1 ^ ^ Λ1 ||Λ1|| = 0.5847 ^ ^ Λ2 ||Λ2|| = 0.8805 ^ Λ ||Λ^ || = 0.8439 ~3 ~3 Λ1 ||Λ1|| = 0.8751 −0.03 −0.02 −0.01 0.00 0.01 0.02 0.03 0.0 0.1 0.2 0.3 0 50 100 150 200 −2−1012345 x
Daily increments of GBPEUR FX log−returns
Stable distribution Density 0 20 40 60 80 100 120 140
0.0 0.2 0.4 0.6 0.8 −0.02 −0.01 0.00 0.01 0.02
−2−1012345 Fig. 8 Top plot Consecutive increments of the log-returns of GBP rate against EUR from 2014-01-02 to 2014-10-10. Bottom plot the fitted Fig. 7 Top plot estimated jump measure for a simulated sample with stable distribution to the increment data jump sizes having a normal distribution with the mean 0.5 and variance 0.25. Bottom plot the theoretical Gaussian density and the smoothed ˆ version of Λ3 measure with a Gaussian kernel with the standard devia- tion 0.2 i.i.d. increments, see e.g. Cont (2001). We took FX data of the Great Britain Pound (GBP) against a few popular cur- rencies and chose to work with GBP to EUR exchange rates 6 Currency exchange data application in a period of 200 consecutive days of a relatively stable market from 2014-01-02 to 2014-10-10, see the top plot of Lévy processes are widely used in financial mathematics to Fig. 8. We fitted various, popular among financial analysts, model the dynamics of the log-returns which for a commod- distributions to the daily increments of the log-returns: Gaus- ity with price St and time t is defined to be Wt = log(St /S0). sian, GEV, Weibull and stable distributions. The best fit was For this model, the increments Wh − W0, W2h − Wh,...are obtained by the stable distribution. In order to have a consis- independent and have a common infinitely divisible distribu- tent comparison with our methods, we used the loss function tion. For example, many authors argue that the log-returns of (3) to estimate the parameters of the stable distribution, such the currency exchange rates in a stable market have indeed estimation method goes back to at least Paulson et al. (1975). 123 Stat Comput
ThefittedstableS(1.882, −1, 0.002, 0; 0) distribution (in S0 ●● ●● parametrisation) is resented on the bottom plot of Fig. 8.A formal Chi-square test, however, rejected the null hypothesis that the data are coming from the fitted stable distribution due to the large discrepancies in the tails. The distance between the empirical characteristic function and the fitted stable dis- tribution’s characteristic function measured in terms of the −3 score function LChF was 6.12 × 10 . We then ran our CoF algorithm with k = 1 and obtained a distribution within the distance 5.53 × 10−6 from the empirical characteristic func- tion. Taking the resulting jump measure as a starting point to our ChF method, we arrived at the distribution within the distance 8.71 × 10−7. The observed improvement is due to ● more accurate estimates of the large jumps of the exchange rates (which are not related to global economical or political ● ● ● ● ● ● ● events). 0.0 0.1 0.2 0.3 0.4 0.5 0.6 It may be expected that, as in the case of a linear regres- −0.015 −0.010 −0.005 0.000 0.005 0.010 0.015 x sion, the agreement of the estimated model with the data could be “too good”. To verify stability of our estimates, we ran our algorithms on the data with different time lags: every ●● ●● 2, 4 and 8 days records. It is interesting to note that even at 8 days lag our algorithms attained a distribution at the distance 2.19 × 10−4, an order of magnitude closer to the empiri- cal characteristic function than the fitted stable distribution despite that fact that 8 times less data were used, see Fig. 9. The estimates of the measure Λ obtained for various lags are not that much different, apart from 8 days lag when only 25 observations are available, which reassures that our esti- mation methods give consistent results. These findings are illustrated on the bottom panel of Fig. 9.
● 7 Discussion
● ● ● ● ● ● ● This paper deals with nonparametric inference for compound 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Poisson processes. We proposed and analysed new algo- −0.015 −0.010 −0.005 0.000 0.005 0.010 0.015 x rithms based on the characteristic function fitting (ChF) and convoluted cumulative distribution function fitting (CoF). Fig. 9 Top plot estimated Lévy measure for a GBP/EUR rate log-return The algorithms are based on the recently developed varia- increments. Bottom plot estimated compounding measure for FX data tional analysis of functionals of measures and the correspond- recorded with various lags: 1 (circle), 2 (plus), 4 (cross)and8(triangle) ing steepest descent methods for constraint optimisation on day intervals the cone of measures. CoF methods are capable of producing very accurate estimates, but at the expense of growing com- putational complexity. The ChF method critically depends closest in the total variation to the measure under estimation, on the initial approximation measure due to highly irregular and thus, this is our method of choice. behaviour of the objective function. We have observed that The practical experience we gained during various tests the problems of convergence of the ChF algorithms can often allows us to conclude that the suggested methods are espe- be effectively overcome by choosing the sample measure cially well suited for estimation of discrete jump size distribu- (discretised to the grid) as the initial approximation mea- tions. They work well even with jumps that take both positive sure. However, a better alternative, as we demonstrated in and negative values, not necessarily belonging to a regu- the paper, is to use the measure obtained by the simplest lar lattice, demonstrating a clear advantage over the existing (k = 1) CoF algorithm. This combined CoF–ChF algorithm methods, see Buchmann and Grübel (2003), Buchmann and is fast and in majority of cases produces a measure which is Grübel (2004). The use of our algorithms for continuous 123 Stat Comput compounding distributions requires more trial and error in for all sufficiently small tn. Thus, in any ball around η there choosing the right discretisation grid and smoothing proce- exists an ηn ∈ A such that L(ηn)
(η)[η − η]= (η ) − (η) + (η − η ) DL n L n L o n Dz G(ϕ) := G(ϕ + δz) − G(ϕ), ϕ ∈ N . = L(ηn) − L(η) + o(tn), For the iterations of such difference operators, implying
(η ) − (η) =− ε( + ( )) < − ε/ = ( ), ( ,..., ) ∈ Rn, L n L tn 1 o 1 tn 2 Dz1,...,zn G Dzn Dz1,...,zn−1 G z1 zn 123 Stat Comput
δ Π = ( , ) it can be checked that where for each realisation j z j of with z j t j x j ,we Π Π [ , ]×R denote by t the restriction of onto 0 t .Forafixed −| | (ν) = (− )n J ν + δ , arbitrary y ∈ R and a point configuration ϕ = δ( , ), Dz1,...,zn G 1 G j∈J z j j ti x j J⊆{1,2,...,n} consider a functional G y defined by where |J| stands for the cardinality of J, so that if J is an G y(ϕ) = 1I x j ≤ y empty set, then |J|=0. Define (t j ,x j )∈ϕ
TμG(z ,...,z ) := Eμ D ,..., G(Π). 1 n z1 zn and notice that for any z = (t, x), Suppose that the functional G is such that there exists a con- (ϕ + δ ) = ≤ − = (ϕ). stant c > 0 satisfying G y z 1I x j y x G y−x (17) (t j ,x j )∈ϕ
n n |G = δz |≤c for all n ≥ 1 and all (z1,...zn). j 1 j Expressing the cumulative distribution function F(y) = P{W ≤ y} as an expectation It was proved in Molchanov and Zuyev (2000a, Theorem 2.1) h that if μ, μ are finite measures, the expectation Eμ+μ G(Π) exists and F(y) = Pμ x j ≤ y = EμG y(Πh), (t j ,x j )∈Πh Eμ+μ G(Π) = EμG(Π)
∞ and putting μ =[0, h]×Λ, μ =[h, 2h]×Λ, we find 1 + TμG(z1,...,zi )μ (dz1)...μ(dzi ). (15) ! Ri ∗ = i (Π) = { ≤ }= { + ≤ }= 2( ), i 1 Eμ +μ G y P W2h y P Wh Wh y F y
Generalisations of this formula to infinite and signed mea- = − where Wh W2h Wh. Observe also that by iteration of sures for square integrable functionals can be found in Last (17), (2014). A finite-order expansion formula can be obtained by representing the expectation above in the form Tμ G y(z1,...,zn)
= Eμ Dz ,...,z G y(Π) Eμ+μ G(Π) = EμEμ [G(Π + Π ) Π], 1 n −| | = (− )n J Π + δ 1 Eμ G y j∈J z j where Π and Π are independent Poisson processes with J⊆{1,2,...,n} intensity measures μ and μ , respectively, and then applying = (− )n−|J| ( − ) = ( ). 1 F y j∈J x j Ux1,...,xn F y the moment expansion formula by Błaszczyszyn et al. (1997, J⊆{1,2,...,n} Theorem 3.1) to G(Π + Π ) viewed as a functional of Π Π withagiven . This gives us To finish the proof, it now remains to apply expansion (15):
(Π) = (Π) ∗2 Eμ+μ G EμG F (y) = Eμ +μ G y(Π) = F(y) k ∞ 1 1 + ( ,..., )μ ( )...μ( ) + U ,..., F(y)μ (dt dx )...μ (dt dx ) TμG z1 zi dz1 dzi ! x1 xi 1 1 n i i! Ri = i (R+×R)i i=1 i 1 ∞ 1 = i ( ,Λ, ). + Tμ+μ G(z1,...,zk+1)μ (dz1)... h i F y ( + )! + k 1 Rk 1 i=0
μ (dzk+1). (16) Gradient of ChF loss function To prove Theorem 2, we use a coupling of the compound Poisson process (Wt )t≥0 with a Poisson process Π on R+×R The ChF method is based on the loss function LChF given by driven by the intensity measure μ = × Λ, where is the (3), which is everywhere differentiable in Fréchet sense with Lebesgue measure on [0, +∞). Clearly, respect to the measure Λ. Aiming at the steepest descent gra- dient method described in Sect. 3 for obtaining the minimum t of the loss function, we compute here the gradient of LChF W = ( , )∈Π x = xΠ( s x), t t j x j t j d d in terms of the following functions 0 R 123 Stat Comput
k (θ, ) := (θ ) − , (θ, ) := (θ ) − θ , ( ) q1 x cos x 1 q2 x sin x x 1I {|x|<ε} ∇ k ( ; Λ) = i ( ˆ ,Λ, ) − ˆ ∗2( ) LCoF x 2h h i Fn y Fn y i=0 Qi (θ, Λ) := q2(θ, x)Λ(dx), i = 1, 2. k−1 j ˆ × h Ξ j (Fn,Λ,y, x)ω(y)dy, Using this notation, the real and imaginary parts of an j=0 infinitely divisible distribution characteristic function ϕ = ϕ + ϕ 1 i 2 can be written down as where (θ,Λ) ϕ (θ, Λ) = ehQ1 cos{hQ (θ, Λ)}, Ξ ( ,Λ, , ) = 1 ( )Λ( )...Λ( ). 1 2 j F y x Ux,x1,...,x j F y dx1 dx j j! R j hQ1(θ,Λ) ϕ2(θ, Λ) = e sin{hQ2(θ, Λ)}. This formula follows from the Chain rule (5) and the After noticing that ϕˆn =ˆϕn,1 + iϕˆn,2, with equality n n k j ϕˆ (θ) = 1 (θ ), ϕˆ (θ) = 1 (θ ), ∇ h ( )Λ( )...Λ( ) ( ; Λ) n,1 cos X j n,2 sin X j Ux1,...,x j F y dx1 dx j x n n j! R j j=1 j=1 j=1 k−1 j = h ( )Λ( )...Λ( ). the loss functional LChF can be written as h Ux,x1,...,x j F y dx1 dx j j! R j j=0 2 LChF(Λ) = ϕ1(θ, Λ) −ˆϕn,1(θ) ω(θ)dθ To justify the last identity, it suffices to see that for any inte- 2 grable symmetric function u(x1,...,x j ) of j ≥ 1 variables, + ϕ2(θ, Λ) −ˆϕn,2(θ) ω(θ)dθ. ∇ u(x1,...,x j )Λ(dx1)...Λ(dx j ) (x; Λ) From this representation, the gradient function corre- Rj sponding to the Fréchet derivative with respect to the measure = ( , ,..., )Λ( )...Λ( ). Λ j u x x1 x j−1 dx1 dx j−1 is obtained using the Chain rule (5): R j−1
∇L (x; Λ) This is due to ChF = 2 {ϕ (θ, Λ) −ˆϕ , (θ)]}∇ϕ (θ)[x,Λ]ω(θ)dθ 1 n 1 1 u(x ,...,x )(Λ + ν)(dx )...(Λ+ ν)(dx ) 1 j 1 j R j + 2 {ϕ2(θ, Λ) −ˆϕn,2(θ)}∇ϕ2(θ)[x,Λ]ω(θ)dθ, − u(x1,...,x j )Λ(dx1)...Λ(dx j ) j (18) R j = u(x1,...,x j )Λ(dx1)...Λ(dxk−1) where the gradients of ϕi (θ) := ϕi (θ, Λ), i = 1, 2, with R j k=1 respect to the measure Λ,aregivenby × ν(dxk)Λ(dxk+1)...Λ(dx j ) + o(ν), (θ,Λ) ∇ϕ (θ)(x; Λ) = hehQ1 1 where the last sum equals × cos hQ (θ, Λ) q (θ, x) 2 1 − sin hQ2(θ, Λ) q2(θ, x) , j u(x, x1,...,x j−1)ν(dx)Λ(dx1)...Λ(dx j−1). (θ,Λ) ∇ϕ (θ)(x; Λ) = hehQ1 R j 2 × sin hQ (θ, Λ) q (θ, x) 2 1 For example, the cost function (12) with k = 1 and ω(y) ≡ 1 + cos hQ2(θ, Λ) q2(θ, x) . has the gradient
( ) Gradient of CoF loss function ∇L 1 (x; Λ) CoF = ˆ ( ) − ˆ ∗2( ) As with the ChF method, the CoF algorithm relies on the 2h Fn y Fn y steepest descent approach. The needed gradient function has + ˆ ( − )Λ( ) ˆ ( − ) . the form Fn y z dz Fn y x dy 123 Stat Comput
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