# Behrens–Fisher distribution

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In statistics, the **Behrens–Fisher distribution**, named after Ronald Fisher and W. V. 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. The Behrens–Fisher problem is that of statistical inference concerning the difference between the means of two normally distributed populations when the ratio of their variances is not known (and in particular, it is not known that their variances are equal).

## Definition

The Behrens–Fisher distribution is the distribution of a random variable of the form

where *T*_{1} and *T*_{2} are independent random variables each with a Student's t-distribution, with respective degrees of freedom *ν*_{1} = *n*_{1} − 1 and *ν*_{2} = *n*_{2} − 1, and *θ* is a constant. Thus the family of Behrens–Fisher distributions is parametrized by *ν*_{1}, *ν*_{2}, and *θ*.

## Derivation

Suppose it were known that the two population variances are equal, and samples of sizes *n*_{1} and *n*_{2} are taken from the two populations:

where "i.i.d" are independent and identically distributed random variables and *N* denotes the normal distribution. The two sample means are

The usual "pooled" unbiased estimate of the common variance *σ*^{2} is then

where *S*_{1}^{2} and *S*_{2}^{2} are the usual unbiased (Bessel-corrected) estimates of the two population variances.

Under these assumptions, the pivotal quantity

has a t-distribution with *n*_{1} + *n*_{2} − 2 degrees of freedom. Accordingly, one can find a confidence interval for *μ*_{2} − *μ*_{1} whose endpoints are

where *A* is an appropriate percentage point 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{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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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 cosine and 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{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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### Fiducial intervals versus confidence intervals

Bartlett{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## Further reading

- Kendall, Maurice G., Stuart, Alan (1973)
*The Advanced Theory of Statistics, Volume 2: Inference and Relationship, 3rd Edition*, Griffin. ISBN 0-85264-215-6 (Chapter 21)

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