<s>
In	O
statistics	O
,	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
,	O
named	O
after	O
Walter	O
Behrens	O
and	O
Ronald	O
Fisher	O
,	O
is	O
the	O
problem	O
of	O
interval	O
estimation	O
and	O
hypothesis	O
testing	O
concerning	O
the	O
difference	O
between	O
the	O
means	O
of	O
two	O
normally	O
distributed	O
populations	O
when	O
the	O
variances	O
of	O
the	O
two	O
populations	O
are	O
not	O
assumed	O
to	O
be	O
equal	O
,	O
based	O
on	O
two	O
independent	O
samples	O
.	O
</s>
<s>
One	O
difficulty	O
with	O
discussing	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
and	O
proposed	O
solutions	O
,	O
is	O
that	O
there	O
are	O
many	O
different	O
interpretations	O
of	O
what	O
is	O
meant	O
by	O
"	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
"	O
.	O
</s>
<s>
Lehmann	O
states	O
that	O
"	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
"	O
is	O
used	O
both	O
for	O
this	O
general	O
form	O
of	O
model	O
when	O
the	O
family	O
of	O
distributions	O
is	O
arbitrary	O
and	O
for	O
when	O
the	O
restriction	O
to	O
a	O
normal	O
distribution	O
is	O
made	O
.	O
</s>
<s>
While	O
Lehmann	O
discusses	O
a	O
number	O
of	O
approaches	O
to	O
the	O
more	O
general	O
problem	O
,	O
mainly	O
based	O
on	O
nonparametrics	B-General_Concept
,	O
most	O
other	O
sources	O
appear	O
to	O
use	O
"	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
"	O
to	O
refer	O
only	O
to	O
the	O
case	O
where	O
the	O
distribution	O
is	O
assumed	O
to	O
be	O
normal	O
:	O
most	O
of	O
this	O
article	O
makes	O
this	O
assumption	O
.	O
</s>
<s>
Solutions	O
to	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
have	O
been	O
presented	O
that	O
make	O
use	O
of	O
either	O
a	O
classical	B-General_Concept
or	O
a	O
Bayesian	O
inference	O
point	O
of	O
view	O
and	O
either	O
solution	O
would	O
be	O
notionally	O
invalid	O
judged	O
from	O
the	O
other	O
point	O
of	O
view	O
.	O
</s>
<s>
If	O
consideration	O
is	O
restricted	O
to	O
classical	B-General_Concept
statistical	O
inference	O
only	O
,	O
it	O
is	O
possible	O
to	O
seek	O
solutions	O
to	O
the	O
inference	O
problem	O
that	O
are	O
simple	O
to	O
apply	O
in	O
a	O
practical	O
sense	O
,	O
giving	O
preference	O
to	O
this	O
simplicity	O
over	O
any	O
inaccuracy	O
in	O
the	O
corresponding	O
probability	O
statements	O
.	O
</s>
<s>
Where	O
exactness	O
of	O
the	O
significance	B-General_Concept
levels	I-General_Concept
of	O
statistical	O
tests	O
is	O
required	O
,	O
there	O
may	O
be	O
an	O
additional	O
requirement	O
that	O
the	O
procedure	O
should	O
make	O
maximum	O
use	O
of	O
the	O
statistical	O
information	O
in	O
the	O
dataset	O
.	O
</s>
<s>
It	O
is	O
well	O
known	O
that	O
an	O
exact	O
test	O
can	O
be	O
gained	O
by	O
randomly	O
discarding	O
data	O
from	O
the	O
larger	O
dataset	O
until	O
the	O
sample	O
sizes	O
are	O
equal	O
,	O
assembling	O
data	O
in	O
pairs	O
and	O
taking	O
differences	O
,	O
and	O
then	O
using	O
an	O
ordinary	O
t-test	B-General_Concept
to	O
test	O
for	O
the	O
mean-difference	O
being	O
zero	O
:	O
clearly	O
this	O
would	O
not	O
be	O
"	O
optimal	O
"	O
in	O
any	O
sense	O
.	O
</s>
<s>
The	O
task	O
of	O
specifying	O
interval	O
estimates	O
for	O
this	O
problem	O
is	O
one	O
where	O
a	O
frequentist	B-General_Concept
approach	O
fails	O
to	O
provide	O
an	O
exact	O
solution	O
,	O
although	O
some	O
approximations	O
are	O
available	O
.	O
</s>
<s>
Thus	O
study	O
of	O
the	O
problem	O
can	O
be	O
used	O
to	O
elucidate	O
the	O
differences	O
between	O
the	O
frequentist	B-General_Concept
and	O
Bayesian	O
approaches	O
to	O
interval	O
estimation	O
.	O
</s>
<s>
Ronald	O
Fisher	O
in	O
1935	O
introduced	O
fiducial	B-Error_Name
inference	I-Error_Name
in	O
order	O
to	O
apply	O
it	O
to	O
this	O
problem	O
.	O
</s>
<s>
where	O
and	O
are	O
the	O
two	O
sample	O
means	O
,	O
and	O
s1	O
and	O
s2	O
are	O
their	O
standard	B-General_Concept
deviations	I-General_Concept
.	O
</s>
<s>
See	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
distribution	O
.	O
</s>
<s>
Fisher	O
approximated	O
the	O
distribution	O
of	O
this	O
by	O
ignoring	O
the	O
random	O
variation	O
of	O
the	O
relative	O
sizes	O
of	O
the	O
standard	B-General_Concept
deviations	I-General_Concept
,	O
</s>
<s>
Fisher	O
's	O
solution	O
provoked	O
controversy	O
because	O
it	O
did	O
not	O
have	O
the	O
property	O
that	O
the	O
hypothesis	O
of	O
equal	O
means	O
would	O
be	O
rejected	B-General_Concept
with	I-General_Concept
probability	I-General_Concept
α	I-General_Concept
if	O
the	O
means	O
were	O
in	O
fact	O
equal	O
.	O
</s>
<s>
Under	O
the	O
null	O
hypothesis	O
of	O
equal	O
expectations	O
,	O
,	O
the	O
distribution	O
of	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
statistic	O
T	O
,	O
which	O
also	O
depends	O
on	O
the	O
variance	O
ratio	O
σ12/σ22	O
,	O
could	O
now	O
be	O
approximated	O
by	O
Student	O
's	O
t	O
distribution	O
with	O
these	O
ν	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
Nevertheless	O
,	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
T	O
can	O
be	O
compared	O
with	O
a	O
corresponding	O
quantile	O
of	O
Student	O
's	O
t	O
distribution	O
with	O
these	O
estimated	O
numbers	O
of	O
degrees	O
of	O
freedom	O
,	O
,	O
which	O
is	O
generally	O
non-integer	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
population	O
variances	O
are	O
equal	O
,	O
or	O
if	O
the	O
samples	O
are	O
rather	O
small	O
and	O
the	O
population	O
variances	O
can	O
be	O
assumed	O
to	O
be	O
approximately	O
equal	O
,	O
it	O
is	O
more	O
accurate	O
to	O
use	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
.	O
</s>
<s>
For	O
several	O
decades	O
,	O
it	O
was	O
commonly	O
believed	O
that	O
no	O
exact	O
solution	O
to	O
the	O
common	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
existed	O
.	O
</s>
<s>
In	O
2018	O
the	O
probability	O
density	O
function	O
of	O
a	O
generalized	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
distribution	O
of	O
m	O
means	O
and	O
m	O
distinct	O
standard	O
errors	O
from	O
m	O
samples	O
of	O
distinct	O
sizes	O
from	O
independent	O
normal	O
distributions	O
with	O
distinct	O
means	O
and	O
variances	O
was	O
proved	O
and	O
the	O
paper	O
also	O
examined	O
its	O
asymptotic	O
approximations	O
.	O
</s>
<s>
A	O
follow-up	O
paper	O
showed	O
that	O
the	O
classic	O
paired	B-General_Concept
t-test	I-General_Concept
is	O
a	O
central	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
with	O
a	O
non-zero	O
population	O
correlation	O
coefficient	O
and	O
derived	O
its	O
corresponding	O
probability	O
density	O
function	O
by	O
solving	O
its	O
associated	O
non-central	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
with	O
a	O
nonzero	O
population	O
correlation	O
coefficient	O
.	O
</s>
<s>
It	O
also	O
solved	O
a	O
more	O
general	O
non-central	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
with	O
a	O
non-zero	O
population	O
correlation	O
coefficient	O
in	O
the	O
appendix	O
.	O
</s>
<s>
A	O
minor	O
variant	O
of	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
has	O
been	O
studied	O
.	O
</s>
<s>
One	O
generalisation	O
of	O
the	O
problem	O
involves	O
multivariate	O
normal	O
distributions	O
with	O
unknown	O
covariance	O
matrices	O
,	O
and	O
is	O
known	O
as	O
the	O
multivariate	B-General_Concept
Behrens	I-General_Concept
–	I-General_Concept
Fisher	I-General_Concept
problem	I-General_Concept
.	O
</s>
<s>
The	O
nonparametric	B-General_Concept
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
does	O
not	O
assume	O
that	O
the	O
distributions	O
are	O
normal	O
.	O
</s>
<s>
Tests	O
include	O
the	O
Cucconi	B-General_Concept
test	I-General_Concept
of	O
1968	O
and	O
the	O
Lepage	B-General_Concept
test	I-General_Concept
of	O
1971	O
.	O
</s>
