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Transmuted Sine-Dagum Distribution and its Properties
K.M. Sakthivel and K. Dhivakar •
Department of Statistics, Bharathiar University, Coimbatore - 641 046, Tamil Nadu, India.
Abstract
In this paper, we introduce a new four parameters continuous probability distribution called transmuted sine-Dagum distribution obtained through the transmuted Sine-G family introduced by Sakthivel et al. [13]. We have obtained some distributional properties including moments, inverted moments, incomplete moments, central moments and order statistics for proposed model. The reliability measures such as reliability function, hazard rate function, reversed hazard rate function, cumulative distribution function, mean waiting time and mean residual life time are studied in this paper. Further, we have discussed some income inequality measures including Lorenz curve, Bonferroni index and Zenga index.
The maximum likelihood method is used to estimate the parameters of the proposed probability distribution. Finally, we demonstrated goodness of fit the proposed model with other suitable models in the literature using real life data sets.
Keywords: Dagum distribution, Sine-G family, Reliability function, Order statistics, Lorenz curve, Maximum likelihood method.
I. Introduction
The lifetime model is playing a vital role in different fields such as life sciences, biological sciences, environmental sciences, medicine, finance and actuarial science. The last three decades, the development and applications of new probability distributions for lifetime data are remarkable in the literature. In this information era, the data generated from different fields are voluminous and dynamic in nature. Therefore, the need for generating new family of probability distributions is inevitable. As a result, the generating new family of probability distribution has attracted many researchers. The main advantage of generating new family of probability distributions is provide better flexibility and better fit at the cost of one or more additional parameters. The following are a list of few well-known generating new family of probability distributions: exponential family is introduced by Gupta et al. [6], Marshall-Olkin family is introduced by Marshal and Olkin [9], transmuted family is introduced by Shaw and Bucklay [14], Kumaraswamy-G family is introduced by Cordeiro and Castro [3], Topp-Leone family is introduced by Al-Shomrani et al. [1], Power Lindley-G family is introduced by Hassan and Nassr [7] and gamma-G family is introduced by Cordeiro et al. [4], to mention a few.
Dagum distribution is introduced by Camilo Dagum [5] in the year 1977 for modeling income data. It has been extensively used in various fields including wealth data, reliability analysis, survival analysis and meteorological data. The Dagum distribution is an alternative to log-normal, Pareto and generalized beta distributions. This distribution is also called Burr-XII distribution, particularly in the actuarial literature.
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A continuous random variable X is said to have Dagum distribution with three parameters а, в and в if its probability density function and cumulative distribution function are given respectively as
f (x; а, в, в) = авв х-в-1 (l + ах—) в 1; x > 0, а > 0, в > 0 and в > 0. (1)
and
F(x; а, в,в) = (1 + ах-в) в; х > 0, а > 0, в > 0 and в > 0. (2)
where а is scale parameter, while в and в are shape parameters. It is to be noted that if а=1 the Dagum distribution becomes Burr III distribution and if в=1, the Dagum distribution becomes log-logistic or Fisk distribution.
In this paper, we introduce a new four parameter continious probability distribution namely transmuted Sine-Dagum distribution. This generating probability distribution provides better fit and flexibility for real life problem.
This paper is organized as follows: In Section 1, a brief introduction and need for the generating family of distributions is given. In Section 2, we present the transmuted Sine-G family and the proposed probability distribution namely transmuted Sine-Dagum distribution. In Section 3, we discuss some reliability measures like reliability function, hazard rate function, reversed hazard rate function, cumulative hazard function, mean waiting time, mean residual life function and mean past life time. In Section 4, we present some distributional properties including moments, inverted moments, incomplete moments, central moments and order statistics. The income inequality measures are discussed in Section 5. The method of maximum likelihood estimation is presented in Section 6. In Section 7, the real time application is presented. Finally, the concluding remarks are presented in Section 8.
II. Transmuted Sine-G Family
Transmuted Sine-G family is introduced by Sakthivel and Rajkumar [13]. This transmuted Sine-G family is the mixture of Sine function and quadratic rank transmuted map. The probability density function of transmuted Sine-G family of distributions is given by
f (x; Я) = Пh(x)cos ПH(x) j (1 + A) - 2Asin ^ПH(x)
; x > 0, A > 0.
and the corresponding cumulative distribution function is given by
F(x; A) = (1 + A)sin ПH(x)j - A sin ^ПH(x)j
n2
; x > 0, A > 0.
(3)
(4)
where, A is the parameter of transmuted Sine-G family of distributions. If A = 0, the transmuted Sine-G family is becomes Sine-G family.
I. Transmuted Sine-Dagum Distribution
A continuous random variable X is said to be follow the transmuted Sine-Dagum distribution with parameters а, в, в and A, (i.e.,) X ~ TSD(X; а, в, в, A), then the probability density function of X is of the form
f (x; а, в, в, A) = Павв x-в-1 ^1 + аx-в j - cos ^П ^1 + аx-вЛJ ^
(1 + A) - 2A sin ^2 ^1 + аx в x > 0, а > 0, в > 0, в > 0 and - 1 < A < 1
п
(5)
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The above equation can be rewritten as
f (x;a,в,в, А) =(1 + A)Пx—в-1 ^1 + ax-^j ^ cos ^^2(1 + ax~e^j ^ — hnadfi x—в—1
-в \ — в
' ^ cos ( П (1 + ax в
sin | -2(1 + ax-
x 1 + a x
The cumulative distribution function is given by
(1 + A A) sin ^ П ^1 + ax—в j ^ — A ^s in П (
x > 0, a > 0, в > 0, в > 0 and — 1 < A < 1.
F (x; a, в, в, A) =
. п в \ — в
sin — 1 + ax
(6)
(7)
where в is scale parameter; a and в are shape parameters and A is the parameter of quadratic rank transmutation map.
of A.
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of a.
a.
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Figure 4: Cdfs of transmuted Sine-Dagum distribution for fixed values of a = 4,0 = 2, A = 0.7 and different values of в.
Figure 5: Reliability function of transmuted Sine-Dagum distribution for fixed values of a = 3.5,0 = 2.2, A = 0.8 and different values of ft.
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III. Reliability Measures
In this Section, we deal with some reliability measures for transmuted Sine-Dagum distribution. If X^TSD (X;u,0,j8,A), then the reliability measures of random variable X are given by;
I. Reliability function
The reliability function is defined by
R(x; а, в, в, Л) = 1 - F(x; а, в, в, Л)
= 1 -
(1 + Л)sin ^ у (1 + ах в^ ^ - Л in -2 ^1 + ах в
II. Hazard rate function
The hazard rate function is defined by
h(x, а, в, в, Л)
f (x, а, в, в, Л)
1 - F(x, а, в, в, Л)
(8)
2авв x в 1 (1 + аx в) в 1 cos ^2 (1 + аx в) ^ (1 + Л) - 2Л sin 2 (1 + аx в)
1
(1 + Л^in (f (1 + аx-в) ^ - Л [sin2 (1 + аx-в)
(9)
2
III. Reversed hazard rate function
The reversed hazard rate function is given by
r(x, а, в, в, Л)
2авв x в 1 (1 + аx в) в 1 cos (2 (1 + аx в) в j (1 + Л) - 2Лsin (2 (1 + аx в)
(1 + Л)sin ^f (1 + аx- в) в j - Л sin2 (1 + аx-в)
(10)
IV. Cumulative hazard function
The cumulative hazard function is defined by
H(x,а,в,в,Л) = -log R(x,а,в,в,Л)
f (x, а, в, в, Л) F(x, а, в, в, Л)
= -log 1
(1 + Л)sin ^~2 (1 + аx в^ ^ - Л ^sin~2 (1 + аx в
(11)
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V. Mean waiting time
The mean waiting time is defined by
<p(x) = x -
F(x) Jo
sf (s)ds
<p(x) =x -
f(x) Г fавв s-e l1+as-")-1 cos(211+т~е) 0
x 1(1 + Я) - 2As in [ П (1 + as 9)
-в \-в
ds
Therefore, the mean waiting time of transmuted Sine-Dagum distribution is given by
Г
L-t\
(-1)n(П)2n r
n=0 (2n)!
(-1)n( f )2n+1
b (1 - 1,2en + в +1;y)
- Г“=о (-1()2n„(+f1))!n 2Ява1B (1 - \,4fin + 2в + 1; y)
=x
(1 + Я)sin [2 (1 + ax-e) ^ - Я [sinП (1 + ax-e)
VI. Mean residual life function
The mean residual life function is defined by
1
1 r“
^(x) = s(x) L xf(x)dx - x
/0“ x Пx-e-1 (1 + ^x-e) в 1 cos (П2 (1 + ax-e) в)
x f(1 + Я) - 2Яsin [П (1 + ax в) -
dx x
1
(1 + Я)sin [П (1 + ax-e) ^ - Я [sin2 (1 + ax-e)
(12)
Therefore, the mean residual life function of transmuted Sine-Dagum distribution is given by
Г
L-t\
(-1)n(2)2n Г
n=0 (2n)!
v-“ г„=0
(1 + Я) П£a1B (1 - \,2fin + в + ^
(2n+1)! Яв^5 B (1 - 0,4en + 2в + l)
(-1)n (f )2n+1
1
(1 + Я)sin [П (1 + ax-e) ^ - Я [sinП (1 + ax-e)
n— x.
(13)
VII. Mean past lifetime
The mean past lifetime of the component can be defined as
K(x) = E [x - X|X < x] =
fo F(t)dt F(x)
x
/px tf (t)dt
F(x)
/0x t f авр t в 1 (1 + at в) в 1 cos [ f (1 + at в) в j
K(x) = x - ^
x f(1 + Я) - 2Яsin [2 (1 + at в) -
dt
(1 + Я)s in [ 2 (1 + ax-e) ^ - Я [sin [ 2 (1 + ax-e)
1
2
2
2
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Therefore, the mean past life time is given by
Его
n=I
(1 + A) § fiae B (1 - 1,2^n + e + 1; y)
ЕГО=о (-1()2>П+1)!Я+1 A£a1B (l - 1 Авп + §e + 1;y)
(-i)n(П)2n Г
(2n)!
= X--F
(1 + A)sin § (l + ax-1) ^ — А {sin(l + ax-1) ^
2
(14)
Figure 6: Reliability function of transmuted Sine-Dagum distribution for fixed values of a = 1.4, f = 2.4, А = 0.7 and different values of 1.
Figure 7: Hazard rate function of transmuted Sine-Dagum distribution for fixed values of 1 = 4, f = 6, А = 0.5 and different values of a.
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IV. Distributional Properties
I. Moments
The rth moment of transmuted Sine-Dagum distribution of the random variable X is obtained as
\ir =J xr (1 + Я) Падв x-2-1 (1 + ux-2 j - cos ^П (1 + ux-2 j ^
-Япадв x-2-1 ^1 + ux-2 j - cos ^ПП (1 + ax—j ^ sin ^П (1 + ux-2 j -
Using the Taylor series of the sine and cosine functions for moments, we have
“ (-1)nx2n+l “ (-1)nx2n
sin x = > 2---Y—, cos x = > —
n> (2n +1)! , n==0 (2n)!
dx
Therefore, we have
cos
n
2
1 + ax д
в
TO
>
n=0
(-1)n(n)2n (1 + ax-d) (2n)!
2fin
sin ( П (1 + ax 2 j
дj-e\ “ (-1)n(%)2n+1 (1 + ax-d) 2ne в
>
n=0 (2n +1)!
Hence, the rth moment of transmuted Sine-Dagum distribution is given by
V-r = £
(-1)n (f)
n( n \2n _
n=0 (2n)!
(1 + Я)Пвад B (1 - д,2/in + в + Г)
(-1)n ( f )2n + 1
n=0 (2n + !)!
- Явад B (1 - - ,4вn + 2в + ^
We have obtained the mean and variance of this distribution as
(-1)n (f )2n
(1 + А)2ва'1 B (1 - 1,2вn + в + в)
V1 = £ ^
to ( I )n ( n )2n+1 1
- £ ( (2„ +1), ЯпвадB | 1 - в,4в» + 2в + д
TO
and
v(x)
to (__-\ ^nf 2n
£0 (2")!
( 1V(2Г 2(1 + я)2ва2B (1 - |,2ви + в + 2
to (__1)n ( n )2n+1
n>=0 (2n +1)!
2
2
2Япвад B 1 - -, 4вn + 2в + ^
д
д
>
n=0
(-1)n (f )2n
(2n)!
((1 + Я)f ва1 B (1 - 1^n + в + 1)
(-1)n (|)2n+1 1
2
n>=0 (2n +1)!
Аква2 B (1 - в, 4вn + 2в + в
д
TO
(15)
(16)
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The moment generating function of transmuted Sine-Dagum distribution is given by
\n( n\2n
(h + a)
tr
MX (t) = E r!
r=0
E
n=0
( 1Г(2^ ((1 + A)(1 - Г,2вп + в + 0
(2n)! (-1)n (f )2n+1
n=0 (2n + !)!
- E ( 1E(ЕЕ 2\пваЬB (1 - -,4en + 2в + ^
The characteristic function is given by
Фх (t) = E
TO (it)r
r=0
Г!
to
E
n=0
n
(-1)n(2)n ((1 + A)Пe.iB (1 - ',2в„ + в + Г)
(2n)!
“ (-1)n(П)2n+1 . , , ,
- E ( ,L(,2^ 2АвanB (1 - -,4en + 2в + ^
n=0 (2n + 1)!
and the cumulant generating function is given by
KX(t) = log
TO ,r
E -
L-1 r!
r=0
TO
TO
E
n=0 n
(-1)n (f )2n
(2n)!
j n r / r r \
(1 + A)-e^B (1 - -Q,2en + в + g)
TO (_ 1 )n(n )2n+1 ( r r)
- E ( (2n+ 1)! A^B (1 - ё^ + 2в + ё)
(17)
(18)
(19)
II. Inverted moments
The inverted moment is defined by
ц* = / x rf(x)dx
-TO
Thus, the inverted moment for this distribution is given by
-ё)-в-* (п Л , _-ё)-в
ц* =J x r (1 + A)П^в x ё 1 ^1 + ax ^ - cos ^П ^1 + ax ^ ^ - Акаёв x-
х (1 + ax-) - cos f П (1 + ax-) sin (П (1 + ax-)
ё-1
dx
TO
E
n=0
(-1)n (f)
n ( n\ 2n n
(2n)!
(1 + А) П2вarB (1 + ё^n + в - r)
TO (_ 1 )n (n )2n+1 . r r.
- E ( (jn +1)! Апва-rB (1 + ё^ + 2в - ё)
III. Incomplete moments
The rth incomplete moment is defined by
mr(x) =/ srf(s)ds
(20)
m
(x) =J sr Паёв s ё 1 (1 + as ё) ^ cos ^2^1 + as ё) ^- Апаёв s-х (1 + as -в-x) cos f ~2 (1 + as- ё) ^'j sin ГП (1 + as- ё) -
ё-1
ds
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Hence, the rth incomplete moments of transmuted Sine-Dagum distribution is given by
E
(-1)n (n)
n ( n\ 2n -
n=0 (2n)!
(1 + А)П fieriВ (1 - 0,2fiie + fi + 0;y)
- nE ^р^Г*2^0 B l1 - -o ■4in+2e+3 ^У
o
IV. Central moments
The central moment is defined by
V = I (x - *)rf (x)dx = E (L) (-1)m(^1)т*
J-то m=0 Vm/
Therefore, the central moments of transmuted Sine-Dagum distribution is given by
* = E (J(-1)
m=0
E (-1)” (2 )2n r ,S=0 (2n)!
(1 + A) 2В [1 - 1,20n + в + 1
то ( 1)n( n )2n+1 1
- E ( ,!(,2 L Апвае В ( 1 - o,4^n + 2в + o
n=0
(2n + 1)!
TO
(-1)n (2 )2n
_ n=0 то / 1 \n
(2n)!
r=mf^ r - m „ r - m\
(1 + A) -0 B --------3^-,2fin + e + —Q—J
E \ ч м
E (2i
~1W2 ) Anfia^B (1 - r-^m,4 fin + 2^ +
e (-1)n(2)2”+i
nE0 (2n + 1)!
r - m
0
V. Order statistics
The pdf of the jth order statistics for transmuted Sine-Dagum distribution is given by n!
f%,(x)
(j - 1)(n - j)!
2авв x-0-1 (1 + ax-0'}
-e-1 / П ( o\ -в
cos( - (1 + ax 0'
21j-1
x ^(1 + A) - 2Asin ^-2(1 + ax 0j -
(1 + A)sin ^2 (1 + ax-0j - A^sin ^у (1 + ax-0^
x 1 - ^(1 + A)sin ^2 (1 + ax-0 j ^ - A^sin^2 (1 + ax-0 j The pdf of the smallest order statistics X(1) is given by
2n
fX(1)(x) =n
1 - ^(1 + A)sin ^2 (1 + ax 0 j ^ - A (sin ^2 ^1 + ax 0 j ^ ^ 2a0j8 x-0-1 ^1 + ax-0^ - cos ^2 (1 + ax-0 j ^
2n
x |(1 + A) - 2Asin[ 2 (1 + ax 0
rm
m
(21)
(22)
1
(23)
-1
(24)
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The pdf of the largest order statistics X(n) is given by
/X(„)M =n
^(1 + A)sin ^П (1 + ax 0) - A^sinП (1 + ax 0 ) ^
-| n—1
- x 0 1 (1 + ax
1 (n Л -0)-в
cos 2 (^1 + ax 0'
f (x) _(2m + 1) /m+1:n (x)_ m!m!
x ^(1 + A) - 2Asin уП ^1 + ax 0j The pdf of the median order statistics is given by
(1 + A)sin ^ 2 ^1 + ax-e^j — A ^sin 2 ^1 + ax-0^j ^
x 1 — ^(1 + A)sin ^ 2 (1 + ax—0) — A ^sin 2 (1 + ax—0) ^
x 2a0e x—0—1 (1 + ax-0) - cos ^2 (1 + ax-0) ^ x |(1 + A) — 2Asin (~2 (1 + ax-'0j
(25)
(26)
V. Income inequality measures
In this section, we deal with different basic income inequality measures including Lorenz curve, Bonferroni index and Zenga index. The following measures are given below.
I. Lorenz curve
The lorenz curve was introduced by Lorenz [8] in the year 1905. it is widely used in economic and many other fields and its defined by
1 Г x
L(x) _ 7 s/(s)ds
7 Jo
Hence, the Lorenz curve of transmuted Sine-Dagum distribution is given by
L(x)_717s[2a0es-Ч1+as-0P1 cos(ft1+as-0 0
x Г(1 + A) - 2As inf 2 ^1 + as-'0j dx
(-1)n(2)2n г
Г1 _____________
rn_0 (2n)!
(1 + A)2ea-0B (1 - r,2en + в + 0;y)
Г“_о (-1)(n2(n+/12))!2ni12AearB (1 - 0,4fin + 2в + 0; y)
2n+1
Г
L-t\
(-1)n( § )
n( n\2n г
(1 + A)2ea0B (1 - 1,2en + в + ))
n_0 (2n)!
Г“_0 (-1)(n2(n+/12))!2n+12Aea1 B (1 - 1,4en + 2в + 1
(27)
II. Bonferroni index
The Bonferroni index was introduced by Bonferroni [2] in the year 1930. it is ratio of Lorenz curve and cumulative distribution function of the distribution. The Bonferroni index is defined as
B(x)
LM
F(x)
0
1 m
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Hence, the Bonferroni index of transmuted Sine-Dagum distribution is given by
B(x) = -
where
- = E
n ( n\ 2n -
(-1)n (П)
(2n)!
(1 + Я)П2(l - r,4^n + 2в + r;y)
n=0
to (_1)n ( n )2n+1
- E ( ,L(2in ЯпраЬB (1 - -,4en + 2в+ ^;y
n=0
(2n + 1)!
(-1)n (f)
n / n\2n г
(1 + Я) 2 B ^1 - 1,201 + в + 1)
n = E ^
to ( 1 )n ( n )2n+1 1
- E ( (2,1 + 1)! 2Яв"'B 11 - j■ 4в" + 2e + г
(1 + A)sin ^П ^1 + ax e j - Я (sinП ^1 + ax e j ^
TO
2
III. Zenga index
Zenga index is introduced by Zenga [15] in the year 1980. The Zenga index is defined by
where
Z
1 V(x)
^(X)
1 fX
V(x) = F(Xy Jo sf(s)ds
1 Г TO
'U = т=щ1 xf (x)dx
Therefore, we get
V (x)
E'
n=0
(-1)n(2)2n -(2n)!
(1 + Я)2ва-rB (1 - r,2en + в + r;y)
(_ 1)n( П )2n+1 r , .
- ETO=0 ( (Л2!), Яжва-eB (1 - j,4en + 2в + j; y)
(1 + Я)sin ^ 2 (1 + ax-e) - Я (sin ^ 2 (1 + ax-e)
2
V+x)
v-iTO
En=0
E
-
(-1)n (f)2n
(2n)!
(1 + Я)2ea1 B (1 - 1,2en + в + 1)
„=о (-1()2n„(++1)!n+ ЯжваъB (1 - 1,4en + 2в + 1)
1
(1 + Я)sin ^2 (1 + ax-e) в j - Я (sin (^ (1 + ax-e) в j
Hence, the Zenga index of transmuted Sine-Dagum distribution is given by
z = 1 - Y
-
2
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where
_ к (-1)”(f )2n 7=к (2”)!
j n r / Г Г \
(1 + A) 2 lia’B (1 - -,2/in + в + g;y)
- к A^B (1 - g ,4в”+2в+g; y)
1 - f(1 + A)sin ^П2 (1 + ax-0j ^ - A^sin ^П (1 + ax-0 j
to /__1 \n( n\2n
s = к ( (2 )
д к (2n)!
(1 + A)nfl„ 1
2
1
^ B ( 1 - g, 2ftn + в + g
(-1)n(2)2n+1 A fla 1
n=0 (2n + 1)!
- L4tr^~A^B ( 1 - -,4en + 2в + ^
(1 + A)sin ^ 2 (1 + ax 0 j ^ - A^sin^ 2 (1 + ax 0 j ^ ^
VI. Parameter Estimation
Let X1, X2, ... , xn be a random sample from the transmuted Sine-Dagum distribution then the likelihood function is given by
L(a0,в A;x)= П 7a0e x(i) Ч1 + aw) - 1 cos(li1 + a«) -)
i=1 L к /
x ^(1 + A) - 2Asin ^2 (1 + a--)0j ^
Hence, the log likelihood function is given by
jjg ___(
L(a,0,в, A; x) =n log — + n loga + n log9 + n loge + (-0 - 1) к logx(
2 i=1
i)
+ (-в - 1) к log (1 + ax-)0 j + к logcos (7 (1 + j ^
i=1 i=1 к /
+ к log cos ^(1 + A) - 2A sin ^2 (1 + ax-) j
The MLE of parameters a, 0, в and A are obtained from the following equations
d,ogL- 0, ^ = 0, ^ = 0 and ^ = 0
a
g
в
A
That is,
-0
ЭlogL _ n к ( в 1 )x(i) к1"
da = a + к (1 + ax-0) + к
sin (f (1 + ax-)0 j fв (1 + ax-j
cos (П2 (1 + ax-2 -
-0j в 1x-0
) x(i)
(28)
0 (29)
2
2
n
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and
MogLn + £(<№) + £ ( - 1)x—)logx(i)
дв в
i=1
1 0+cx—>)
д\ —в\ ( дЧ — в — 1
+ £
(тт (1 + cx ^^ в (1 + — 1 cx—)log V(i)
i=1
cos ( 2 (1 + Cx—)
£
2Aco^T2 (1 + cx—) ^ e (1 + cx—) — ( о-Х(—в|ogV(i)
i=1 cos ^(1 + Я) — 2A sin 1 ^1 + cx^ ^ ^
0 (30)
dlogL n П1 (л -в) — e
~W =J - £ l°g l1+сх— ^
i=1
£
i=1
sin (2 (1+CV—i )) 2 (1 + сх— )) — log (1 + сх— ))
cos(+ сх—)в)
1 — 2sin T2 (1 + сх—)в j ^
(1 + A) — 2Asin(T2 (1 + сх—" j j
0 (31)
dlogL
dA
0.
(32)
n
The above mentioned four nonlinear equations are difficult to solve analytically. Therefore, these equations can be solved through iteration methods like Newton-Raphshon method etc., However, we estimate the parameters using R software.
VII. Applications
The data set is about the time-to-failure of a 100 cm polyster/viscose in a textile experiment to evaluate the tensile fatigue characteristics of the yarn when its strain level is 2.3 percentage. This data set is used early by Quesenberry and Kent [12], Pal and Tiensuwan [11] and Nasiru et al. [10].
We have fitted the model based on the minimum value of different goodness of fit measures values represented by -2log likelihood, corrected Akaike Information Criterion (CAIC), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC). We have compared the Topp-Leone generated Dagum distribution with other competitive statistical models like Exponentiated generalized exponential Dagum distribution (EGEDD), Exponentiated generalized Dagum distribution (EGDD), Dagum distribution (DD), Exponentiated generalized exponential Burr distribution (EGEBD), Exponentiated generalized Burr distribution (EGBD), Exponentiated generalized exponential Frechet distribution (EGEFD), Exponentiated generalized Frechet distribution (EGFD), Mc-Dagum distribution (McD), exponentiated Kumaraswamy Dagum distribution (EKDD) for this yarn data. The transmuted Sine-Dagum distribution provides better fit for tensile fatigue characteristics of the yarn data compared to other above mentioned competitive statistical models. The details are given in the following tables.
Table 1: Summary of statistics for tensile fatigue characteristic of the yarn data
n Mean Median Minimum Maximum Q1 Q3
100 222.0 195.5 15.0 829.0 129.2 283.0
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Table 2: The values of estimated parameters for tensile fatigue characteristic of the yarn data
Model Estimated value of the parameters
TSDD (7=10868.26, 0=1.732, /3=1.0574, A=-0.473
EGEDD 7=0.026, 7=75 .310, /3=0.017, 0=3.513, c=45.692, d=0.090
EGDD A=1.992, в=10.480, 0=4.733, C=75.487, cf=0.223
DD a=19.749, $=11.599,01=1.126
EGEBD A=35.463, в=35.965, 0=4.859, C=15.667, d=0.070
EGBD ^=24.801, 0=4.196, C=73.9120, d=0.258
EGEFD a=20.662, A=34.477, 0=5.217, c=16.438, d=0.65
EGFD a=10.537, 0=5.239, c=21.341, d=0.140
McD A=0.027, <3=0.600, ^=98.780, a=0.333, b=25.042, C=46.276
EKD a=546.109, A=39.413, <=5.188, ф=0.203, 0=31.169
Table 3: Statistical model selection for tensile fatigue characteristic of the yarn data
Model -2LL aic aicc bic
TSDD 1252.689 1260.689 1261.11 1271.11
EGEDD 1256.34 1268.336 1269.553 1283.967
EGDD 1306.14 1316.137 1317.040 1329.163
DD 1298.52 1304.517 1304.938 1312.333
EGEBD 1261.74 1271.745 1272.648 1284.771
EGBD 1306.06 1314.056 1314.694 1324.447
EGEFD 1261.52 1271.523 1272.426 1284.549
EGFD 1333.76 1341.757 1342.395 1352.177
McD 1256.4 1268.399 1269.616 1284.030
EKD 1307.92 1317.913 1318.816 1330.938
VIII. Conclusion
In this paper, we have presented a new transmuted Sine-Dagum distribution using transmuted Sine-G family of distributions. We have studied some reliability measures like reliability function, hazard rate function, reverse hazard rate function, cumulative hazard rate function, second failure rate function, mean waiting time, mean past lifetime and mean residual life. We have obtained some distributional properties like moments, moment generating function, characteristic function, cumulant generating function, incomplete moments, central moments and order statistics. We have also investigated some income inequality measures including Lorenz curve, Bonferroni index and Zenga index for proposed new probability distribution. The maximum likelihood method is used to estimate the parameters of proposed new probability distribution. Finally, we have analysed a real lifetime data set for proposed probability distribution. The proposed distribution fits well for this data compared to other competitive models.
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