A formula for finding the approximate distribution of a linear combination of independent chi-squared variables. Define the random variable S2 by
where c1, c2,… are known positive constants, and the chi-squared random variable Sj2 has νj degrees of freedom. Satterthwaite’s suggestion, made in 1946, was that the distribution of νS2/σ2 is approximately chi-squared with ν degrees of freedom, where , and ν is the nearest integer to
where sj2 is the observed value of Sj2.
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