<s>
In	O
statistical	O
inference	O
,	O
the	O
concept	O
of	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
(	O
CD	O
)	O
has	O
often	O
been	O
loosely	O
referred	O
to	O
as	O
a	O
distribution	O
function	O
on	O
the	O
parameter	O
space	O
that	O
can	O
represent	O
confidence	O
intervals	O
of	O
all	O
levels	O
for	O
a	O
parameter	O
of	O
interest	O
.	O
</s>
<s>
Historically	O
,	O
it	O
has	O
typically	O
been	O
constructed	O
by	O
inverting	O
the	O
upper	O
limits	O
of	O
lower	O
sided	O
confidence	O
intervals	O
of	O
all	O
levels	O
,	O
and	O
it	O
was	O
also	O
commonly	O
associated	O
with	O
a	O
fiducial	O
interpretation	O
(	O
fiducial	B-Error_Name
distribution	I-Error_Name
)	O
,	O
although	O
it	O
is	O
a	O
purely	O
frequentist	B-General_Concept
concept	O
.	O
</s>
<s>
A	O
confidence	B-General_Concept
distribution	I-General_Concept
is	O
NOT	O
a	O
probability	O
distribution	O
function	O
of	O
the	O
parameter	O
of	O
interest	O
,	O
but	O
may	O
still	O
be	O
a	O
function	O
useful	O
for	O
making	O
inferences	O
.	O
</s>
<s>
In	O
recent	O
years	O
,	O
there	O
has	O
been	O
a	O
surge	O
of	O
renewed	O
interest	O
in	O
confidence	B-General_Concept
distributions	I-General_Concept
.	O
</s>
<s>
In	O
the	O
more	O
recent	O
developments	O
,	O
the	O
concept	O
of	O
confidence	B-General_Concept
distribution	I-General_Concept
has	O
emerged	O
as	O
a	O
purely	O
frequentist	B-General_Concept
concept	O
,	O
without	O
any	O
fiducial	O
interpretation	O
or	O
reasoning	O
.	O
</s>
<s>
Conceptually	O
,	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
is	O
no	O
different	O
from	O
a	O
point	O
estimator	O
or	O
an	O
interval	O
estimator	O
(	O
confidence	O
interval	O
)	O
,	O
but	O
it	O
uses	O
a	O
sample-dependent	O
distribution	O
function	O
on	O
the	O
parameter	O
space	O
(	O
instead	O
of	O
a	O
point	O
or	O
an	O
interval	O
)	O
to	O
estimate	O
the	O
parameter	O
of	O
interest	O
.	O
</s>
<s>
A	O
simple	O
example	O
of	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
,	O
that	O
has	O
been	O
broadly	O
used	O
in	O
statistical	O
practice	O
,	O
is	O
a	O
bootstrap	B-Application
distribution	O
.	O
</s>
<s>
The	O
development	O
and	O
interpretation	O
of	O
a	O
bootstrap	B-Application
distribution	O
does	O
not	O
involve	O
any	O
fiducial	O
reasoning	O
;	O
the	O
same	O
is	O
true	O
for	O
the	O
concept	O
of	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
.	O
</s>
<s>
But	O
the	O
notion	O
of	O
confidence	B-General_Concept
distribution	I-General_Concept
is	O
much	O
broader	O
than	O
that	O
of	O
a	O
bootstrap	B-Application
distribution	O
.	O
</s>
<s>
In	O
particular	O
,	O
recent	O
research	O
suggests	O
that	O
it	O
encompasses	O
and	O
unifies	O
a	O
wide	O
range	O
of	O
examples	O
,	O
from	O
regular	O
parametric	O
cases	O
(	O
including	O
most	O
examples	O
of	O
the	O
classical	O
development	O
of	O
Fisher	B-Error_Name
's	I-Error_Name
fiducial	I-Error_Name
distribution	I-Error_Name
)	O
to	O
bootstrap	B-Application
distributions	O
,	O
p-value	B-General_Concept
functions	O
,	O
normalized	O
likelihood	O
functions	O
and	O
,	O
in	O
some	O
cases	O
,	O
Bayesian	O
priors	O
and	O
Bayesian	O
posteriors	O
.	O
</s>
<s>
Just	O
as	O
a	O
Bayesian	O
posterior	O
distribution	O
contains	O
a	O
wealth	O
of	O
information	O
for	O
any	O
type	O
of	O
Bayesian	O
inference	O
,	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
contains	O
a	O
wealth	O
of	O
information	O
for	O
constructing	O
almost	O
all	O
types	O
of	O
frequentist	B-General_Concept
inferences	I-General_Concept
,	O
including	O
point	O
estimates	O
,	O
confidence	O
intervals	O
,	O
critical	O
values	O
,	O
statistical	B-General_Concept
power	I-General_Concept
and	O
p-values	B-General_Concept
,	O
among	O
others	O
.	O
</s>
<s>
Neyman	O
(	O
1937	O
)	O
introduced	O
the	O
idea	O
of	O
"	O
confidence	O
"	O
in	O
his	O
seminal	O
paper	O
on	O
confidence	O
intervals	O
which	O
clarified	O
the	O
frequentist	B-General_Concept
repetition	O
property	O
.	O
</s>
<s>
According	O
to	O
Fraser	O
,	O
the	O
seed	O
(	O
idea	O
)	O
of	O
confidence	B-General_Concept
distribution	I-General_Concept
can	O
even	O
be	O
traced	O
back	O
to	O
Bayes	O
(	O
1763	O
)	O
and	O
Fisher	O
(	O
1930	O
)	O
.	O
</s>
<s>
Some	O
researchers	O
view	O
the	O
confidence	B-General_Concept
distribution	I-General_Concept
as	O
"	O
the	O
Neymanian	O
interpretation	O
of	O
Fisher	B-Error_Name
's	I-Error_Name
fiducial	I-Error_Name
distributions	I-Error_Name
"	O
,	O
which	O
was	O
"	O
furiously	O
disputed	O
by	O
Fisher	O
"	O
.	O
</s>
<s>
It	O
is	O
also	O
believed	O
that	O
these	O
"	O
unproductive	O
disputes	O
"	O
and	O
Fisher	O
's	O
"	O
stubborn	O
insistence	O
"	O
might	O
be	O
the	O
reason	O
that	O
the	O
concept	O
of	O
confidence	B-General_Concept
distribution	I-General_Concept
has	O
been	O
long	O
misconstrued	O
as	O
a	O
fiducial	O
concept	O
and	O
not	O
been	O
fully	O
developed	O
under	O
the	O
frequentist	B-General_Concept
framework	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
confidence	B-General_Concept
distribution	I-General_Concept
is	O
a	O
purely	O
frequentist	B-General_Concept
concept	O
with	O
a	O
purely	O
frequentist	B-General_Concept
interpretation	O
,	O
and	O
it	O
also	O
has	O
ties	O
to	O
Bayesian	O
inference	O
concepts	O
and	O
the	O
fiducial	O
arguments	O
.	O
</s>
<s>
Classically	O
,	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
is	O
defined	O
by	O
inverting	O
the	O
upper	O
limits	O
of	O
a	O
series	O
of	O
lower-sided	O
confidence	O
intervals	O
.	O
</s>
<s>
Then	O
,	O
Hn( •	O
)	O
=	O
ξn−1( •	O
)	O
is	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
forθ	O
.	O
</s>
<s>
In	O
the	O
classical	O
literature	O
,	O
the	O
confidence	B-General_Concept
distribution	I-General_Concept
function	O
is	O
interpreted	O
as	O
a	O
distribution	O
function	O
of	O
the	O
parameter	O
θ	O
,	O
which	O
is	O
impossible	O
unless	O
fiducial	O
reasoning	O
is	O
involved	O
since	O
,	O
in	O
a	O
frequentist	B-General_Concept
setting	O
,	O
the	O
parameters	O
are	O
fixed	O
and	O
nonrandom	O
.	O
</s>
<s>
To	O
interpret	O
the	O
CD	O
function	O
entirely	O
from	O
a	O
frequentist	B-General_Concept
viewpoint	O
and	O
not	O
interpret	O
it	O
as	O
a	O
distribution	O
function	O
of	O
a	O
(	O
fixed/nonrandom	O
)	O
parameter	O
is	O
one	O
of	O
the	O
major	O
departures	O
of	O
recent	O
development	O
relative	O
to	O
the	O
classical	O
approach	O
.	O
</s>
<s>
The	O
nice	O
thing	O
about	O
treating	O
confidence	B-General_Concept
distributions	I-General_Concept
as	O
a	O
purely	O
frequentist	B-General_Concept
concept	O
(	O
similar	O
to	O
a	O
point	O
estimator	O
)	O
is	O
that	O
it	O
is	O
now	O
free	O
from	O
those	O
restrictive	O
,	O
if	O
not	O
controversial	O
,	O
constraints	O
set	O
forth	O
by	O
Fisher	O
on	O
fiducial	B-Error_Name
distributions	I-Error_Name
.	O
</s>
<s>
A	O
function	O
Hn( •	O
)	O
=	O
Hn( Xn	O
,	O
•	O
)	O
on	O
χ×Θ→[0,1]	O
is	O
called	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
(	O
CD	O
)	O
for	O
a	O
parameter	O
θ	O
,	O
if	O
it	O
follows	O
two	O
requirements	O
:	O
</s>
<s>
In	O
nontechnical	O
terms	O
,	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
is	O
a	O
function	O
of	O
both	O
the	O
parameter	O
and	O
the	O
random	O
sample	O
,	O
with	O
two	O
requirements	O
.	O
</s>
<s>
based	O
on	O
the	O
confidence	B-General_Concept
distribution	I-General_Concept
have	O
desired	O
frequentist	B-General_Concept
properties	O
.	O
</s>
<s>
A	O
confidence	B-General_Concept
distribution	I-General_Concept
derived	O
by	O
inverting	O
the	O
upper	O
limits	O
of	O
confidence	O
intervals	O
(	O
classical	O
definition	O
)	O
also	O
satisfies	O
the	O
requirements	O
in	O
the	O
above	O
definition	O
and	O
this	O
version	O
of	O
the	O
definition	O
is	O
consistent	O
with	O
the	O
classical	O
definition	O
.	O
</s>
<s>
Unlike	O
the	O
classical	O
fiducial	B-Error_Name
inference	I-Error_Name
,	O
more	O
than	O
one	O
confidence	B-General_Concept
distributions	I-General_Concept
may	O
be	O
available	O
to	O
estimate	O
a	O
parameter	O
under	O
any	O
specific	O
setting	O
.	O
</s>
<s>
Also	O
,	O
unlike	O
the	O
classical	O
fiducial	B-Error_Name
inference	I-Error_Name
,	O
optimality	O
is	O
not	O
a	O
part	O
of	O
requirement	O
.	O
</s>
<s>
Depending	O
on	O
the	O
setting	O
and	O
the	O
criterion	O
used	O
,	O
sometimes	O
there	O
is	O
a	O
unique	O
"	O
best	O
"	O
(	O
in	O
terms	O
of	O
optimality	O
)	O
confidence	B-General_Concept
distribution	I-General_Concept
.	O
</s>
<s>
But	O
sometimes	O
there	O
is	O
no	O
optimal	O
confidence	B-General_Concept
distribution	I-General_Concept
available	O
or	O
,	O
in	O
some	O
extreme	O
cases	O
,	O
we	O
may	O
not	O
even	O
be	O
able	O
to	O
find	O
a	O
meaningful	O
confidence	B-General_Concept
distribution	I-General_Concept
.	O
</s>
<s>
A	O
confidence	B-General_Concept
distribution	I-General_Concept
for	O
a	O
parameter	O
in	O
a	O
measurable	O
space	O
is	O
a	O
distribution	O
estimator	O
with	O
for	O
a	O
family	O
of	O
confidence	O
regions	O
for	O
with	O
level	O
for	O
all	O
levels	O
.	O
</s>
<s>
If	O
only	O
exists	O
for	O
,	O
then	O
is	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
with	O
level	O
set	O
.	O
</s>
<s>
If	O
the	O
defining	O
requirement	O
holds	O
with	O
equality	O
,	O
then	O
the	O
confidence	B-General_Concept
distribution	I-General_Concept
is	O
by	O
definition	O
exact	O
.	O
</s>
<s>
satisfy	O
the	O
two	O
requirements	O
in	O
the	O
CD	O
definition	O
,	O
and	O
they	O
are	O
confidence	B-General_Concept
distribution	I-General_Concept
functions	O
forμ	O
.	O
</s>
<s>
satisfies	O
the	O
definition	O
of	O
an	O
asymptotic	O
confidence	B-General_Concept
distribution	I-General_Concept
when	O
n	O
→	O
∞	O
,	O
and	O
it	O
is	O
an	O
asymptotic	O
confidence	B-General_Concept
distribution	I-General_Concept
for	O
μ	O
.	O
</s>
<s>
For	O
the	O
parameter	O
μ	O
,	O
since	O
involves	O
the	O
unknown	O
parameter	O
σ	O
and	O
it	O
violates	O
the	O
two	O
requirements	O
in	O
the	O
CD	O
definition	O
,	O
it	O
is	O
no	O
longer	O
a	O
"	O
distribution	O
estimator	O
"	O
or	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
forμ	O
.	O
</s>
<s>
is	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
function	O
for	O
σ2	O
.	O
</s>
<s>
In	O
the	O
case	O
when	O
the	O
variance	O
σ2	O
is	O
unknown	O
,	O
is	O
an	O
optimal	O
confidence	B-General_Concept
distribution	I-General_Concept
for	O
μ	O
.	O
</s>
<s>
It	O
is	O
well	O
known	O
that	O
Fisher	O
's	O
z	O
defined	O
by	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
:	O
</s>
<s>
has	O
the	O
limiting	O
distribution	O
with	O
a	O
fast	O
rate	O
of	O
convergence	O
,	O
where	O
r	B-Language
is	O
the	O
sample	O
correlation	O
and	O
n	O
is	O
the	O
sample	O
size	O
.	O
</s>
<s>
is	O
an	O
asymptotic	O
confidence	B-General_Concept
distribution	I-General_Concept
for	O
ρ	O
.	O
</s>
<s>
This	O
formula	O
was	O
derived	O
by	O
C	O
.	O
R	B-Language
.	O
Rao	O
.	O
</s>
<s>
The	O
distribution	O
of	O
defines	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
for	O
.	O
</s>
<s>
The	O
confidence	B-General_Concept
distribution	I-General_Concept
is	O
in	O
this	O
case	O
binormal	O
with	O
mean	O
,	O
and	O
the	O
confidence	O
regions	O
can	O
be	O
chosen	O
in	O
many	O
other	O
ways	O
.	O
</s>
<s>
The	O
confidence	B-General_Concept
distribution	I-General_Concept
coincides	O
in	O
this	O
case	O
with	O
the	O
Bayesian	O
posterior	O
using	O
the	O
right	O
Haar	O
prior	O
.	O
</s>
<s>
The	O
argument	O
generalizes	O
to	O
the	O
case	O
of	O
an	O
unknown	O
mean	O
in	O
an	O
infinite-dimensional	O
Hilbert	O
space	O
,	O
but	O
in	O
this	O
case	O
the	O
confidence	B-General_Concept
distribution	I-General_Concept
is	O
not	O
a	O
Bayesian	O
posterior	O
.	O
</s>
<s>
Point	O
estimators	O
can	O
also	O
be	O
constructed	O
given	O
a	O
confidence	B-General_Concept
distribution	I-General_Concept
estimator	O
for	O
the	O
parameter	O
of	O
interest	O
.	O
</s>
<s>
Certain	O
confidence	B-General_Concept
distributions	I-General_Concept
can	O
give	O
optimal	O
frequentist	B-General_Concept
estimators	O
.	O
</s>
<s>
One	O
can	O
derive	O
a	O
p-value	B-General_Concept
for	O
a	O
test	O
,	O
either	O
one-sided	O
or	O
two-sided	O
,	O
concerning	O
the	O
parameterθ	O
,	O
from	O
its	O
confidence	B-General_Concept
distribution	I-General_Concept
Hn(θ )	O
.	O
</s>
<s>
Denote	O
by	O
the	O
probability	O
mass	O
of	O
a	O
set	O
C	O
under	O
the	O
confidence	B-General_Concept
distribution	I-General_Concept
function	O
This	O
ps(C )	O
is	O
called	O
"	O
support	O
"	O
in	O
the	O
CD	O
inference	O
and	O
also	O
known	O
as	O
"	O
belief	O
"	O
in	O
the	O
fiducial	O
literature	O
.	O
</s>
<s>
Thus	O
,	O
ps(C )	O
=	O
Hn(C )	O
is	O
the	O
corresponding	O
p-value	B-General_Concept
of	O
the	O
test	O
.	O
</s>
<s>
Thus	O
,	O
2min{	O
ps(Clo )	O
,	O
ps(Cup )	O
}	O
=	O
2min{	O
Hn(b )	O
,	O
1Hn(b )	O
}	O
is	O
the	O
corresponding	O
p-value	B-General_Concept
of	O
the	O
test	O
.	O
</s>
<s>
A	O
few	O
statistical	O
programs	O
have	O
implemented	O
the	O
ability	O
to	O
construct	O
and	O
graph	O
confidence	B-General_Concept
distributions	I-General_Concept
.	O
</s>
