Nonparametric estimation for compound Poisson process via variational analysis on measures

Lindo, Alexey; Zuyev, Sergei; Sagitov, Serik

doi:10.1007/s11222-017-9748-4

Stat Comput

2017

DOI: 10.1007/s11222-017-9748-4

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Nonparametric estimation for compound Poisson process via variational analysis on measures

Alexey Lindo

Sergei Zuyev

Serik Sagitov

Abstract: The paper develops new methods of nonparametric estimation of a compound Poisson process. Our key estimator for the compounding (jump) measure is based on series decomposition of functionals of a measure and relies on the steepest descent technique. Our simulation studies for various examples of such measures demonstrate flexibility of our methods. They are particularly suited for discrete jump distributions, not necessarily concentrated on a grid nor on the positive or negative semi-axis. Our estimators also … Show more

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Cited by 3 publications

(2 citation statements)

References 28 publications

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“…Buchmann and Grübel (2004) establish weak convergence of their modified estimator, but on the downside its asymptotic distribution is unwieldy to give confidence statements on p. Most importantly, the plug-in approaches in Buchmann and Grübel (2003) and Buchmann and Grübel (2004) do not allow obvious generalisations to non-uniformly spaced observation times {t i }. In Lindo et al (2018), another frequentist estimator of the jump measure is introduced, that is obtained via the steepest descent technique as a solution to an optimisation problem over the cone of positive measures. The emphasis in Lindo et al (2018) is on numerical aspects; again, no obvious generalisation to the case of non-uniform {t i } is available.…”

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confidence: 99%

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Decompounding discrete distributions: A nonparametric Bayesian approach

Gugushvili

Mariucci

Meulen

2019

Scandinavian J Statistics

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Suppose that a compound Poisson process is observed discretely in time and assume that its jump distribution is supported on the set of natural numbers. In this paper we propose a non-parametric Bayesian approach to estimate the intensity of the underlying Poisson process and the distribution of the jumps. We provide a MCMC scheme for obtaining samples from the posterior. We apply our method on both simulated and real data examples, and compare its performance with a frequentist plug-in estimator from Buchmann and Grübel (2004). On a theoretical side, we study the posterior from the frequentist point of view and prove that as the sample size n → ∞, it contracts around the 'true', data-generating parameters at rate 1/ √ n, up to a log n factor.

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confidence: 99%

“…In Lindo et al (2018), another frequentist estimator of the jump measure is introduced, that is obtained via the steepest descent technique as a solution to an optimisation problem over the cone of positive measures. The emphasis in Lindo et al (2018) is on numerical aspects; again, no obvious generalisation to the case of non-uniform {t i } is available.…”

mentioning

confidence: 99%

Decompounding discrete distributions: A nonparametric Bayesian approach

Gugushvili

Mariucci

Meulen

2019

Scandinavian J Statistics

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An inverse problem for infinitely divisible moving average random fields

Karcher

Roth

Spodarev

et al. 2018

Stat Inference Stoch Process

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Given a low frequency sample of an infinitely divisible moving average random field { R d f (x−t)Λ(dx); t ∈ R d } with a known simple function f , we study the problem of nonparametric estimation of the Lévy characteristics of the independently scattered random measure Λ. We provide three methods, a simple plug-in approach, a method based on Fourier transforms and an approach involving decompositions with respect to L 2 -orthonormal bases, which allow to estimate the Lévy density of Λ. For these methods, the bounds for the L 2 -error are given. Their numerical performance is compared in a simulation study.An Inverse Problem for ID Moving Averages with Lévy characteristics (a 1 , b 1 , v 1 ), where f = n k=1 f k 1I ∆ k is a simple function. Suppose a sample (X(t 1 ), . . . , X(t N )) from X is available. The problem studied in this paper is the nonparametric estimation of (a 0 , b 0 , v 0 ). For any simple function f with congruent sets ∆ k , X(t) in (1) has the same distribution as a linear combination of i.i.d. infinitely divisible random variables. Therefore, existence and uniqueness of a characteristic triplet (a 0 , b 0 , v 0 ) with the property that a certain linear combination of independent random variables with the corresponding infinitely divisible distribution leading to a random variable with Lévy characteristics (a 1 , b 1 , v 1 ) becomes a characterization problem for such distributions. For certain distributions, namely the Poisson and the Gaussian one as well as a mixture of both, all possible distributions for the summands in the linear combination can be described (see e.g. [1]). The disadvantage of those characterization theorems is that they do not give any information about the involved parameters (expectation and variance of each summand) and so it is not possible to derive sufficient conditions for the existence of a solution in terms of the kernel function f . Therefore, to solve the inverse problem, we prefer to use concrete relations between the characteristic triplets of X and Λ (Section 3) given in terms of f . The recent preprint [2] covers the case d = 1 estimating the Lévy density v 0 of the integrator Lévy process {L s } of a moving average processThe estimate is based on the inversion of the Mellin transform of the second derivative of the cumulant of X(0). A uniform error bound as well as the consistency of the estimate are given. It is not assumed that f is simple, however, main results are subject to a number of quite restricting integrability assumptions onto x 2 v 0 (x) and f as well as mixing properties of {L s } that are tricky to check. Additionally, the logarithmic convergence rate shown there (cf. [2, Corollary 1]) is too slow.In our approach, we develop the ideas of [3] and use Banach fixed-point theorem combined with a recursive iteration procedure (Theorem 4.1) to give sufficient conditions for the existence of a (unique) solution of our (generally speaking, illposed) inverse problem v 1 → v 0 . We consider simple functions f since 1. in applications, f is mainly discr...

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The Infinitely Divisible Characteristic Function of Compound Poisson Distribution as the Sum of Variational Cauchy Distribution

Devianto

Sarah

Yozza

et al. 2019

J. Phys.: Conf. Ser.

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The new particular compound Poisson distribution is introduced as the sum of independent and identically random variables of variational Cauchy distribution with the number of random variables has Poisson distribution. This compound Poisson distribution is characterized by using characteristic function that is obtained by using Fourier-Stieltjes transform. The infinite divisibility of this characteristic function is constructed by introducing the specific function that satisfied the criteria of characteristic function. This characteristic function is employing the properties of continuity and quadratic form in term of real and nonnegative function such that its convolution has the characteristic function of compound Poisson distribution as the sum of variational Cauchy distribution.

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Nonparametric estimation for compound Poisson process via variational analysis on measures

Cited by 3 publications

References 28 publications

Decompounding discrete distributions: A nonparametric Bayesian approach

Decompounding discrete distributions: A nonparametric Bayesian approach

An inverse problem for infinitely divisible moving average random fields

The Infinitely Divisible Characteristic Function of Compound Poisson Distribution as the Sum of Variational Cauchy Distribution

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