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

We consider the nonparametric regression estimation problem of recovering an unknown response function f on the basis of spatially inhomogeneous data when the design points follow a known density g with a finite number of well-separated zeros. In particular, we consider two different cases: when g has zeros of a polynomial order and when g has zeros of an exponential order. These two cases correspond to moderate and severe data losses, respectively. We obtain asymptotic (as the sample size increases)...

This work deals with a multivariate random coefficient autoregressive model (RCA) of the first order. A class of modified least-squares estimators of the parameters of the model, originally proposed by Schick for univariate first-order RCA models, is studied under more general conditions. Asymptotic behavior of such estimators is explored, and a lower bound for the asymptotic variance matrix of the estimator of the mean of random coefficient is established. Finite sample properties are demonstrated...

A kernel estimator of the squared $L_2$-norm of the intensity function of a Poisson random field is defined. It is proved that the estimator is asymptotically unbiased and strongly consistent. The problem of estimating the squared $L_2$-norm of a function disturbed by a Wiener random field is also considered.

In the problem of signal detection in gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal $L_2$-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that the $L_2$-norms of signal smoothed by the kernels exceed some constants ${\rho }_{ϵ}\>0$. The constant ${\rho }_{ϵ}$ depends on the power $ϵ$ of noise and ${\rho }_{ϵ}\to 0$ as $ϵ\to 0$. Similar statements are proved also if an additional information on a signal smoothness is given....

In the problem of signal detection in Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L2-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that the L2-norms of signal smoothed by the kernels exceed some constants pε > 0. The constant pε depends on the power ϵ of noise and pε → 0 as ε → 0. Similar statements are proved also if an additional information on a signal...

We study the impact of certain transformations within the class of Archimedean copulas. We give some admissibility conditions for these transformations, and define some equivalence classes for both transformations and generators of Archimedean copulas. We extend the r-fold composition of the diagonal section of a copula, from r ∈ N to r ∈ R. This extension, coupled with results on equivalence classes, gives us new expressions of transformations and generators. Estimators deriving directly from these...

A method of estimation of intrinsic volume densities for stationary random closed sets in ${ℝ}^d$ based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the...

We present a review of several results concerning invariant density estimation by observations of ergodic diffusion process and some related problems. In every problem we propose a lower minimax bound on the risks of all estimators and then we construct an asymptotically efficient estimator.

The problem of nonparametric estimation of a bounded regression function $f\in L²\left({[a,b]}^d\right)$, [a,b] ⊂ ℝ, d ≥ 1, using an orthonormal system of functions $e_k$, k=1,2,..., is considered in the case when the observations follow the model $Y_i=f\left(X_i\right)+{\eta }_i$, i=1,...,n, where $X_i$ and ${\eta }_i$ are i.i.d. copies of independent random variables X and η, respectively, the distribution of X has density ϱ, and η has mean zero and finite variance. The estimators are constructed by proper truncation of the function $f̂ₙ\left(x\right)={\sum }_{k=1}^{N\left(n\right)}c{̂}_k{e}_k\left(x\right)$, where the coefficients $ĉ₁,...,c{̂}_{N\left(n\right)}$ are determined...