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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1971, Volume 16, Issue 1, Pages 3–20 (Mi tvp1950)

Some tests of chi-square type for continuous distributions

D. M. Čibisov

Moscow

Abstract: In testing the hypothesis that a sample $X_1,…,X_n$ is drawn from a d.f. $F(x,\theta)$ where $\theta\in R^s$ is an unspecified parameter, the following three test statistics are considered. 1. The $\chi^2$-statistic $X^2(\widehat\theta)$ with class boundaries fixed in advance and class probabilities $p_i(\widehat\theta)$ determined by an estimate $\widehat\theta$ (cf. [2]). 2. The $\chi^2$-statistic $X^2(\theta^*,\widehat\theta)$ with class boundaries $(a^*_{i-1},a^*_i)$ determined by $F(a^*_i,\theta^*)-F(a^*_{i-1},\theta^*)=p_i$, $p_1,…,p_k$ being prescribed probabilities and $\theta^*$ an estimate of $\theta$ (cf. [4]). 3. $Z^2(\widehat\theta)=n\sum p_i^{-1}[p_i-(F(Y_i,\widehat\theta)-F(Y_{i-1},\widehat\theta))]^2$, $Y_i$ being the sample $(p_1+…+p_i)$-quantile. It is proved, under certain regularity conditions, that $X^2(\theta^*,\widehat\theta)-X^2(\widehat\theta)\to0$ and $Z^2(\widehat\theta)-X^2(\widehat\theta)\to0$ provided $\theta^*$ is a consistent and $\widehat\theta$ a root $n$ consistent estimate and $p_i(\theta_0)=p_i$, $\theta_0$ being the true value of $\theta$. Therefore asymptotic results on $X^2(\widehat\theta)$ hold true for $X^2(\theta^*,\widehat\theta)$ and $Z^2(\widehat\theta)$. It is shown that the minimization of any of the three statistics gives estimates equivalent to the multinomial ML estimate, and that the use of the ML estimate based on the whole sample can decrease as well as increase the power.

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English version:
Theory of Probability and its Applications, 1971, 16:1, 1–22

Bibliographic databases:

Citation: D. M. Čibisov, “Some tests of chi-square type for continuous distributions”, Teor. Veroyatnost. i Primenen., 16:1 (1971), 3–20; Theory Probab. Appl., 16:1 (1971), 1–22

Citation in format AMSBIB
\Bibitem{Chi71} \by D.~M.~{\v C}ibisov \paper Some tests of chi-square type for continuous distributions \jour Teor. Veroyatnost. i Primenen. \yr 1971 \vol 16 \issue 1 \pages 3--20 \mathnet{http://mi.mathnet.ru/tvp1950} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=283914} \zmath{https://zbmath.org/?q=an:0234.62019} \transl \jour Theory Probab. Appl. \yr 1971 \vol 16 \issue 1 \pages 1--22 \crossref{https://doi.org/10.1137/1116001} 

• http://mi.mathnet.ru/eng/tvp1950
• http://mi.mathnet.ru/eng/tvp/v16/i1/p3

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. È. V. Khmaladze, “Some applications of the theory of martingales to statistics”, Russian Math. Surveys, 37:6 (1982), 215–237
2. J. Appl. Industr. Math., 3:4 (2009), 462–475