Estymacja frakcji

• Autorzy: Zieliński R.

Warianty tytułu

EN

Estimating proportion

• Języki publikacji: PL

Abstrakty

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.

Słowa kluczowe

PL

frakcja; prawdopodobieństwo sukcesu w doświadczeniu Bernoulliego; estymator nieobciążony; estymator o jednostajnie minimalnym błędzie średniokwadratowym; estymator bayesowski; losowanie warstwowe; randomizowane odpowiedzi; przedział ufności

EN

fraction; probability of success in Bernoulli scheme; unbiased estimator; uniformly minimum variance estimator; Bayesian estimator; stratified sampling; randomized response; confidence interval

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Matematyka Stosowana : matematyka dla społeczeństwa
• Rocznik: 2008
• Tom: Nr 9
• Strony: 76--90
• Opis fizyczny: bibliogr. 7 poz., tab., wykr.

Twórcy

autor

Zieliński R.

• Ryszard Zieliński Instytut Matematyczny PAN ul. Śniadeckich 8 00-956 Warszawa 1, Poland,

Bibliografia

• [1] Bartoszewicz, J. (1996): Wykłady ze statystyki matematycznej. Warszawa, PWN
• [2] Bracha, Cz. (1996): Teoretyczne podstawy metody reprezentacyjnej. Warszawa, WNT
• [3] DeGroot, M.H. ((1981): Optymalne decyzje statystyczne. Warszawa, PWN
• [4] Jakubowski J., Sztencel, R. (2001): Wstęp do teorii prawdopodobieństwa. Wyd. SCRIPT, Warszawa
• [5] Lehmann, E.L. (1968): Testowanie hipotez statystycznych, Warszawa, PW
• [6] Warner, S. (1965): Randomized response: a survey technique for eliminating evasive answer bias. JASA, March 1965, 63-69
• [7] Zasępa, R. (1972): Metoda reprezentacyjna. Warszawa, PWE