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1

Wishart Laws and Variance Function on Homogeneous Cones

Graczyk Piotr, Ishi Hideyuki, Kołodziejek Bartosz

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 2

337--360

EN

We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.

2

Performance of variance function estimators for autoregressive time series of order one: asymptotic normality and numerical study

Borkowski P., Mielniczuk J.

Control and Cybernetics

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2012

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Vol. 41, no. 2

415-441

EN

We study performance of several conditional variance estimators for an autoregressive time series which include local linear smoothers with various bandwidths, local likelihood and difference-based estimators. In the theoretical part, asymptotic normality of the local linear estimator of variance with no mixing assumptions imposed on the underlying process is proved. Moreover, numerical examples performed reveal that a two-stage local linear smoother with a bandwidth, proposed by Ruppert, Sheather and Wand, used to estimate the regression function and a simple rule of thumb bandwidth for variance estimation performs best for variances without much structure, whereas the bandwidth considered by Fan and Yao works very well for much more variable variances.

3

Bivariate natural exponential families with linear diagonal variance functions

Chachulska J.

Probability and Mathematical Statistics

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2010

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Vol. 30, Fasc. 1

121--139

EN

It is well known that natural exponential families (NEFs) are uniquely determined by their variance functions (VFs). However, there exist examples showing that even an incomplete knowledge of a matrix VF can be sufficient to determine a multivariate NEF. Following such an idea, in this paper a complete description of bivariate NEFs with linear diagonal of the matrix VF is given. As a result we obtain the families of distributions with marginals that are some combinations of Poisson and normal distributions. Furthermore, the characterization extends (in two-dimensional case) the classification of NEFs with linear matrix VF obtained by Letac [11]. The main result is formulated in terms of regression properties.

4

Reliability of rigid piles subjected to lateral loads. a revised approach

Puła W., Różański A., Shahrour I.

Prace Naukowe Instytutu Geotechniki i Hydrotechniki Politechniki Wrocławskiej. Konferencje

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2005

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Vol. 75, nr 41

463-472

EN

In this paper the reliability of laterally loaded rigid piles in case of piles capacity is considered. A revised approach utilising the response surface method is proposed and compared with the solution presented in earlier papers. In both cases the problems were limited to non-cohesive soils only. However the expansion of the new approach on cohesive soils is going to be a near future studies. Additionally the influence of spatial averaging is also analysed in this paper.

PL

W pracy przedstawiono nową metodę oceny miar niezawodności dla przypadku nośności bocznej sztywnego pala obciążonego siłami poprzecznymi. Nowe podejście oparte na metodzie powierzchni odpowiedzi zestawiono z wcześniejszym stosującym obliczenia symboliczne w programie MATHEMATICA. Obie metody dały prawie identyczne rezultaty. Nowa metoda jest jednak prostsza pod względem obliczeń numerycznych, a ponadto daje możliwość uogólnienia na przypadek gruntów spoistych. Innym nowym elementem jest analiza niezawodności z uwzględnieniem uśrednień przestrzennych.