If a sufficiently large random sample is taken from a population with known distribution, depending upon a couple $\zeta$ of parameters, so that Pearson $\chi^2$ criterion is applicable to test the agreement between the observed and the expected sample class frequencies, and if the $\chi^2$ statistic is considered to be a random function defined on the space of all admisible $\zeta$ values, then the region in on which $\chi^2$ is less than its $100\alpha$ per cent critical value, constitutes an approximately $100(1-\alpha)$ per cent level confidence region for the true population value $\zeta_0$ of $\zeta$. Under certain general conditions this region always exists and lies within a closed curve the graphic construction of which is not very difficult if the expected sample class frequencies in a sufficiently large area in , surrounding the maximum likelihood or the $\chi^2$ minimum estimate of $\zeta_0$, are known.
If a sufficiently large random sample is taken from a population with known distribution, depending upon a couple $\zeta$ of parameters, so that Pearson $\chi^2$ criterion is applicable to test the agreement between the observed and the expected sample class frequencies, and if the $\chi^2$ statistic is considered to be a random function defined on the space of all admisible $\zeta$ values, then the region in on which $\chi^2$ is less than its $100\alpha$ per cent critical value, constitutes an approximately $100(1-\alpha)$ per cent level confidence region for the true population value $\zeta_0$ of $\zeta$. Under certain general conditions this region always exists and lies within a closed curve the graphic construction of which is not very difficult if the expected sample class frequencies in a sufficiently large area in , surrounding the maximum likelihood or the $\chi^2$ minimum estimate of $\zeta_0$, are known.
@article{10_21136_AM_1970_103301,
author = {Pavl{\'\i}k, Milo\v{s}},
title = {Approximate construction of a two-dimensional confidence region},
journal = {Applications of Mathematics},
pages = {305--309},
year = {1970},
volume = {15},
number = {5},
doi = {10.21136/AM.1970.103301},
mrnumber = {0266362},
zbl = {0212.50701},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1970.103301/}
}
TY - JOUR
AU - Pavlík, Miloš
TI - Approximate construction of a two-dimensional confidence region
JO - Applications of Mathematics
PY - 1970
SP - 305
EP - 309
VL - 15
IS - 5
UR - http://geodesic.mathdoc.fr/articles/10.21136/AM.1970.103301/
DO - 10.21136/AM.1970.103301
LA - en
ID - 10_21136_AM_1970_103301
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%0 Journal Article
%A Pavlík, Miloš
%T Approximate construction of a two-dimensional confidence region
%J Applications of Mathematics
%D 1970
%P 305-309
%V 15
%N 5
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.1970.103301/
%R 10.21136/AM.1970.103301
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%F 10_21136_AM_1970_103301
Pavlík, Miloš. Approximate construction of a two-dimensional confidence region. Applications of Mathematics, Tome 15 (1970) no. 5, pp. 305-309. doi: 10.21136/AM.1970.103301
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[3] Wilks S. S.: Shortest average confidence intervals for large samples. Ann. Math. Stat. 9, 166-175, 1938. | DOI
[4] Wilks S. S.: Optimum fiducial regions for simultaneous estimation of several population parameters from large samples. (Abstract.) Ann. Math. Stat. 10, 85-86, 1939.
[5] Wilks S. S. J. F. Daly: An optimum property of confidence regions associated with the likelihood function. Ann. Math. Stat. 10, 225 - 235, 1939. | DOI | MR
