Difference between revisions of "Omega-squared distribution"
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| − | \lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \int\limits _ { 0 } ^ { 1 } Z _ {n} ^ {2} ( t) dt | + | \lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \int\limits _ { 0 } ^ { 1 } Z _ {n} ^ {2} ( t) dt < \lambda \right \} = \ |
| − | {\mathsf P} \{ \omega ^ {2} | + | {\mathsf P} \{ \omega ^ {2} < \lambda \} = |
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\frac{e ^ {- t ^ {2} \lambda / 2 } | \frac{e ^ {- t ^ {2} \lambda / 2 } | ||
}{\sqrt {- t \sin t } } | }{\sqrt {- t \sin t } } | ||
| − | dt,\ \lambda | + | dt,\ \lambda > 0 . |
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====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[1]</td> <td valign="top"> N.V. Smirnov, "On the $\omega ^ { 2 }$-distribution" ''Mat. Sb.'' , '''2''' (1937) pp. 973–993 (In Russian)</td></tr><tr><td valign="top">[2]</td> <td valign="top"> T.W. Anderson, D.A. Darling, "Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes" ''Ann. Math. Stat.'' , '''23''' (1952) pp. 193–212</td></tr></table> |
====Comments==== | ====Comments==== | ||
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====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> D.A. Darling, "The Cramér–Smirnov test in the parametric case" ''Ann. Math. Stat.'' , '''26''' (1955) pp. 1–20</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> J. Durbin, "Distribution theory for tests based on the sample distribution function" , SIAM (1973)</td></tr></table> |
Revision as of 16:57, 1 July 2020
$ \omega ^ {2} $-
distribution
The probability distribution of the random variable
$$ \omega ^ {2} = \int\limits _ { 0 } ^ { 1 } Z ^ {2} ( t) dt, $$
where $ Z( t) $ is a conditional Wiener process (conditioned on $ Z ( 1) = 0 $). The characteristic function of the "omega-squared" distribution is expressed by the formula
$$ {\mathsf E} e ^ {it \omega ^ {2} } = \prod _ { k= } 1 ^ \infty \left ( 1 - \frac{2it }{\pi ^ {2} k ^ {2} } \right ) ^ {-} 1/2 . $$
In mathematical statistics, the "omega-squared" distribution is often found in the following circumstances. Let $ X _ {1} \dots X _ {n} $ be independent random variables, uniformly distributed on $ [ 0, 1] $, according to which an empirical distribution function $ F _ {n} ( \cdot ) $ is constructed. In this case, the process
$$ Z _ {n} ( t) = \sqrt n ( F _ {n} ( t) - t) $$
converges weakly to a conditional Wiener process, from which it follows that
$$ \lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \int\limits _ { 0 } ^ { 1 } Z _ {n} ^ {2} ( t) dt < \lambda \right \} = \ {\mathsf P} \{ \omega ^ {2} < \lambda \} = $$
$$ = \ 1 - \frac{2} \pi \sum _ { k= } 1 ^ \infty (- 1) ^ {k-} 1 \int\limits _ {( 2k- 1) \pi } ^ { {2k } \pi } \frac{e ^ {- t ^ {2} \lambda / 2 } }{\sqrt {- t \sin t } } dt,\ \lambda > 0 . $$
See also Cramér–von Mises test.
References
| [1] | N.V. Smirnov, "On the $\omega ^ { 2 }$-distribution" Mat. Sb. , 2 (1937) pp. 973–993 (In Russian) |
| [2] | T.W. Anderson, D.A. Darling, "Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes" Ann. Math. Stat. , 23 (1952) pp. 193–212 |
Comments
The "conditional Wiener process" $ Z $ is usually referred to in the Western literature as tied-down Brownian motion, pinned Brownian motion or as the Brownian bridge.
The pioneering paper is [a1].
References
| [a1] | D.A. Darling, "The Cramér–Smirnov test in the parametric case" Ann. Math. Stat. , 26 (1955) pp. 1–20 |
| [a2] | J. Durbin, "Distribution theory for tests based on the sample distribution function" , SIAM (1973) |
Omega-squared distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Omega-squared_distribution&oldid=48039