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 constants for the shape factor extremes. The theory is illustrated on examples of distributions belonging to Gumbel and Fréchet domain of attraction. We discuss the ML estimator based on the largest observations and hence the possible statistical applications at the end.
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 Nt that solves the Engelbert–Schmidt stochastic differential equation (2). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coeffients depend on the size Nt. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer simulations performed.
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 and profile shape factor. Comparing the tail behaviour of the distribution of the profile and spheroid shape factor we show the relation between the n.c. of the profile shape factor (which can be estimated) and the n.c. of the spheroid shape factor (cannot be estimated directly) which are needed for the prediction of the tail behaviour of the shape factor. The paper completes the study \cite{hlubinka:06} for prolate spheroids.
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 distribution functions and its results are employed for the estimation of the spheroid size. Examples covering families of Gamma, Pareto and Weibull distributions are provided. A short discussion of maximum likelihood estimators of the normalizing constants is also included.
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.