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Globální obálkové testy aneb jak otestovat vhodnost statistického modelu na základě funkcionální charakteristiky

Tomáš Mrkvička — 2017

Pokroky matematiky, fyziky a astronomie

Obálkové metody představují populární nástroj pro testování hypotéz o vhodnosti statistického modelu. Tyto testy graficky porovnávají funkci T : I vypočtenou ze statistických dat s jejím protějškem získaným simulacemi. Chyba prvního druhu α , tj. pravděpodobnost zamítnutí platné hypotézy, je obvykle kontrolována pouze pro fixní hodnotu r I , zatímco funkce T je definována na intervalu hodnot I . V tomto článku představíme nový globální obálkový test, který umožňuje kontrolovat chybu prvního druhu současně...

On testing of general random closed set model hypothesis

Tomáš Mrkvička — 2009

Kybernetika

A new method of testing the random closed set model hypothesis (for example: the Boolean model hypothesis) for a stationary random closed set Ξ d with values in the extended convex ring is introduced. The method is based on the summary statistics – normalized intrinsic volumes densities of the ε -parallel sets to Ξ . The estimated summary statistics are compared with theirs envelopes produced from simulations of the model given by the tested hypothesis. The p-level of the test is then computed via approximation...

Estimation of intersection intensity in a Poisson process of segments

Tomáš Mrkvička — 2007

Commentationes Mathematicae Universitatis Carolinae

The minimum variance unbiased estimator of the intensity of intersections is found for stationary Poisson process of segments with parameterized distribution of primary grain with known and unknown parameters. The minimum variance unbiased estimators are compared with commonly used estimators.

Estimation variances for parameterized marked Poisson processes and for parameterized Poisson segment processes

Tomáš Mrkvička — 2004

Commentationes Mathematicae Universitatis Carolinae

A complete and sufficient statistic is found for stationary marked Poisson processes with a parametric distribution of marks. Then this statistic is used to derive the uniformly best unbiased estimator for the length density of a Poisson or Cox segment process with a parametric primary grain distribution. It is the number of segments with reference point within the sampling window divided by the window volume and multiplied by the uniformly best unbiased estimator of the mean segment length.

On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane

Tomáš MrkvičkaJan Rataj — 2009

Kybernetika

A method of estimation of intrinsic volume densities for stationary random closed sets in 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...

On the Bayesian estimation for the stationary Neyman-Scott point processes

Jiří KopeckýTomáš Mrkvička — 2016

Applications of Mathematics

The pure and modified Bayesian methods are applied to the estimation of parameters of the Neyman-Scott point process. Their performance is compared to the fast, simulation-free methods via extensive simulation study. Our modified Bayesian method is found to be on average 2.8 times more accurate than the fast methods in the relative mean square errors of the point estimates, where the average is taken over all studied cases. The pure Bayesian method is found to be approximately as good as the fast...

Spatial prediction of the mark of a location-dependent marked point process: How the use of a parametric model may improve prediction

We discuss the prediction of a spatial variable of a multivariate mark composed of both dependent and explanatory variables. The marks are location-dependent and they are attached to a point process. We assume that the marks are assigned independently, conditionally on an unknown underlying parametric field. We compare (i) the classical non-parametric Nadaraya-Watson kernel estimator based on the dependent variable (ii) estimators obtained under an assumption of local parametric model where explanatory...

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