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Koncepcja odporności według Ryszarda Zielińskiego. Odporność w modelach parametrycznych

Boratyńska A.

Mathematica Applicanda

2012

Vol. 40, No. 2

83–-92

The concept of robustness of statistical procedures is one of the most important subject in Zielinski's papers. In this article the development of the idea of robustness as introduced in Zielinski's papers is presented. The definitions of a supermodel and a robustness function are given. The problem of the robust estimation of a scale parameter in an exponential model and the robustness of tests for comparison of means in two or more populations are described. Robustness in Bayesian statistical models is connected with an unexactly specified prior distribution. Here the following Zielinski's results in Bayesian robustness are presented: the most stable estimator in the Poisson model and the Bayes optimal stopping rule in a homogeneous Poisson process with conjugate classes of priors, the optimal experimental designs in Bayesian linear models under variation in the prior, an upper bound for the Kolmogorov distance between the posterior distributions in terms of that between the prior distributions.

Estymacja frakcji

Zieliński R.

Matematyka Stosowana : matematyka dla społeczeństwa

2008

Nr 9

76-90

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.

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.