Geodesic
Limit Distributions of the $\chi^2$ Statistic of K.~Pearson in a Sequence of Independent Trials
Matematičeskie zametki, Tome 83 (2008) no. 6, pp. 899-911

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the $\chi^2$ statistic of K. Pearson in a sequence of independent and, generally, inhomogeneous trials with a fixed number of outcomes. It is assumed that the probabilities of occurrence of outcomes of the trials satisfy certain conditions. This problem statement embraces familiar results for the $\chi^2$ statistic in the case of multinomial trials. We obtain explicit expressions and estimates for the expectation and the variance of the $\chi^2$ statistic. For the $\chi^2$ statistic centered and normalized in a suitable way, we find limit distributions (the normal one, the distribution of the sum of the squares of normal random variables and, in particular, the $\chi^2$ distribution). Conditions for the convergence to the corresponding limit distributions are given.
Keywords: $\chi^2$ statistic of K. Pearson, $\chi^2$ distribution, goodness-of-fit test, multinomial trials, (in)homogenous trials, asymptotically normal random variable.
Mots-clés : normal distribution 
@article{MZM_2008_83_6_a8,
     author = {B. I. Selivanov},
     title = {Limit {Distributions} of the $\chi^2$ {Statistic} of {K.~Pearson} in a {Sequence} of {Independent} {Trials}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {899--911},
     publisher = {mathdoc},
     volume = {83},
     number = {6},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_83_6_a8/}
}
TY  - JOUR
AU  - B. I. Selivanov
TI  - Limit Distributions of the $\chi^2$ Statistic of K.~Pearson in a Sequence of Independent Trials
JO  - Matematičeskie zametki
PY  - 2008
SP  - 899
EP  - 911
VL  - 83
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2008_83_6_a8/
LA  - ru
ID  - MZM_2008_83_6_a8
ER  - 
%0 Journal Article
%A B. I. Selivanov
%T Limit Distributions of the $\chi^2$ Statistic of K.~Pearson in a Sequence of Independent Trials
%J Matematičeskie zametki
%D 2008
%P 899-911
%V 83
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2008_83_6_a8/
%G ru
%F MZM_2008_83_6_a8
B. I. Selivanov. Limit Distributions of the $\chi^2$ Statistic of K.~Pearson in a Sequence of Independent Trials. Matematičeskie zametki, Tome 83 (2008) no. 6, pp. 899-911. http://geodesic.mathdoc.fr/item/MZM_2008_83_6_a8/