In statistics, the Behrens–Fisher distribution, named after Ronald Fisher and Walter Behrens, is a parameterized family of probability distributions arising from the solution of the Behrens–Fisher problem proposed first by Behrens and several years later by Fisher.
[1] The Behrens–Fisher distribution is the distribution of a random variable of the form where T1 and T2 are independent random 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.
The two sample means are The usual "pooled" unbiased estimate of the common variance σ2 is then where S12 and S22 are the usual unbiased (Bessel-corrected) estimates of the two population variances.
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.
Fisher then found the "fiducial interval" whose endpoints are where A is the appropriate percentage point of the Behrens–Fisher distribution.