ARTICLE
ABSTRACT
The article presents the exact bootstrap method, which can be used to estimate the parameters of the estimators of random variables with unknown distribution. The method allows to determine an estimate of any parameter, the error of estimation, the distribution of the estimator and confidence intervals. Traditionally this task is carried out using the bootstrap method, which consists of resampling of the original sample. Random sampling is necessary if examining the entire population data is impossible or too costly. Note that the fundamental sample property is of finite size and we know its distribution – it is the empirical distribution. Rather than driving a resample, we can generate automatically the entire resample space and calculate the values of a statistic which is looking for a parameter estimator. This article describes a method for performing exact algorithm for bootstrapping, which correctness was verified on an example of the unbiased estimator of variance. It is shown that the expected value of the estimator calculated with exact bootstrap is exactly equal the variance of the sample. The method does not introduce bias of the resampling, as it may be for the classic bootstrap. The distribution of the estimator determined by the exact bootstrap compared with the limit distribution for estimator of variance, which does not require assumptions of normality of the original sample. Research has shown that if we estimate variance we cannot ignore the issue of normality of the original sample.
REFERENCES
[1] Boot J.G., Sarkar S. [1998]: Monte Carlo Approximation of Bootstrap Variances. The American Statistician. Nr. 52, Assue 4. 354-357.
[2] Boot J.G.: Monte Carlo and the boothstrap. Prezentacja elektroniczna: http://www.stat.ufl.edu/~jbooth/documents/talks/bootse.pdf (2010.1.3).
[3] Domański C., Pruska K. [2000]: Nieklasyczne metody statystyczne. Polskie Wydawnictwo Ekonomiczne, Warszawa.
[4] Efron B. [1979]: Bootstrap methods: another look at the jackknife. The Annals of Statistics. Vol. 7, No. 1, 1-26.
[5] Efron B. [1987]: Better Bootstrap Confidence Intervals (with discussion). Journal of the American Statistical Association. Vol. 82, No. 397, 171-185.
[6] Efron B., Tibshirani R.J. [1993]: An introduction to the Bootstrap. Chapman & Hall. London.
[7] Feller W. [1950]: An introduction to probability theory and its application. John Wiley & Sons. New York, London, Sydney.
[8] Smirnow N.W., Dunin-Barkowski I.W. [1973]: Kurs rachunku prawdopodobieństwa i statystyki matematycznej dla zastosowań technicznych. Państwowe Wydawnictwo Naukowe, Warszawa.
[9] Zieliński R. [2004]: Siedem wykładów wprowadzających do statystyki matematycznej. Publikacja elektroniczna: http://www.impan.pl/~rziel/7ALL.pdf (2009.08.09)
