<s>
The	O
ratio	O
of	O
the	O
density	O
functions	O
above	O
is	O
increasing	O
in	O
the	O
parameter	O
,	O
so	O
satisfies	O
the	O
monotone	B-General_Concept
likelihood	I-General_Concept
ratio	I-General_Concept
property	I-General_Concept
.	O
</s>
<s>
In	O
statistics	O
,	O
the	O
monotone	B-General_Concept
likelihood	I-General_Concept
ratio	I-General_Concept
property	I-General_Concept
is	O
a	O
property	O
of	O
the	O
ratio	O
of	O
two	O
probability	O
density	O
functions	O
(	O
PDFs	O
)	O
.	O
</s>
<s>
For	O
two	O
distributions	O
that	O
satisfy	O
the	O
definition	O
with	O
respect	O
to	O
some	O
argument	O
x	O
,	O
we	O
say	O
they	O
"	O
have	O
the	O
MLRP	B-General_Concept
in	O
x.	O
"	O
</s>
<s>
The	O
MLRP	B-General_Concept
is	O
used	O
to	O
represent	O
a	O
data-generating	O
process	O
that	O
enjoys	O
a	O
straightforward	O
relationship	O
between	O
the	O
magnitude	O
of	O
some	O
observed	O
variable	O
and	O
the	O
distribution	O
it	O
draws	O
from	O
.	O
</s>
<s>
If	O
satisfies	O
the	O
MLRP	B-General_Concept
with	O
respect	O
to	O
,	O
the	O
higher	O
the	O
observed	O
value	O
,	O
the	O
more	O
likely	O
it	O
was	O
drawn	O
from	O
distribution	O
rather	O
than	O
.	O
</s>
<s>
If	O
the	O
MLRP	B-General_Concept
holds	O
for	O
the	O
distribution	O
of	O
q	O
conditional	O
on	O
your	O
effort	O
,	O
the	O
higher	O
the	O
quality	O
the	O
more	O
likely	O
you	O
worked	O
hard	O
.	O
</s>
<s>
Suppose	O
satisfies	O
the	O
MLRP	B-General_Concept
.	O
</s>
<s>
which	O
,	O
thanks	O
to	O
the	O
MLRP	B-General_Concept
,	O
is	O
monotonically	O
increasing	O
in	O
(	O
because	O
is	O
decreasing	O
in	O
)	O
.	O
</s>
<s>
This	O
task	O
is	O
simplified	O
if	O
the	O
family	O
has	O
the	O
monotone	B-General_Concept
likelihood	I-General_Concept
ratio	I-General_Concept
property	I-General_Concept
(	O
MLRP	B-General_Concept
)	O
.	O
</s>
<s>
A	O
family	O
of	O
density	O
functions	O
indexed	O
by	O
a	O
parameter	O
taking	O
values	O
in	O
an	O
ordered	O
set	O
is	O
said	O
to	O
have	O
a	O
monotone	B-General_Concept
likelihood	I-General_Concept
ratio	I-General_Concept
(	O
MLR	O
)	O
in	O
the	O
statistic	O
if	O
for	O
any	O
,	O
</s>
<s>
If	O
the	O
family	O
of	O
random	O
variables	O
has	O
the	O
MLRP	B-General_Concept
in	O
,	O
a	O
uniformly	B-General_Concept
most	I-General_Concept
powerful	I-General_Concept
test	I-General_Concept
can	O
easily	O
be	O
determined	O
for	O
the	O
hypothesis	O
versus	O
.	O
</s>
<s>
Example	O
:	O
Let	O
be	O
an	O
input	O
into	O
a	O
stochastic	O
technology	O
–	O
worker	O
's	O
effort	O
,	O
for	O
instance	O
–	O
and	O
its	O
output	O
,	O
the	O
likelihood	O
of	O
which	O
is	O
described	O
by	O
a	O
probability	O
density	O
function	O
Then	O
the	O
monotone	B-General_Concept
likelihood	I-General_Concept
ratio	I-General_Concept
property	I-General_Concept
(	O
MLRP	B-General_Concept
)	O
of	O
the	O
family	O
is	O
expressed	O
as	O
follows	O
:	O
for	O
any	O
,	O
the	O
fact	O
that	O
implies	O
that	O
the	O
ratio	O
is	O
increasing	O
in	O
.	O
</s>
<s>
Monotone	O
likelihood	O
functions	O
are	O
used	O
to	O
construct	O
uniformly	B-General_Concept
most	I-General_Concept
powerful	I-General_Concept
tests	I-General_Concept
,	O
according	O
to	O
the	O
Karlin	O
–	O
Rubin	O
theorem	O
.	O
</s>
<s>
If	O
a	O
family	O
of	O
distributions	O
has	O
the	O
monotone	B-General_Concept
likelihood	I-General_Concept
ratio	I-General_Concept
property	I-General_Concept
in	O
,	O
</s>
<s>
But	O
not	O
conversely	O
:	O
neither	O
monotone	O
hazard	O
rates	O
nor	O
stochastic	O
dominance	O
imply	O
the	O
MLRP	B-General_Concept
.	O
</s>
