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A statistical test for verifying the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e0355801.png" /> that an independent sample belongs to a given distribution. More explicitly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e0355802.png" /> be an independent sample taken from a continuous distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e0355803.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e0355804.png" /> are chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e0355805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e0355806.png" />. The test is constructed on the basis of the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e0355807.png" /> equal to the number of half-intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e0355808.png" /> in which there is not a single observation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e0355809.png" />. This test has the following form: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e03558010.png" />, then the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e03558011.png" /> is accepted; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e03558012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e03558013.png" /> is rejected. The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e03558014.png" /> is chosen from the condition that the error of the first kind, that is, the probability that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e03558015.png" /> is rejected while it is true, is equal to a given value. One may calculate the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e03558016.png" /> and estimate the power of the empty-boxes test for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e03558017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035580/e03558018.png" /> by using limit theorems for random distributions.
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A statistical test for verifying the hypothesis $H_0$ that an independent sample belongs to a given distribution. More explicitly, let $X_1,\ldots,X_n$ be an independent sample taken from a continuous distribution $F(x)$. The points $z_0=-\infty<z_1<\ldots<z_{N-1}<z_n=\infty$ are chosen so that $F(z_k)-F(z_{k-1})=1/N$, $k=1,\ldots,N$. The test is constructed on the basis of the statistic $\mu_0$ equal to the number of half-intervals $(z_{k-1},z_k]$ in which there is not a single observation $x_i$. This test has the following form: If $\mu_0\leq C$, then the hypothesis $H_0$ is accepted; if $\mu_0>C$, then $H_0$ is rejected. The constant $C$ is chosen from the condition that the error of the first kind, that is, the probability that $H_0$ is rejected while it is true, is equal to a given value. One may calculate the constant $C$ and estimate the power of the empty-boxes test for large $n$ and $N$ by using limit theorems for random distributions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. [V.F. Kolchin] Kolčin,  B.A. [B.A. Sevast'yanov] Sevast'janov,  V.P. [V.P. Chistyakov] Čistyakov,  "Random allocations" , Winston  (1978)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. [V.F. Kolchin] Kolčin,  B.A. [B.A. Sevast'yanov] Sevast'janov,  V.P. [V.P. Chistyakov] Čistyakov,  "Random allocations" , Winston  (1978)  (Translated from Russian)</TD></TR></table>

Revision as of 10:54, 10 August 2014

A statistical test for verifying the hypothesis $H_0$ that an independent sample belongs to a given distribution. More explicitly, let $X_1,\ldots,X_n$ be an independent sample taken from a continuous distribution $F(x)$. The points $z_0=-\infty<z_1<\ldots<z_{N-1}<z_n=\infty$ are chosen so that $F(z_k)-F(z_{k-1})=1/N$, $k=1,\ldots,N$. The test is constructed on the basis of the statistic $\mu_0$ equal to the number of half-intervals $(z_{k-1},z_k]$ in which there is not a single observation $x_i$. This test has the following form: If $\mu_0\leq C$, then the hypothesis $H_0$ is accepted; if $\mu_0>C$, then $H_0$ is rejected. The constant $C$ is chosen from the condition that the error of the first kind, that is, the probability that $H_0$ is rejected while it is true, is equal to a given value. One may calculate the constant $C$ and estimate the power of the empty-boxes test for large $n$ and $N$ by using limit theorems for random distributions.

References

[1] V.F. [V.F. Kolchin] Kolčin, B.A. [B.A. Sevast'yanov] Sevast'janov, V.P. [V.P. Chistyakov] Čistyakov, "Random allocations" , Winston (1978) (Translated from Russian)
How to Cite This Entry:
Empty-boxes test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Empty-boxes_test&oldid=15185
This article was adapted from an original article by B.A. Sevast'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article