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8/10/2019 Maximum Likelihood Estimation of the.pdf
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American Society for Quality
Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their BiasAuthor(s): S. C. Choi and R. WetteReviewed work(s):Source: Technometrics, Vol. 11, No. 4 (Nov., 1969), pp. 683-690Published by: American Statistical Associationand American Society for QualityStable URL: http://www.jstor.org/stable/1266892.
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VOL.
11,
No. 4
Maximum Likelihood Estimation
of
the
Parameters
of
the
Gamma
Distribution
and
Their
Bias
S. C. CHOI AND
R. WETTE
Washington
University,
St. Louis
The
numerical
echnique
of the
maximum ikelihood
method
to
estimate the
param-
eters
of Gamma
distribution
is
examined.
A
convenient table
is obtained to
facilitate
the
maximum
likelihood estimation
of the
parameters
and the
estimates of the
var-
iance-covariance
matrix.
The
bias of the
estimates
is
investigated
numerically.
The
empirical
result indicates that the
bias
of both
parameter
estimates
produced
by
the
maximum likelihood
method is
positive.
1.
INTRODUCTION
Several authors
have
considered
the
problem
of
estimating
the
parameters
of the Gamma distribution. Fisher [5] showed that the method of moments may
be
inefficient
for
estimating
the
parameters
of
Pearson
type
III
distributions
and
suggested
use of
the
maximum
likelihood
(M.L.)
method. For
example,
it
has
been
shown
[9]
that
the
efficiency
of
the
estimated
shape
parameter
of
a
Gamma
distribution
by
the
method of
moments
may
be
as
low as
22
percent.
Chapman
[2],
Des
Raj
[4]
Stacy
et
al.
[12]
and
Harter et
al.
[7]
have
applied
the
M.L.
principle
to
study
the Gamma
parameters.
Estimation
by
the
method
of
moments
has
been
considered
by
Cohen
[3].
In
this
paper
we
examine
two
numerical
methods
to obtain
the
M.L. esti-
mates of the parameters of the Gamma distribution. Both methods can be con-
veniently
employed
by
the use
of
an
electronic
computer.
A
table
for
the
M.L.
estimates of
the
parameters
was
given
by
Greenwood
et al.
[6].
A
new
table
to
facilitate
M.L.
estimation
of
the
parameters
and
also
to
estimate their
variances
and
covariance is
presented.
2.
MATHEMATICAL
FORMULATION
The
probability
density
function of
the
random
variable T
having
a
Gamma
distribution with parameters Xand
,u
(the latter one called shape parameter ) is
f
(t;
X
)
=
-r)
t~-1
exp(--Xt)
X,
,
>
0,
t
>
.
(1)
Let
ti,
t2,
**
*,
tn(n
>
1)
represent
a
random
sample
of
values of
T. If
L
de-
Received
Jan.
1968
683
TECH
NOMETRICS
NOVEMBER969
8/10/2019 Maximum Likelihood Estimation of the.pdf
3/9
notes the
log
likelihood
function,
then
L
=n{,u
log
X
-
log
r()}
+
(
-
1)
log
ti
-
X
E
ti.
(2)
i-i
i-1
For
simplicity
of notation
let
m('L,
X)
=
aL/a
and
g(,i, X)
=
aL/x.
From
(2)
we
have:
m(,u,
X)
=
n{log
X
-
,I()}
+
E
log
t,
(3)
i-1
and
g(,
X)
=
n(g/X)
-
E
t,
,
(4)
i-1
where
d
()
=
-
log
r(u).
Analytically
closed
expressions
for
the
M.L.
estimators
cannot
be
obtained
and
the
solution
of
m(A,
[)
=
0 and
g(a,
S)
=
0
yielding
the
parameter
estimates
j,
aP
equires
numerical
techniques.
3. NUMERICALSOLUTIONS
3.1
Newton-Raphson
Method
Let
A
and
[
denote
the
M.L. estimators
of
,u
and
X.
Simultaneous
solution
of
the
system
of
equations,
m(ta, X)
=
0 and
g(Af,
X)
=
0 in terms
of
Pi
yields
the
implicit
equation
log
,
-
,)
=
log
I
--
w,
(5)
where
1
=
E
t,/n
(6)
*-1
and
=
(
log
t/n.
(7)
It
is
noted
that
once
P
is
determined,
X is estimated
by
X
=
A/l.
(8)
For
simplicity
of
notation
let
M
=
log
-
w. (9)
The
Newton-Raphson
iteration
method
then
gives
k lk-1
log
p,-l
-
9k-)
-
M
(10)
-
1-Ak
1
-
'
1(k-1)'
684
S. C.
CHOI AND
R.
WETTE
8/10/2019 Maximum Likelihood Estimation of the.pdf
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ESTIMATION
F
THE
PARAMETERSF
THEGAMMADISTRIBUTION
where
Ik
denotes the
kth
estimate
starting
with
initial trial value
Po
and
V'(()
represents
dJ(M)/dd.
The
functions
'(g)
and
V'(j,)
are tabulated
in
Pairman
[10]
in
the
form
of
the
Digamma
and
Trigamma
functions
and
can
be
expressed
in power series as
co
()
-
-
+
E {i(i
+
)}1
(11)
i-1
and
'I )
= (i
+
)-2
(12)
i-0
where
y
=
Euler's
Constant
=
0.5772157
**
.
Using
Bernoulli series
expansions
of
(11)
and
(12), (see
Jordan
[8]) very
good
approximations
of
these
functions
can
be obtained as
I(,u)
-
log,
-
{1
+
[1-
(1/10
-
1/(21/2))//2]/(6Qz)}/(2u)
(13)
and
/'(u)
~
{1
+
{1
+
[1-
(1/5-
l/(7?2))/,2]/(3A)}/2,u)}/
(14)
if X
>
8. For X
1.
Secondly,
both
numerator and
denominator
of
the last term in
(10)
are
monotone
functions of
,u.
Thus
convergence
of
the
method is
assured
and
the
process
has
second
order
convergence.
3.2
M.L.
Scoring
Method
The
M.L.
scoring
method is
again
a
Newton's
iteration
technique
and
the
concept
is
discussed in
[11].
Although
the
method of
Section
3.1
would
require
less
computation
than
this
technique,
the
M.L.
scoring
method is
appealing
because it
provides
a
statistical
criterion
for
stopping
the
iterations.
First
we
evaluate
685
8/10/2019 Maximum Likelihood Estimation of the.pdf
5/9
S.
C.
CHOI AND R.
WETTE
82L
-
-
q'((19)
02L
2
d
X
=
-n/X, (20)
and
-
L-
=
n/X2.
(21)
Let
D
=
n(t'(.)
-
1).
(22)
The
asymptotic
variances and covariance
of
,u
and
X
are
derived
as
var (,)
=
.//D,
(23)
cov
(jA)
=
X/D,
(24)
and
var (
2)
=
X2I'(g)/D.
(25)
In
large
samples
it
is
appropriate
to
replace
,u
and
X
of
(23-25)
by
their M.L.
estimates
to
obtain
the variances
and covariances.
Now,
the
M.L.
scoring
method
is given by the following
iteration
scheme
/-k
A=
-k-
+
vark-l(A)
COVk-l(A)
mk-l,(A,
)
(26)
J_k-
_Ak-I
_coVk-l(A)
vark-._()
__
9k-1(A,
)-
where
mk_lC(Q,
^),
gk-1(,
i),
vark_l(y),
covk_l()
and
vark-l(X)
are obtained
from
(3),
(4), (23), (24),
and
(25)
respectively
by
replacing
the
parameters
by
their
k-lst estimates.
It should
be
noted
that the
M.L.
scoring
method
would
be undefined
if D
defined
in
(22)
were
identically
equal
to zero
for some 0