Editing Behrens–Fisher distribution

{{Short description|Probability distribution}}
{{more citations needed|date=July 2022}}

In [[statistics]], the '''Behrens–Fisher distribution''', named after [[Ronald Fisher]] and [[Walter Behrens (statistician)|Walter Behrens]], is a [[parameter]]ized family of [[probability distribution]]s 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 [[normal distribution|normally distributed]] [[Statistical population|populations]] when the [[ratio]] of their [[variance]]s is not known (and in particular, it is not known that their variances are equal).<ref>{{Cite journal |last1=Kim |first1=Seock-Ho |last2=Cohen |first2=Allan S. |date=December 1998 |title=On the Behrens-Fisher Problem: A Review |url=http://journals.sagepub.com/doi/10.3102/10769986023004356 |journal=Journal of Educational and Behavioral Statistics |language=en |volume=23 |issue=4 |pages=356–377 |doi=10.3102/10769986023004356 |s2cid=85462934 |issn=1076-9986}}</ref>

== Definition ==

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

: <math> T_2 \cos\theta - T_1\sin\theta \, </math>

where ''T''<sub>1</sub> and ''T''<sub>2</sub> are [[independence (probability theory)|independent]] [[random variable]]s each with a Student's [[t-distribution]], with respective degrees of freedom ''&nu;''<sub>1</sub>&nbsp;=&nbsp;''n''<sub>1</sub>&nbsp;&minus;&nbsp;1 and ''&nu;''<sub>2</sub>&nbsp;=&nbsp;''n''<sub>2</sub>&nbsp;&minus;&nbsp;1, and ''&theta;'' is a constant.  Thus the family of Behrens–Fisher distributions is parametrized by ''&nu;''<sub>1</sub>,&nbsp;''&nu;''<sub>2</sub>,&nbsp;and&nbsp;''&theta;''.

== Derivation ==

Suppose it were known that the two population variances are equal, and samples of sizes ''n''<sub>1</sub> and ''n''<sub>2</sub> are taken from the two populations:

: <math>
\begin{align}
X_{1,1},\ldots,X_{1,n_1} & \sim \operatorname{i.i.d.} N(\mu_1,\sigma^2), \\[6pt]
X_{2,1},\ldots,X_{2,n_2} & \sim \operatorname{i.i.d.} N(\mu_2,\sigma^2).
\end{align}
</math>

where "i.i.d" are [[independent and identically distributed random variables]] and ''N'' denotes the [[normal distribution]]. The two sample [[Arithmetic mean|mean]]s are

: <math>
\begin{align}
\bar{X}_1 & = (X_{1,1}+\cdots+X_{1,n_1})/n_1 \\[6pt]
\bar{X}_2 & = (X_{2,1}+\cdots+X_{2,n_2})/n_2
\end{align}
</math>

The usual "[[pooled variance|pooled]]" [[Sampling bias|unbiased]] estimate of the common variance ''&sigma;''<sup>2</sup> is then

: <math> S_\mathrm{pooled}^2 = \frac{\sum_{k=1}^{n_1}(X_{1,k}-\bar X_1)^2 + \sum_{k=1}^{n_2}(X_{2,k}-\bar X_2)^2}{n_1+n_2-2} = \frac{(n_1-1)S_1^2 + (n_2-1)S_2^2}{n_1+n_2-2} </math>

where ''S''<sub>1</sub><sup>2</sup> and ''S''<sub>2</sub><sup>2</sup> are the usual unbiased ([[Bessel's correction|Bessel-corrected]]) estimates of the two population variances.

Under these assumptions, the [[pivotal quantity]]

: <math> \frac{(\mu_2-\mu_1)-(\bar X_2 - \bar X_1)}{\displaystyle\sqrt{\frac{S^2_\mathrm{pooled}}{n_1} + \frac{S^2_\mathrm{pooled}}{n_2} }} </math>

has a [[Student's t-distribution|t-distribution]] with ''n''<sub>1</sub>&nbsp;+&nbsp;''n''<sub>2</sub>&nbsp;&minus;&nbsp;2 [[Degrees of freedom (statistics)|degrees of freedom]].  Accordingly, one can find a [[confidence interval]] for ''&mu;''<sub>2</sub>&nbsp;&minus;&nbsp;''&mu;''<sub>1</sub> whose endpoints are

: <math> \bar{X}_2 - \bar{X_1} \pm A \cdot S_\mathrm{pooled} \sqrt{\frac{1}{n_1} +\frac{1}{n_2}}, </math>

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|date=August 2012}} the pivotal quantity

: <math> \frac{(\mu_2-\mu_1)-(\bar X_2 - \bar X_1)}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }}. </math>

This can be written as

: <math> T_2\cos\theta - T_1\sin\theta, \, </math>

where

: <math> T_i = \frac{\mu_i - \bar{X}_i}{S_i/\sqrt{n_i}}\text{ for }i=1,2 \, </math>

are the usual one-sample t-statistics and

: <math> \tan\theta = \frac{S_1/\sqrt{n_1}}{S_2/\sqrt{n_2}} </math>

and one takes ''&theta;'' to be in the first quadrant.  The algebraic details are as follows:

: <math>
\begin{align}
\frac{(\mu_2-\mu_1)-(\bar X_2 - \bar X_1)}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} & = \frac{\mu_2-\bar{X}_2}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} - \frac{\mu_1-\bar{X}_1}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} \\[10pt]
& = \underbrace{\frac{\mu_2-\bar{X}_2}{S_2/\sqrt{n_2}}}_{\text{This is }T_2} \cdot \underbrace{\left( \frac{S_2/\sqrt{n_2}}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} \right)}_{\text{This is }\cos\theta} - \underbrace{\frac{\mu_1-\bar{X}_1}{S_1/\sqrt{n_1}}}_{\text{This is }T_1}\cdot\underbrace{\left( \frac{S_1/\sqrt{n_1}}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} \right)}_{\text{This is }\sin\theta}.\qquad\qquad\qquad (1)
\end{align}
</math>

The fact that the sum of the squares of the expressions in parentheses above is&nbsp;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&nbsp;''&theta;'' and sin&nbsp;''&theta;''.  In effect, Fisher [[conditioning on an ancillary statistic|conditions on ancillary information]].

Fisher then found the "[[fiducial inference|fiducial]] interval" whose endpoints are

: <math> \bar{X}_2-\bar{X}_1 \pm A \sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} } </math>

where ''A'' is the appropriate percentage point of the Behrens–Fisher distribution.  Fisher claimed{{citation needed|date=August 2012}} that the probability that ''&mu;''<sub>2</sub>&nbsp;&minus;&nbsp;''&mu;''<sub>1</sub> is in this interval, given the data (ultimately the ''X''s) is the probability that a Behrens–Fisher-distributed random variable is between &minus;''A'' and&nbsp;''A''.

=== Fiducial intervals versus confidence intervals ===

Bartlett{{citation needed|date=August 2012}} 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|date=August 2012}}

<!-- ==See also==
* [[Fisher transformation]]
* [[Fisher information]] -->

== 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)

{{ProbDistributions|continuous-infinite}}

==References==
{{Reflist}}

{{DEFAULTSORT:Behrens-Fisher distribution}}
[[Category:Continuous distributions]]