<s>
In	O
statistics	O
,	O
the	O
score	B-General_Concept
test	I-General_Concept
assesses	O
constraints	B-Application
on	O
statistical	O
parameters	O
based	O
on	O
the	O
gradient	O
of	O
the	O
likelihood	O
function	O
—	O
known	O
as	O
the	O
score	O
—	O
evaluated	O
at	O
the	O
hypothesized	O
parameter	O
value	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
While	O
the	O
finite	O
sample	O
distributions	O
of	O
score	B-General_Concept
tests	I-General_Concept
are	O
generally	O
unknown	O
,	O
they	O
have	O
an	O
asymptotic	O
χ2-distribution	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
as	O
first	O
proved	O
by	O
C	O
.	O
R	O
.	O
Rao	O
in	O
1948	O
,	O
a	O
fact	O
that	O
can	O
be	O
used	O
to	O
determine	O
statistical	B-General_Concept
significance	I-General_Concept
.	O
</s>
<s>
Since	O
function	O
maximization	O
subject	O
to	O
equality	O
constraints	B-Application
is	O
most	O
conveniently	O
done	O
using	O
a	O
Lagrangean	O
expression	O
of	O
the	O
problem	O
,	O
the	O
score	B-General_Concept
test	I-General_Concept
can	O
be	O
equivalently	O
understood	O
as	O
a	O
test	O
of	O
the	O
magnitude	O
of	O
the	O
Lagrange	O
multipliers	O
associated	O
with	O
the	O
constraints	B-Application
where	O
,	O
again	O
,	O
if	O
the	O
constraints	B-Application
are	O
non-binding	O
at	O
the	O
maximum	O
likelihood	O
,	O
the	O
vector	O
of	O
Lagrange	O
multipliers	O
should	O
not	O
differ	O
from	O
zero	O
by	O
more	O
than	O
sampling	O
error	O
.	O
</s>
<s>
The	O
equivalence	O
of	O
these	O
two	O
approaches	O
was	O
first	O
shown	O
by	O
S	O
.	O
D	O
.	O
Silvey	O
in	O
1959	O
,	O
which	O
led	O
to	O
the	O
name	O
Lagrange	B-General_Concept
multiplier	I-General_Concept
test	I-General_Concept
that	O
has	O
become	O
more	O
commonly	O
used	O
,	O
particularly	O
in	O
econometrics	O
,	O
since	O
Breusch	O
and	O
Pagan	O
's	O
much-cited	O
1980	O
paper	O
.	O
</s>
<s>
The	O
main	O
advantage	O
of	O
the	O
score	B-General_Concept
test	I-General_Concept
over	O
the	O
Wald	B-General_Concept
test	I-General_Concept
and	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
is	O
that	O
the	O
score	B-General_Concept
test	I-General_Concept
only	O
requires	O
the	O
computation	O
of	O
the	O
restricted	O
estimator	O
.	O
</s>
<s>
Further	O
,	O
because	O
the	O
score	B-General_Concept
test	I-General_Concept
only	O
requires	O
the	O
estimation	O
of	O
the	O
likelihood	O
function	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
it	O
is	O
less	O
specific	O
than	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
about	O
the	O
alternative	O
hypothesis	O
.	O
</s>
<s>
While	O
asymptotically	O
identical	O
,	O
calculating	O
the	O
LM	O
statistic	O
using	O
the	O
outer-gradient-product	B-Algorithm
estimator	I-Algorithm
of	O
the	O
Fisher	O
information	O
matrix	O
can	O
lead	O
to	O
bias	O
in	O
small	O
samples	O
.	O
</s>
<s>
where	O
is	O
the	O
likelihood	O
function	O
,	O
is	O
the	O
value	O
of	O
the	O
parameter	O
of	O
interest	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
and	O
is	O
a	O
constant	O
set	O
depending	O
on	O
the	O
size	O
of	O
the	O
test	O
desired	O
(	O
i.e.	O
</s>
<s>
The	O
score	B-General_Concept
test	I-General_Concept
is	O
the	O
most	O
powerful	O
test	O
for	O
small	O
deviations	O
from	O
.	O
</s>
<s>
If	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
,	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
,	O
the	O
Wald	B-General_Concept
test	I-General_Concept
,	O
and	O
the	O
Score	B-General_Concept
test	I-General_Concept
are	O
asymptotically	O
equivalent	O
tests	O
of	O
hypotheses	O
.	O
</s>
<s>
If	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
not	O
true	O
,	O
however	O
,	O
the	O
statistics	O
converge	O
to	O
a	O
noncentral	O
chi-squared	O
distribution	O
with	O
possibly	O
different	O
noncentrality	O
parameters	O
.	O
</s>
<s>
A	O
more	O
general	O
score	B-General_Concept
test	I-General_Concept
can	O
be	O
derived	O
when	O
there	O
is	O
more	O
than	O
one	O
parameter	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
the	O
maximum	O
likelihood	O
estimate	O
of	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
while	O
and	O
are	O
respectively	O
,	O
the	O
score	O
and	O
the	O
Fisher	O
information	O
matrices	O
under	O
the	O
alternative	O
hypothesis	O
.	O
</s>
<s>
In	O
linear	B-General_Concept
regression	I-General_Concept
,	O
the	O
Lagrange	B-General_Concept
multiplier	I-General_Concept
test	I-General_Concept
can	O
be	O
expressed	O
as	O
a	O
function	O
of	O
the	O
F-test	B-General_Concept
.	O
</s>
<s>
When	O
the	O
data	O
consists	O
of	O
binary	O
observations	O
,	O
the	O
score	O
statistic	O
is	O
the	O
same	O
as	O
the	O
chi-squared	O
statistic	O
in	O
the	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
.	O
</s>
