# Omega-squared distribution

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-distribution

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

#### References

 [1] N.V. Smirnov, "On the -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