In this paper we consider the problem of estimating the intensity of a spatial homogeneous Poisson process if a part of the observations (quadrat counts) is censored. The actual problem has occurred during a court case when one of the authors was a referee for the defense.

In this paper we derive conditions upon the nonnegative random variable under which the inequality $Dg\left(\xi \right)\le cE{\left[g^{\text{'}}\left(\xi \right)\right]}^2\xi$ holds for a fixed nonnegative constant $c$ and for any absolutely continuous function $g$. Taking into account the characterization of a Gamma distribution we consider the functional $U_{\xi }={sup}_g\frac{Dg\left(\xi \right)}{E{\left[g^{\text{'}}\left(\xi \right)\right]}^2\xi }$ and establishing some of its properties we show that $U_{\xi }\ge 1$ and that $U_{\xi }=1$ iff the random variable $\xi$ has a Gamma distribution.

Assuming that $C_{X,Y}$ is the copula function of $X$ and $Y$ with marginal distribution functions $F_X\left(x\right)$ and $F_Y\left(y\right)$, in this work we study the selection distribution $Z\stackrel{\mathrm{d}}{=}\left(X\right|Y\in T)$. We present some special cases of our proposed distribution, among them, skew-normal distribution as well as normal distribution. Some properties such as moments and moment generating function are investigated. Also, some numerical analysis is presented for illustration.

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...

The maximum likelihood scale invariant estimator of the shape parameter of the gamma distribution, proposed by the authors [Statist. Probab. Lett. 78 (2008)], is considered. The asymptotics of the mean square error of this estimator, with respect to that of the usual maximum likelihood estimator, is established.

An explicit formula for the correlation coefficient in a two-dimensional AR(1) process is derived. Approximate critical values for the correlation coefficient between two one-dimensional AR(1) processes are tabulated. They are based on Bartlett’s approximation and on an asymptotic distribution derived by McGregor. The results are compared with critical values obtained from a simulation study.

The paper gives some basic ideas of both the construction and investigation of the properties of the Bayesian estimates of certain parametric functions of the parent exponential distribution under the model of random censorship assuming the Koziol–Green model. Various prior distributions are investigated and the corresponding estimates are derived. The stress is put on the asymptotic properties of the estimates with the particular stress on the Bayesian risk. Small sample properties are studied...

This paper gives a generalization of results presented by ten Berge, Krijnen,Wansbeek & Shapiro. They examined procedures and results as proposed by Anderson & Rubin, McDonald, Green and Krijnen, Wansbeek & ten Berge.We shall consider the same matter, under weaker rank assumptions. We allow some moments, namely the variance Ω of the observable scores vector and that of the unique factors, Ψ, to be singular. We require T' Ψ T > 0, where T Λ T' is a Schur decomposition of Ω. As...