Omega-squared distribution

$ \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.

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

[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
[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)