ESTIMATION OF PARAMETERS OF GENERALIZED BETA OF THE SECOND KIND (GB2) DISTRIBUTION BY MAXIMUM LIKELIHOOD ESTIMATION (MLE) AND NEWTON-RAPHSON ITERATION
Far East Journal of Mathematical Sciences (FJMS) © 2018 Pushpa Publishing House, Allahabad, India http://www.pphmj.com http://dx.doi.org/10.17654/MS105010001 Volume 105, Number 1, 2018, Pages 1-20
ISSN: 0972-0871
ESTIMATION OF PARAMETERS OF GENERALIZED
BETA OF THE SECOND KIND (GB2) DISTRIBUTION BY
MAXIMUM LIKELIHOOD ESTIMATION (MLE)
AND NEWTON-RAPHSON ITERATION
Warsono, Waryoto, Mustofa Usman and Wamiliana Department of Mathematics University of Lampung Indonesia
Abstract
The aim of this study is to derive the estimates of the parameters
of generalized beta of the second kind (GB2) distribution by using
the maximum likelihood estimation (MLE). Due to the difficulty
in finding the analytical solution by MLE approach, the estimate
is determined numerically by using iteration and Newton-Raphson
methods. The Newton-Raphson method is used to estimate
parameters, to find a confidence interval, to estimate the bias, and to
estimate the variance for some difference sample size configuration
n viz. for 20 , 30 , 50 , 100 and 500. The estimation of the parameters a
, b, p and q attains the values close to the real value of the
parameters. If the size of the sample increases, the confidence interval
becomes narrower and the bias and variance become smaller.Received: September 19, 2017; Revised: January 17, 2018; Accepted: March 17, 2018
2 Warsono, Waryoto, Mustofa Usman and Wamiliana
1. Introduction
Many papers which have discussed the distribution of income used betadistribution [1]; gamma distribution [2-5]; and Weibull model [6]. The
generalized beta distribution of the second kind (GB2) is a very flexible
four-parameter distribution. It is used a lot to analyze income distribution.
References [7] and [8] suggested the generalized beta of the second kind
(GB2) as a model for the size distribution of income and indicators
of poverty. It captures the characteristics of income distribution including
skewness, peakness in low-middle range, and long right hand tail. This
distribution therefore provides good description of income distribution
[7, 9-11]. GB2 is used in mathematical economic, in insurance company,
health science and in industry. Although a large number of functional forms
have been proposed, the four-parameter generalized beta of the second kind
(GB2) model is now widely acknowledged to give an excellent description
of income distributions, providing the goodness-of-fit with relative
parsimony, while also including many other models as special or limiting
cases [7, 12-15]. [16] addressed issues of time-inconsistency in top-coded
US Current Population Survey earnings data by fitting GB2 distributions
that account for top-coding, and derive a consistent time series of Gini
coefficients from the estimates. In [17], the model of optimization of the
behavior predicts that the earnings distribution has the GB2 shape.A random variable has a distribution of generalized beta of the second
kind (GB2) with parameter a , b , p , q if the probability density function is
of the form: ap 1 ax f x , x ; a , b , p , q . (1)
p q a
ap x
b B p , q 1
b
Parameter b is a scale parameter, a, p and q are each shape parameter, p q
B p , q is the beta function, and is the gamma
p q
Estimation of Parameters of Generalized Beta …
n i q p a i n nap n i ap i n b x
B q p b x a (2) Applying the natural logarithm to equation (2) above, we have
n q p a i n nap n i ap i n x x a
L x q p b a
1
1
3 q p a k q a k p b k
and exists only if .
aq k ap Tail behavior of
the distribution depends on ap (lower tail) and aq (upper tail), with larger
values of a reducing the density at both tails, and the relative sizes of p and q
affecting skewness [10, 18].The aim of this study is to estimate the parameters of the GB2
distribution by using the maximum likelihood estimation (MLE) method and
then the simulation by using Software R is used to estimate the parameters
for some difference sample size configuration, namely for the sample sizes:
20, 30, 50 100 and 500.
2. The Estimation
The estimation of parameters GB2 by MLE To estimate the parameter by using the maximum likelihood estimation (MLE) method, first we define the likelihood function as follows:
1
n i n i q p a i ap ap i i b x
B q p b x a L x f x q p b a
1
1
1
1 , , , ,
. 1 ,
1
1
, ln , , ln
Warsono, Waryoto, Mustofa Usman and Wamiliana
1
n i n i a i i b x
q p
a b p n x p a n1
ˆ
ˆ ln ˆ
1 ln ˆ ln
,
a L x q p b a
L
The first derivative with respect to a is set equal to zero,n i n i a i i a i i x b x b x q p b p n x p a n
ˆ
ˆ ln ˆ
ˆ ˆ ˆ ln
ˆ ln ˆ
1 ˆ ˆ .
1
, , , , ln
Next, we set the first derivatives with respect to the parameters of interest equal to zero as follows:
. ln 4
where
, 1 ln (3)
1
n
i
a i b x q p
B q p
1 ln
1 , ln 1 ln
B q p n nap x ap a n
n i i
, , q p q p
q p B , is the beta function and p is the gamma function, then equation (3) can be written as:
1 ln (4)
1 .
n
i
a i b x q p
ln ln
q p n q n
ln ln ln , 1 ln , , ln
1
L p n b nap x ap a n x q p b a
n i i
(5)
Estimation of Parameters of Generalized Beta …
(6) The first derivative with respect to p is set equal to zero,
p L x q p b a
, , , , ln
ˆ ˆ
ˆ ˆ ˆ , , , ln
1 ˆ ˆ ˆ
ˆ
1 ˆ .
2
1 ˆ
L x q p b a
n i a i a i i b x b b x x a
q p
b p n a b
ˆ ln ˆ
p q p q p q p q p q p where p
ˆ ˆ ln
, , ˆ ˆ
ln
,
(7) Since p p p p p
ˆ ˆ ln
ˆ ˆ ˆ
1 ˆ ˆ
ˆ
1 .
1 ˆ
n i a i n i i b x n q p q p n p p n b n a x a
5 The first derivative with respect to b is set equal to zero,
1 ˆ
, ˆ
Let
n
i
a i b x b q p b p n a
, 1 ln ˆ
1
ˆ ˆ ˆ
1 ln ˆ ˆ
.
b L x q p b a
, , , , ln
ˆ ˆ
2
S
a i a
i
i a i bx
bb
x
x a b s b x
1 ˆ
is a function psi (digamma).
6 Equation (7) can be written as follows:
(9) Equations (5), (6), (8) and (9) are very difficult to be solved by
analytical method. Thus, the Newton-Raphson method will be used to
estimate the parameters a, b, p and q. To estimate the parameters a, b, p and
q by using Newton-Raphson method, first we find the gradient vector and the
first derivative vector from the logarithm function with respect to a, b, p and
q and define g as follows:n
i
a i b x q p q p n q q n
n i a i b x q p n q n
1 ˆ .
ˆ 1 ln ˆ ˆ ˆ
. , , , ln , , , ln , , , ln , , , ln
, , , ln
q L x q p b a p L x q p b a b L x q p b a a L x q p b a
L x q p b a g (10) The Hessian matrix which was found from the second derivative of the
logarithm function with respect to the parameters a, b, p and q and is defined
Warsono, Waryoto, Mustofa Usman and Wamiliana
ˆ
n i a i n i i b x q p n p n b n a x a
1 ˆ
1 .
ˆ 1 ln ˆ ˆ ˆ
ˆ ln ˆ ln
(8) The first derivative with respect to q is set equal to zero,
, , , , ln
p L x q p b a
, ˆ 1 ln
ˆ ˆ ˆ ˆ
ˆ ˆ
1 ˆ
Estimation of Parameters of Generalized Beta …
ˆ ˆ ˆ
2
2
ˆ 2 ˆ
2 ˆ
2 ˆ
1
1 ˆ ˆ ln
ˆ ˆ
1 ˆ ln
, ˆ
n i a i i a i b x b x b x a q p a n
7
2
1
ˆ ˆ
The iteration process by using the Newton-Raphson method is: ,
(11)
q q L x q p b a a q L x q p b a q a L x q p b a a a L x q p b a
2
2
2
. , , , ln , , , ln , , , ln , , , ln
2
, , , ln
L x q p b a H
1
H g i i where . ... , 3 ,
a a L x q p b a ,
2
1
ˆ
1 ln ˆ ˆ
2 , 1 i (12) To find the gradient vector
2
, , , , ln
The second derivative of equation (5) with respect to a is
H by using the second derivative of the logarithm function with respect to the respective parameters.
Hessian matrix ,
, g we use the first derivative from
respective parameters. From equations (5), (6), (8) and (9), we find the
n i a i i a i a i a i i b x b x b x b x b x b x q p a n
Warsono, Waryoto, Mustofa Usman and Wamiliana
S Then
a i i a i b x b x b x
1 ln
Let .
1 ˆ ˆ ˆ ln
2 ˆ b b x b x
x a
b b x b i a i i a i1 ˆ
2
ˆ ˆ
ˆ ˆ
ˆ ln ˆ
. ˆ
2 ˆ ˆ ˆ
1 ln
a i a i i a i
a
i i a i x b x b b x b x b x b x b x b b
a a i i a i b x a b x b x b b y
8
2
2
2
ˆ ˆ
2 ˆ
2 ˆ
1
1 ˆ ˆ ˆ ˆ ln ˆ
1 ˆ ln ˆ ˆ
. ˆ
1
ˆ 1 ˆ ˆ ˆ ˆ ˆ
ln
ˆ ˆThen
b x b x y i a i
Let . ln
n i a i i a i b x b x b x
b
q p b p n
1
ˆ
1 ln ˆ ˆ ˆ
b a L x q p b a .
2
, , , , ln
(13) The second derivative of equation (5) with respect to b is
n i a i i a i a i a i i b x b x b x b x b x b x q p a n
S
Estimation of Parameters of Generalized Beta …
ˆ 1 ˆ
2
2 ˆ
ˆ ln ˆ
ˆ ˆ
1 ˆ
ˆ
ˆ ˆ .
ˆ ˆ ˆ
ˆ ln ˆ
ˆ ˆ
1 ˆ
ˆ
2 ˆ
2
, , , , ln
(15) The second derivative of equation (5) with respect to q is
ˆ ˆ ln ln
1 ˆ ln
ˆ
1
ˆ ˆ .1
n i
n
i
a i i a i i b x b x b x b n x
a i a i i i a i b x x b b x x a b x b x
p a L x q p b a
2
, , , , ln
(14) The second derivative of equation (5) with respect to p is
1
ˆ
1 ˆ
9
x b b x x a b x b x a i i i a i
ˆ ˆ
ˆ ˆ ln
1 ˆ
ˆ
1 ˆ
2 ˆ
a i a i i a i i b x b b x b b x b x x a ˆ
n i a i a i i
a
i i b x b b x b b x b x x a q p b p n
ˆ
ˆ ˆ ln
ˆ ˆ
ˆ
1
ˆ2
q a L x q p b a
10 Warsono, Waryoto, Mustofa Usman and Wamiliana
a
ˆ
x x
i i ln
ˆ ˆ
n
b b
. (16)
ˆ
a
1 i
x
i 1
ˆ b
The second derivative of equation (6) with respect to a is
2
ln L a , b , p , q x ,
b a
1 x
a
i
n ax
i ˆ b n p
ˆ ˆ
p q .
ˆ a
a b x
2 i i
1 b
1 b
a
1 x
i Let w ax . Then i b
ˆ ˆ a 1 a
1 ˆ x x x
w n p i i i ˆ x a x ln i i
ˆ ˆ ˆ ˆ a b b b b
ˆ ˆ
a
1 a 1
x x x
i i i ˆ
n x a x ln
i i ˆ ˆ ˆ
1 b b b
ˆ ˆ
p q
ˆ
2
2 ˆ a
b
i 1 x
i
1
ˆ
b
ˆ ˆ
a
1 a
x x x
i i i ˆ
ˆ i a x ln
a ˆ ˆ ˆ
b b b
x
i
1 ,
ˆ
2
ˆ
a
b
x
i 1
ˆ
b
Estimation of Parameters of Generalized Beta … ˆ ˆ
11 a 1 a
1 x x x
i i i ˆ a x ln x i i
ˆ ˆ ˆ ˆ n n p
b b b ˆ ˆ p q
ˆ ˆ ˆ a a
1 i
b x x
ˆ 2 ˆ
2 i i b 1 b 1
ˆ ˆ
b b
ˆ
2 a
1 x x
i i
ˆ ln a x i
ˆ ˆ
b b . (17)
ˆ
2 a
x
i 1
ˆ b
The second derivative of equation (6) with respect to b is
2
ln L a , b , p , q x
, b b
1 x
a
i n ax i
ˆ ˆ a n p b
ˆ ˆ p q .
ˆ 2 a b
b x
2
i i 1
b
1 b
a
1 x
i ax i b
Let w . Then a
x 2 i b
1
b
ˆ
a 1 a
x x
ˆ
2
i i ˆ
a x b
1
i
ˆ
b b
b
ˆ
a 1 a
x x
i
2 i
b 1
ˆ
b b
b
w
2 b
a
x
ˆ
4 i b
1
ˆ
b
Warsono, Waryoto, Mustofa Usman and Wamiliana
12
1 ˆ ˆ
2 ˆ
1 ˆ ˆ ˆ
ˆ
1
ˆ
ˆ ˆ
2
ˆ 1 ˆˆ
4
2 ˆ
ˆ 1 ˆ
3
a
i i ai
a i i b x b xb
x
a b x x a 2 ˆ
4
2 ˆ
2
2
2 ˆ
1
ˆ ˆ ˆ1 ˆ ˆ ˆ b p n a b x b b x x a b x b b x
x a a
b x b b x x a a i a i i a i a i i a i a i i
2 ˆ
1
ˆ
ˆ
1 ˆ ˆ ˆ
4
a i a i i a i a i i a i a i i b x b b x x a b x
b
b x x a b x b b x x a a
2 ˆ
3
ˆ 1 ˆ
2 ˆ
ˆ
2
2 ˆ
4
2 ˆ
2
2
2 ˆ ˆ ˆ
1 ˆ
a i a i b x b b x b b
2
ˆ4 ˆ
1 ˆ ˆ
2 ˆ
1
2 ˆ
1 ˆ
1 ˆ ˆ
4
2
ˆ
1 ˆ
ˆ ˆ
1 ˆ ˆ