The probability distribution of the random variable
where is a conditional Wiener process (conditioned on ). The characteristic function of the "omega-squared" distribution is expressed by the formula
In mathematical statistics, the "omega-squared" distribution is often found in the following circumstances. Let be independent random variables, uniformly distributed on , according to which an empirical distribution function is constructed. In this case, the process
converges weakly to a conditional Wiener process, from which it follows that
See also Cramér–von Mises test.
|||N.V. Smirnov, "On the -distribution" Mat. Sb. , 2 (1937) pp. 973–993 (In Russian)|
|||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|
The "conditional Wiener process" 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].
|[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. M.S. Nikulin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Omega-squared_distribution&oldid=17313