Maximum Likelihood Estimation of the.pdf

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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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2/9

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


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


4/9

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


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

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