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.

The abstract is available at MR0518661.

A birth and death process with parameters θ=(λ,μ), λ>0, μ>0, is considered. The absolute continuity of measures generated by this process is proved. The Rao-Cramér inequality for the variance of the unbiased estimator of a function h(θ) is derived. Some properties of the estimator attaining the Rao-Cramér lower bound are asserted.

A histogram sieve estimator of the drift function in Ito processes and some semimartingales is constructed. It is proved that the estimator is pointwise and L¹ consistent and its finite-dimensional distributions are asymptotically normal. Our approach extends the results of Leśkow and Różański (1989a).

In the article, we propose a new estimator of the hazard rate function in the framework of the multiplicative point process intensity model. The technique combines the reflection method and the method of transformation. The new method eliminates the boundary effect for suitably selected transformations reducing the bias at the boundary and keeping the asymptotics of the variance. The transformation depends on a pre-estimate of the logarithmic derivative of the hazard function at the boundary.

In the article, we consider construction of prediction intervals for stationary time series using Bühlmann's [8], [9] sieve bootstrapapproach. Basic theoretical properties concerning consistency are proved. We extend the results obtained earlier by Stine [21], Masarotto and Grigoletto [13] for an autoregressive time series of finite order to the rich class of linear and invertible stationary models. Finite sample performance of the constructed intervals is investigated by computer simulations.

Download Results (CSV)