Shota Gugushvili

A kernel type nonparametric density estimator for decompounding

Abstract Given a sample from a discretely observed compound Poisson process, we consider estimation of the density of the jump sizes. We propose a kernel type nonparametric density estimator and study its asymptotic properties. An order bound for the bias and an asymptotic expansion of the variance of the estimator are given. Pointwise weak consistency and asymptotic normality are established. The results show that, asymptotically, the estimator behaves very much like an ordinary kernel estimator. Keywords: asymptotic normality; consistency; decompounding; kernel estimation Full-text: Access by subscription (subscriber: Univ Biblio SZ (UVA))

Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process

Given a discrete time sample X 1, … X n from a Lévy process X=(X t ) t≥0 of a finite jump activity, we study the problem of nonparametric estimation of the characteristic triplet (γ, σ2, ρ) corresponding to the process X. Based on Fourier inversion and kernel smoothing, we propose estimators of γ, σ2 and ρ and study their asymptotic behaviour. The obtained results include derivation of upper bounds on the mean square error of the estimators of γ and σ2 and an upper bound on the mean integrated square error of an estimator of ρ.

Nonparametric inference for discretely sampled Lévy processes

Given a sample from a discretely observed Levy process X = (Xt)t�0 of the finite jump activity, the problem of nonparametric estimation of the Levy density ρ corresponding to the process X is studied. An estimator of ρ is proposed that is based on a suitable inversion of the Levy-Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of ρ over suit- able classes of Levy triplets. The corresponding lower bounds are also discussed.

Deconvolution for an atomic distribution

Let X1, . . . ,Xn be i.i.d. observations, where Xi = Yi + σZi and Yi and Zi are independent. Assume that unobservable Ys are distributed as a random variable UV, where U and V are independent, U has a Bernoulli distribution with probability of zero equal to p and V has a distribution function F with density f. Furthermore, let the random variables Zi have the standard normal distribution and let σ > 0. Based on a sample X1, . . . ,Xn, we consider the problem of estimation of the density f and the probability p. We propose a kernel type deconvolution estimator for f and derive its asymptotic normality at a fixed point. A consistent estimator for p is given as well. Our results demonstrate that our estimator behaves very much like the kernel type deconvolution estimator in the classical deconvolution problem.

Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations

We consider nonparametric Bayesian estimation of the drift coefficient of a multidimensional stochastic differential equation from discrete-time observations on the solution of this equation. Under suitable regularity conditions, we establish posterior consistency in this context.

Deconvolution for an atomic distribution: Rates of convergence

Let X 1, …, X n be i.i.d. copies of a random variable X=Y+Z, where X i =Y i +Z i , and Y i and Z i are independent and have the same distribution as Y and Z, respectively. Assume that the random variables Y i ’s are unobservable and that Y=AV, where A and V are independent, A has a Bernoulli distribution with probability of success equal to 1−p and V has a distribution function F with density f. Let the random variable Z have a known distribution with density k. Based on a sample X 1, …, X n , we consider the problem of nonparametric estimation of the density f and the probability p. Our estimators of f and p are constructed via Fourier inversion and kernel smoothing. We derive their convergence rates over suitable functional classes. By establishing in a number of cases the lower bounds for estimation of f and p we show that our estimators are rate-optimal in these cases.

A non-parametric Bayesian approach to decompounding from high frequency data

Given a sample from a discretely observed compound Poisson process, we consider non-parametric estimation of the density

Asymptotic normality of the deconvolution kernel density estimator under the vanishing error variance

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