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1

Optimality of the auxiliary particle filter

Douc R., Moulines É, Olsson J.

Probability and Mathematical Statistics

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2009

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Vol. 29, Fasc. 1

1--28

EN

In this article we study asymptotic properties of weighted samples produced by the auxiliary particle filter (APF) proposed by Pitt and Shephard [17]. Besides establishing a central limit theorem (CLT) for smoothed particle estimates, we also derive bounds on the Lp error and bias of the same for a finite particle sample size. By examining the recursive formula for the asymptotic variance of the CLT we identify first-stage importance weights for which the increase of asymptotic variance at a single iteration of the algorithm is minimal. In the light of these findings, we discuss and demonstrate on several examples how the APF algorithm can be improved.

2

Estymacja frakcji

Zieliński R.

Matematyka Stosowana : matematyka dla społeczeństwa

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2008

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Nr 9

76-90

PL

W populacji składającej się z N elementów jest nieznana liczba M elementów wyróżnionych. W artykule w przystępny sposób prezentuję różne problemy związane z estymacją frakcji θ= M/N.

EN

A population of N elements contains an unknown number M of marked units. Problems of estimating the fraction θ = M/N are discussed. The well known standard solution is θ= K/n which is the uniformly minimum variance unbiased estimator, maximum likelihood estimator, estimator obtained by the method of moments, and in consequence it shares all advantages of such estimators. In the paper some versions of the estimator are considered which are more adequate in real situations. If we know in advance that the unknown fraction lies in a given interval (t1, t2) and we consider an estimator θ1 as better than the estimator θ2 if the average of its mean square error is smaller on that interval, then the optimal estimator is given by (3). The values of the estimator for (t1, t2) = (0, 0.5) and for (t1, t2) = (0.3, 0.4) in a sample of size n = 10 if the number of marked units in the sample equals K, are given in the table TABELKA and the mean square errors of these estimator, versus the error of the standard estimator θ = K/n are presented in Rys. 2. Averaging the mean square error with a weight function, for example such as in Rys.3, gives us the Bayesian estimator with the mean square error like in Rys. 4 (for n = 10). If in some real situations we are interested in minimizing the mean square error "in the worst possible case", the adequate is the minimax estimator. Another situation appears if the population can be divided in some more homogenous subpopulations, for example in two subpopulations with fractions of marked units close to zero or close to one in each of them. Then stratified sampling is more effective; then the mean square error of estimation may be significantly reduced. In the paper the problem of randomized responses is also presented, very shortly and elementarily. The problem arises if a unit in the sample can not be for sure recognized as "marked"or "not marked" and that can be done with some probability only. The situation is typical for survey interview: it allows respondents to respond to sensitive issues (such as criminal behavior or sexuality) while remaining confidential. The final section of the paper is devoted to some remarks concerning the confidence intervals for the fraction. The exact optimal solution is well known for mathematicians but it is probably not very easy for statistical practitioners to follow all theoretical details, and typically confidence interval based on asymptotic approximation of the binomial distribution by a normal distribution are used. That is neither sufficiently exact nor correct. The proper and exact solution is given by quantiles of a suitable Beta distribution which are easily computable in typical statistical and mathematical computer packages.

3

Size Effects in Problems of Footings on Sand within Micro-Polar Hypoplasticity

Tejchman J., Górski J.

Archives of Hydro-Engineering and Environmental Mechanics

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2008

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Vol. 55, nr 3-4

95-124

EN

Numerical FE investigations of size effects in problems of footings on sand were performed. Micro-polar hypoplastic constitutive model was used to describe a mechanical behaviour of a cohesionless granular material during a monotonic deformation path. The FE analyses were carried out with three different footing widths. In deterministic calculations, a uniform distribution of initial void ratio was used. In statistical calculations, initial void ratios took the form of random spatial fields described by a truncated Gaussian random distribution. In order to reduce the number of stochastic realizations without sacrificing the accuracy of the calculations, a stratified sampling method was applied. The numerical results were compared with corresponding laboratory tests by Tatsuoka et al (1997). The numerical results show that the bearing capacity of footings decreases with increasing specimen size. If the initial void ratio is stochastically distributed, the mean bearing capacity of footings may be larger than the deterministic value. The statistical size effect is smaller than the deterministic one.