Omega-squared distribution
-distribution
The probability distribution of the random variable
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where is a conditional Wiener process (conditioned on
). The characteristic function of the "omega-squared" distribution is expressed by the formula
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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
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converges weakly to a conditional Wiener process, from which it follows that
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See also Cramér–von Mises test.
References
[1] | N.V. Smirnov, "On the ![]() |
[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" 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=17313