In probability theory, the Bapat–Beg theorem[1] gives the joint cumulative distribution function of order statistics of independent but not necessarily identically distributed random variables in terms of the cumulative distribution functions of the random variables. A simple proof of this can be found in [2]
Ordinarily, all elements of the sample are obtained from the same population and thus have the same probability distribution. The Bapat–Beg theorem describes the order statistics when each element of the sample is obtained from a possibly different population with a different probability distribution.
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Let , be independent real valued random variables with cumulative distribution functions . Denote the order statistics by , with . Further let , and
for all For and , the joint cumulative distribution function of the subset of the order statistics satisfies
where
is the permanent of a block matrix with the subscript denoting the dimension of a block obtained by repeating the same entry, and the block row index and block column index .
In the case when the variables , are independent and identically distributed with cumulative probability distribution function for all , the Bapat–Beg theorem reduces to [3]
where again
The Bapat–Beg formula involves exponentially many permanents, and the complexity of the computation of the permanent itself is at least exponential (unless P = NP). Thus, in the general case, the formula is not practical. However, when the random variables have only two possible distributions, the complexity can be reduced to .[3] Thus, in the case of two populations, the complexity is polynomial in m for any fixed number of statistics k.