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The Behrens–Fisher distribution is the distribution of a random variable of the form
where T1 and T2 are independentrandom variables each with a Student's t-distribution, with respective degrees of freedom ν1 = n1 − 1 and ν2 = n2 − 1, and θ is a constant. Thus the family of Behrens–Fisher distributions is parametrized by ν1, ν2, and θ.
Derivation[]
Suppose it were known that the two population variances are equal, and samples of sizes n1 and n2 are taken from the two populations:
where A is an appropriate quantile of the t-distribution.
However, in the Behrens–Fisher problem, the two population variances are not known to be equal, nor is their ratio known. Fisher considered[citation needed] the pivotal quantity
This can be written as
where
are the usual one-sample t-statistics and
and one takes θ to be in the first quadrant. The algebraic details are as follows:
The fact that the sum of the squares of the expressions in parentheses above is 1 implies that they are the squared cosine and squared sine of some angle.
The Behren–Fisher distribution is actually the conditional distribution of the quantity (1) above, given the values of the quantities labeled cos θ and sin θ. In effect, Fisher conditions on ancillary information.
Fisher then found the "fiducial interval" whose endpoints are
where A is the appropriate percentage point of the Behrens–Fisher distribution. Fisher claimed[citation needed] that the probability that μ2 − μ1 is in this interval, given the data (ultimately the Xs) is the probability that a Behrens–Fisher-distributed random variable is between −A and A.
Fiducial intervals versus confidence intervals[]
Bartlett[citation needed] showed that this "fiducial interval" is not a confidence interval because it does not have a constant coverage rate. Fisher did not consider that a cogent objection to the use of the fiducial interval.[citation needed]
Further reading[]
Kendall, Maurice G., Stuart, Alan (1973) The Advanced Theory of Statistics, Volume 2: Inference and Relationship, 3rd Edition, Griffin. ISBN0-85264-215-6 (Chapter 21)