<s>
In	O
statistics	O
,	O
the	O
variance	B-General_Concept
function	I-General_Concept
is	O
a	O
smooth	O
function	O
which	O
depicts	O
the	O
variance	O
of	O
a	O
random	O
quantity	O
as	O
a	O
function	O
of	O
its	O
mean	O
.	O
</s>
<s>
The	O
variance	B-General_Concept
function	I-General_Concept
is	O
a	O
measure	O
of	O
heteroscedasticity	B-General_Concept
and	O
plays	O
a	O
large	O
role	O
in	O
many	O
settings	O
of	O
statistical	O
modelling	O
.	O
</s>
<s>
It	O
is	O
a	O
main	O
ingredient	O
in	O
the	O
generalized	O
linear	O
model	O
framework	O
and	O
a	O
tool	O
used	O
in	O
non-parametric	O
regression	O
,	O
semiparametric	B-General_Concept
regression	I-General_Concept
and	O
functional	B-General_Concept
data	I-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
In	O
parametric	O
modeling	O
,	O
variance	B-General_Concept
functions	I-General_Concept
take	O
on	O
a	O
parametric	O
form	O
and	O
explicitly	O
describe	O
the	O
relationship	O
between	O
the	O
variance	O
and	O
the	O
mean	O
of	O
a	O
random	O
quantity	O
.	O
</s>
<s>
In	O
a	O
non-parametric	O
setting	O
,	O
the	O
variance	B-General_Concept
function	I-General_Concept
is	O
assumed	O
to	O
be	O
a	O
smooth	O
function	O
.	O
</s>
<s>
A	O
main	O
assumption	O
in	O
linear	B-General_Concept
regression	I-General_Concept
is	O
constant	O
variance	O
or	O
(	O
homoscedasticity	B-General_Concept
)	O
,	O
meaning	O
that	O
different	O
response	O
variables	O
have	O
the	O
same	O
variance	O
in	O
their	O
errors	O
,	O
at	O
every	O
predictor	O
level	O
.	O
</s>
<s>
As	O
we	O
will	O
see	O
later	O
,	O
the	O
variance	B-General_Concept
function	I-General_Concept
in	O
the	O
Normal	O
setting	O
is	O
constant	O
;	O
however	O
,	O
we	O
must	O
find	O
a	O
way	O
to	O
quantify	O
heteroscedasticity	B-General_Concept
(	O
non-constant	O
variance	O
)	O
in	O
the	O
absence	O
of	O
joint	O
Normality	O
.	O
</s>
<s>
Variance	B-General_Concept
functions	I-General_Concept
play	O
a	O
very	O
important	O
role	O
in	O
parameter	O
estimation	O
and	O
inference	O
.	O
</s>
<s>
In	O
summary	O
,	O
to	O
ensure	O
efficient	O
inference	O
of	O
the	O
regression	O
parameters	O
and	O
the	O
regression	O
function	O
,	O
the	O
heteroscedasticity	B-General_Concept
must	O
be	O
accounted	O
for	O
.	O
</s>
<s>
Variance	B-General_Concept
functions	I-General_Concept
quantify	O
the	O
relationship	O
between	O
the	O
variance	O
and	O
the	O
mean	O
of	O
the	O
observed	O
data	O
and	O
hence	O
play	O
a	O
significant	O
role	O
in	O
regression	O
estimation	O
and	O
inference	O
.	O
</s>
<s>
The	O
variance	B-General_Concept
function	I-General_Concept
and	O
its	O
applications	O
come	O
up	O
in	O
many	O
areas	O
of	O
statistical	O
analysis	O
.	O
</s>
<s>
When	O
a	O
member	O
of	O
the	O
exponential	O
family	O
has	O
been	O
specified	O
,	O
the	O
variance	B-General_Concept
function	I-General_Concept
can	O
easily	O
be	O
derived	O
.	O
</s>
<s>
The	O
general	O
form	O
of	O
the	O
variance	B-General_Concept
function	I-General_Concept
is	O
presented	O
under	O
the	O
exponential	O
family	O
context	O
,	O
as	O
well	O
as	O
specific	O
forms	O
for	O
Normal	O
,	O
Bernoulli	O
,	O
Poisson	O
,	O
and	O
Gamma	O
.	O
</s>
<s>
In	O
addition	O
,	O
we	O
describe	O
the	O
applications	O
and	O
use	O
of	O
variance	B-General_Concept
functions	I-General_Concept
in	O
maximum	O
likelihood	O
estimation	O
and	O
quasi-likelihood	O
estimation	O
.	O
</s>
<s>
We	O
use	O
the	O
Bartlett	O
's	O
Identities	O
to	O
derive	O
a	O
general	O
expression	O
for	O
the	O
variance	B-General_Concept
function	I-General_Concept
.	O
</s>
<s>
We	O
derive	O
the	O
variance	B-General_Concept
function	I-General_Concept
for	O
a	O
few	O
common	O
distributions	O
.	O
</s>
<s>
The	O
Normal	O
distribution	O
is	O
a	O
special	O
case	O
where	O
the	O
variance	B-General_Concept
function	I-General_Concept
is	O
a	O
constant	O
.	O
</s>
<s>
To	O
calculate	O
the	O
variance	B-General_Concept
function	I-General_Concept
,	O
we	O
first	O
express	O
as	O
a	O
function	O
of	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
variance	B-General_Concept
function	I-General_Concept
is	O
constant	O
.	O
</s>
<s>
A	O
very	O
important	O
application	O
of	O
the	O
variance	B-General_Concept
function	I-General_Concept
is	O
its	O
use	O
in	O
parameter	O
estimation	O
and	O
inference	O
when	O
the	O
response	O
variable	O
is	O
of	O
the	O
required	O
exponential	O
family	O
form	O
as	O
well	O
as	O
in	O
some	O
cases	O
when	O
it	O
is	O
not	O
(	O
which	O
we	O
will	O
discuss	O
in	O
quasi-likelihood	O
)	O
.	O
</s>
<s>
Weighted	O
least	B-Algorithm
squares	I-Algorithm
(	O
WLS	O
)	O
is	O
a	O
special	O
case	O
of	O
generalized	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
As	O
in	O
regular	O
least	B-Algorithm
squares	I-Algorithm
,	O
the	O
goal	O
is	O
to	O
estimate	O
the	O
unknown	O
parameters	O
in	O
the	O
regression	O
function	O
by	O
finding	O
values	O
for	O
parameter	O
estimates	O
that	O
minimize	O
the	O
sum	O
of	O
the	O
squared	O
deviations	O
between	O
the	O
observed	O
responses	O
and	O
the	O
functional	O
portion	O
of	O
the	O
model	O
.	O
</s>
<s>
While	O
WLS	O
assumes	O
independence	O
of	O
observations	O
it	O
does	O
not	O
assume	O
equal	O
variance	O
and	O
is	O
therefore	O
a	O
solution	O
for	O
parameter	O
estimation	O
in	O
the	O
presence	O
of	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
where	O
are	O
defined	O
in	O
the	O
previous	O
section	O
,	O
it	O
allows	O
for	O
iteratively	B-Algorithm
reweighted	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
(	O
IRLS	B-Algorithm
)	O
estimation	O
of	O
the	O
parameters	O
.	O
</s>
<s>
See	O
the	O
section	O
on	O
iteratively	B-Algorithm
reweighted	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
for	O
more	O
derivation	O
and	O
information	O
.	O
</s>
<s>
Because	O
most	O
features	O
of	O
GLMs	O
only	O
depend	O
on	O
the	O
first	O
two	O
moments	O
of	O
the	O
distribution	O
,	O
rather	O
than	O
the	O
entire	O
distribution	O
,	O
the	O
quasi-likelihood	O
can	O
be	O
developed	O
by	O
just	O
specifying	O
a	O
link	O
function	O
and	O
a	O
variance	B-General_Concept
function	I-General_Concept
.	O
</s>
<s>
With	O
a	O
specified	O
variance	B-General_Concept
function	I-General_Concept
and	O
link	O
function	O
we	O
can	O
develop	O
,	O
as	O
alternatives	O
to	O
the	O
log-likelihood	O
function	O
,	O
the	O
score	O
function	O
,	O
and	O
the	O
Fisher	O
information	O
,	O
a	O
quasi-likelihood	O
,	O
a	O
quasi-score	O
,	O
and	O
the	O
quasi-information	O
.	O
</s>
<s>
Obtaining	O
the	O
score	O
function	O
and	O
the	O
information	O
of	O
allows	O
for	O
parameter	O
estimation	O
and	O
inference	O
in	O
a	O
similar	O
manner	O
as	O
described	O
in	O
Application	O
–	O
weighted	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
There	O
are	O
many	O
forms	O
of	O
non-parametric	O
smoothing	B-Application
methods	O
to	O
help	O
estimate	O
the	O
function	O
.	O
</s>
<s>
An	O
interesting	O
approach	O
is	O
to	O
also	O
look	O
at	O
a	O
non-parametric	O
variance	B-General_Concept
function	I-General_Concept
,	O
.	O
</s>
<s>
A	O
non-parametric	O
variance	B-General_Concept
function	I-General_Concept
allows	O
one	O
to	O
look	O
at	O
the	O
mean	O
function	O
as	O
it	O
relates	O
to	O
the	O
variance	B-General_Concept
function	I-General_Concept
and	O
notice	O
patterns	O
in	O
the	O
data	O
.	O
</s>
<s>
An	O
initial	O
scatter	O
plot	O
of	O
the	O
data	O
indicates	O
that	O
there	O
is	O
heteroscedasticity	B-General_Concept
in	O
the	O
data	O
as	O
the	O
variance	O
is	O
not	O
constant	O
at	O
each	O
level	O
of	O
the	O
predictor	O
.	O
</s>
<s>
One	O
can	O
estimate	O
and	O
using	O
a	O
general	O
smoothing	B-Application
method	O
.	O
</s>
<s>
The	O
plot	O
of	O
the	O
non-parametric	O
smoothed	B-Application
variance	B-General_Concept
function	I-General_Concept
can	O
give	O
the	O
researcher	O
an	O
idea	O
of	O
the	O
relationship	O
between	O
the	O
variance	O
and	O
the	O
mean	O
.	O
</s>
<s>
As	O
we	O
saw	O
above	O
,	O
the	O
Gamma	O
variance	B-General_Concept
function	I-General_Concept
is	O
quadratic	O
in	O
the	O
mean	O
.	O
</s>
