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Having Polish spaces $𝕏$, $𝕐$ and $\mathbb{Z}$ we shall discuss the existence of an $𝕏\times 𝕐$-valued random vector $(\xi ,\eta )$ such that its conditional distributions $K_x=ℒ(\eta \mid \xi =x)$ satisfy $e(x,K_x)=c\left(x\right)$ or $e(x,K_x)\in C\left(x\right)$ for some maps $e:𝕏\times {ℳ}_1\left(𝕐\right)\to \mathbb{Z}$, $c:𝕏\to \mathbb{Z}$ or multifunction $C:𝕏\to 2^{\mathbb{Z}}$ respectively. The problem is equivalent to the existence of universally measurable Markov kernel $K:𝕏\to {ℳ}_1\left(𝕐\right)$ defined implicitly by $e(x,K_x)=c\left(x\right)$ or $e(x,K_x)\in C\left(x\right)$ respectively. In the paper we shall provide sufficient conditions for the existence of the desired Markov kernel. We shall discuss some special solutions of the $(e,c)$- or $(e,C)$-problem and illustrate...

The extremal shape factor of spheroidal particles is studied. Three dimensional particles are considered to be observed via their two dimensional profiles and the problem is to predict the extremal shape factor in a given size class. We proof the stability of the domain of attraction of the spheroid’s and its profile shape factor under a tail equivalence condition. We show namely that the Farlie–Gumbel–Morgenstern bivariate distributions gives the tail uniformity. We provide a way how to find normalising...

Microscopic prolate spheroids in a given volume of an opaque material are considered. The extremes of the shape factor of the spheroids are studied. The profiles of the spheroids are observed on a random planar section and based on these observations we want to estimate the distribution of the extremal shape factor of the spheroids. We show that under a tail uniformity condition the Maximum domain of attraction is stable. We discuss the normalising constants (n.c.) for the extremes of the spheroid...

The prediction of size extremes in Wicksell’s corpuscle problem with oblate spheroids is considered. Three-dimensional particles are represented by their planar sections (profiles) and the problem is to predict their extremal size under the assumption of a constant shape factor. The stability of the domain of attraction of the size extremes is proved under the tail equivalence condition. A simple procedure is proposed of evaluating the normalizing constants from the tail behaviour of appropriate...

The generalized FGM distribution and related copulas are used as bivariate models for the distribution of spheroidal characteristics. It is shown that this model is suitable for the study of extremes of the 3D spheroidal particles observed in terms of their random planar sections.

A properly measurable set $𝒫\subset X\times M_1\left(Y\right)$ (where $X,Y$ are Polish spaces and $M_1\left(Y\right)$ is the space of Borel probability measures on $Y$) is considered. Given a probability distribution $\lambda \in M_1\left(X\right)$ the paper treats the problem of the existence of $X\times Y$-valued random vector $(\xi ,\eta )$ for which $ℒ\left(\xi \right)=\lambda$ and $ℒ\left(\eta \right|\xi =x)\in {𝒫}_x$ $\lambda$-almost surely that possesses moreover some other properties such as “$ℒ(\xi ,\eta )$ has the maximal possible support” or “$ℒ\left(\eta \right|\xi =x)$’s are extremal measures in ${𝒫}_x$’s”. The paper continues the research started in [7].

This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale $N_t$ that solves the Engelbert–Schmidt stochastic differential equation (). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size $N_t$. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer...

We propose a new nonparametric procedure to solve the problem of classifying objects represented by $d$-dimensional vectors into $K\ge 2$ groups. The newly proposed classifier was inspired by the $k$ nearest neighbour (kNN) method. It is based on the idea of a depth-based distributional neighbourhood and is called $k$ nearest depth neighbours (kNDN) classifier. The kNDN classifier has several desirable properties: in contrast to the classical kNN, it can utilize global properties of the considered distributions...

Generalised halfspace depth function is proposed. Basic properties of this depth function including the strong consistency are studied. We show, on several examples that our depth function may be considered to be more appropriate for nonsymetric distributions or for mixtures of distributions.