CC BY
20
ON SOME STATISTICAL PROPERTIES AND APPLICATIONS OF THREE-PARAMETER SUJATHA
DISTRIBUTION
Hosenur Rahman Prodhani*
Department of Statistics, Assam University, Silchar, India hosenur72@gmail. com
Rama Shanker
Department of Statistics, Assam University, Silchar, India shankerrama2009@gmail.com
Corresponding Author
Abstract
In this paper some important statistical properties of three-parameter Sujatha distribution including descriptive measures based on moments, reliability properties, mean deviations, stochastic ordering and Bonferroni and Lorenz curves have been discussed. The estimation of parameters using maximum likelihood estimation has been discussed. Finally, the goodness of fit of the distribution has been presented for two real lifetime datasets and compared with several one and two-parameter well-known lifetime distributions.
Keywords: Sujatha distribution, Extended Sujatha distributions, Statistical Properties, Estimation, Applications.
1. Introduction
Due to stochastic nature of lifetime data, it is really very challenging to search a suitable distribution to model lifetime data. The search for a suitable distribution for modeling of lifetime data is very challenging because the lifetime data are stochastic in nature. The analysis and modeling of lifetime data are essential in almost every field of knowledge including medical science, engineering, physical sciences, finance, insurance, demography, social sciences, literature, etc. and during recent eras several researchers in mathematics and statistics tried to introduce lifetime distributions. Recently, Sharma et al. [1] studied comparative study of several one parameter lifetime distributions and observed that there are some datasets which are extreme skewed to the right where these distributions were not giving well fit.
Recently, Nwikpe and Iwok [2] proposed a three-parameter generalization of Sujatha distribution (ATPSD) and studied few of its properties including hazard rate function, moment generating function, moments about origin, distribution of order statistics, and application on a dataset. The
probability density function (pdf) and the cumulative distribution function (cdf) of ATPSD are given by
f (x; G, À,a) = —(---J 2G + 2Xx + da x2 ) X; x > 0, (-, A,a)> 0 (1)
2 (G2 + 2 + aP '
F (x;6, l,a) = 1 -
6 ax2 + 2û(â + a)x
2 (62 + Â + a)
1+
e 6x ; x > 0, (6, Â,a)> 0
(2)
The survival function of ATPSD is given by
\d2ax2 + 26(A + a) x + 26 +A + a)\e~6x
S (xfi,X,a) = 1 - F (xß,X,a) = -----—---; x > 0,6> 0,a> 0
2 (62 +l + a\
(3)
It has been observed that there are several interesting properties of ATPSD including central moments and moments based descriptive measures, reliability properties, mean deviations, stochastic ordering and Bonferroni and Lorenz curves have not been studied. In this paper an attempt has been made to discuss these statistical properties of ATPSD and propose some areas of applications.
The distributions which are particular case of ATPSD are summarized in table 1 along with its introducers.
Table 1: Some particular distributions of ATPSD
Parameter values(distributions) Pdf of distribution Introducer
(a = 2, A = 6) Sujatha distribution 03 f (x,G)= , (l + x + x2 ) e~°X JK ' e2+e+21 ' Shanker [3]
(a = 0, x = e) Lindley distribution f (x, e) = fx (1 + x) e~-X Lindley [4]
(a = 2, A = 0) Akash distribution a3 f (x, 6)= 66 (1 + x2 ) e~6x 6 + 2v ' Shanker [5]
(a = 0, A = l) Shanker distribution f ^(6 + x) e-x Shanker [6]
a = 0 (Quasi Lindley distribution) f (x,6)- 6 (a+6x) e~6x a +1 Shanker and Mishra [7]
(a = 0, A = 0) Exponential distribution f ( x, 6) = 6e~6 x
The behavior of the pdf and the cdf of ATPSD are presented in figures 1 and 2 respectively.
Figure 1: The graphs of the pdf of ATPSD for different values of 0,aand A
CDF of ATPSD
CDF of ATPSD
CDF of ATPSD
ttl8ta = 0.5,alptia45,la[iia=l5 theta = C.S. alpha=1 .lamda=C.S theta =15, alpha=1 B,lamda=0.5 theta=0.5,alpha=2,lamda=0.5
— theta = 0.5,alpha=0 5,la[THla=(}.5 — Iheta = 1 .a lpha=C. S.lamda=0.5 - Iheta =1.£-.alpha=0.B.lamda=0.E — theta=2,alpha=ï 5,1аггк1а=& 5
— ta=0.5,^8=0.5^8=0.5 — Oietë=0.5,alphB=0.5,l3mjB=1 — foeta=0.5,alptia=S.5,lamJa=1.5 — M3=0.5,3lpta=0.5,l3Hli3=2
CDF of ATPSD
CDF ofATPSD
CDF of ATPSD
— Мз = 0.5,з1р1и=0.5,1злй=|].5 theta = 1,alpha=0.5,lamda=1 - Шз=1.5,а1р1и=0.5,1зт1з=1.5 — Ма=2,#а=0.5, lamda=2
— Ma = 0.5,alplia=051l3Bt3=(1.5 Ma = 1,alpha=1,lamla45 — Ma=1.5,3lph3=l.5,l3rala=0.5 — thsta=2,alpha=2,laiiKla=0,5
— thEta = D E.alp-ha=O.E-.lamda=O.E- — 1Ма=0.5,а1рЫ,Ыа=1 — theta =0. b,alpha=1 .E-lamda="1 .E— tWa=0.5,alptia=2,larala=2
Figure 2: The graphs of the cdf of ATPSD for different values of 0,aand A
2. Descriptive Properties Based On moments
The r th moment about origin, ¡ur , of ATPSD is given by
r ¡I2&2 + 2 ( r + l)A + ( r + l)( r + 2) a|
^ =-—(~2-\-; r = 1,2
2d (d2 +A + a)
Thus, the first four moments about origin are obtained as
(4)
,_62 + 21 + 3a , _2 + 31 + 6a) ,_6 + 41 +10a) , _ 24 + 51 +15«) 6(6 + 1 + a I
ln2 , V 2 a2(a2 . V 3 a3ia2 . \'H4 4 / ,2 . \ '
16 + 1 + a) 6 (6 + 1 + a) 6 (6 + 1 + a) 6 (6 + 1 + a)
Now using the relationship between moments about mean and the moments about origin, the moments about the mean of ATPSD are obtained as
12
64 + (8a + 41) 62 + (212 + 3a2 + 61a )
1
62 (62 +i+a
2166 + (15a + 61)64 + ^9a2 + 612 + 211a) 62 + ( 213 + 3a3 + 912a + 91a2
)}
3 <
1
63 (62 + 1 + a~)
368 + (64a + 241) 66 + (102a2 + 1721a + 4412 ) 6 + (72a3 + 3213 + 16012a + 1921a2 ) 62 + (:
814 + 15a4 + 4813a + 601a3 + 8412a2
64 (62 +1 + a)
The coefficients of variation (C.V), skewness (Jß), kurtosis ß ) and index of dispersion (/) of
ATPSD are thus obtained as
J1 6
C.V -
1 ^64 + (8a + 41) 62 + ( 212 + 3a2 + 61a)
1
^2
6 + 21 + 3a
2166 +(15a + 61) 64 +(9a2 + 612 + 211a) 62 +( 213 + 3a3 + 912a + 91a2 )}
1
3/2
(12 ) |64 +(8a+ 41) 62 +(
368 + (64a + 241) 66 + (102a2 + 1721a + 4412 ) 6 + ( 72a3 + 3213 + 16012a+ 1921a2 )62 +(
64 + (8a + 41) 62 + ( 212 + 3a2 + 61a)}
ß -14 P2 = 2 1
32
814 + 15a4 + 4813a + 601a3 + 8412a2
|64 + (8a + 41) 62 + (212 + 3a2 + 61a)}
2 4 2 2 2
c 6 + 4(2« + 1)62 + (21 + 3a + 6a1) r = =-2-2-
JUA 6(6 + 1 + a)(e2 + 21 + 3a)
The nature of the coefficient of variation, skewness, kurtosis and index of dispersion of ATPSD are shown graphically in figure 3.
Covfflclcxt of Variation of ATPSD dl.lrlbullon CroWcl»)« of tforlntlon of ATPSD dl.lrlbullon Coo file I on I fl Violation oi ATPSD di.lribullo.
Figure 3: The nature of the coefficient of variation, skewness, kurtosis and index of dispersion of ATPSD
When a and A are fixed and d increases, the value of the CV is increases till d < 3 and when d > 3, then CV starts decreasing slowly increasing values of d . When d and A is fixed, then CV decreases for increasing values of a . Similarly, for, fixed values of d and a and increasing values of A, CV decreases.
When a and A are fixed and d increases, skewness decreases speedily till d < 1 and when d > 1, it becomes constant. When d and A is fixed, then skewness decreases for increasing values of a . Similarly, for, fixed values of d and a and increasing values of A , skewness increases.
For fixed values of (a, A) and increasing values of d , the kurtosis is decreasing, increasing and again decreasing. For fixed values of (d, a) and increasing values of a, the kurtosis is increasing and then decreasing. And for fixed values of (d, a) and increasing values of A, the kurtosis i decreasing speedily till A < 2 and for A > 2, it starts increasing very slowly.
is
For the nature of index of dispersion, it is always decreasing for increasing values of one parameter and fixed values of another two-parameter.
3. Reliability Properties
3.1. Hazard Rate Function
The hazard rate function of ATPSD is obtained as
f (x; 6,1, a)
h (x; 6,1, a) -
62 (26 + 21x + 6ax2
)
S (x;6,1,a) 62ax2 + 26
,2 2
(1 + a) x + 2 (62 + 1 + a)
; x > 0, (6,1, a) > 0
It can
be easily verified that h (0; 6,1,a)- f (0; 6,1, a) -
f (x;6,1,a) 63
S (x;d, A,a) (d2 +A+a) The behaviors of the hazard rate function of ATPSD are explained in the following figure 4.
(5)
Figure 4: The graphs of the hazard rate function of ATPSD for different values of d, a and A
3.2. Mean Residual Life Function
The mean residual life function of ATPSD can be obtained as
1 <» r
i (x; 0, A, a) =
1 - F (x; 0, A, a) x 1
I [ 1 - F (t;0, A, a)] dt
-0 x
jo2ax2 + 20 (A + a) x + 2 (o2 + A + a)| a (o2x2 + 20x + 2) + 2 (A + a) (Ox + l) + 2 (o2 +A + a)
02at2 + 20 (A + a) t + 2 (o2 + A + a)
-Ot , e dt
0
02ax2 + 20 (A + a) x + 2 (02 +A + a)
(6)
.2
, v d + 2A+ 3a It can be easily verified that m (0; d, A, a) = —j-—-^ = M\ .The behavior of mean residual life
d(d2 +A+a) function is explained in the following figure 5.
Figure 5; The graphs of the mean residual life function of ATPSD for different values of d, a and A
4. Stochastic Ordering
In probability theory and statistics, a stochastic order quantifies the concept of one random variable being bigger than another. A random variable X is said to be smaller than a random variable Y in the:
i. Stochastic order (X <st Y ) if F^ (x) > Fj (y) for all x
u. Hazard rate order (x <hr Y) if hx (x) > hY (y) for all x
iii. Mean residual life order (X <mrj Y) if m^ (x) > my (y) for all x
iv. Likelihood ratio order (X Y) if "x ,v ' decrease in x
« fAx)
fY ( ^ )
The following results due to Shaked and Shantikumar [8] are well known for establishing stochastic ordering of distributions
X < Y ^ X < Y ^ X < ,Y Ir hr mrl
X<stY
Theorem: Let X ~ ATPSD (^, A a) and Y ~ ATPSD (#2, A, a) . If A = = a2 and
> ^2 or =@2,a\ = a2 and A < A2 or = ^2' A = A and a^ < a2 then X Y hence
X <, Y X< , Yand X <.f Y hr , mrl and st .
Proof: We have
3 2
f y (x) вл (в + А + а )
fY ( x) в3 (в2 +Ä + a1)
в^а x + 2 А x + 2 в
1
в^а x + 2 А x + 26 V 2 2 2 2 J
-(в1-в2 ) x
e 11 2 ; x>0
Now
log
f fX ( x)Л V fY ( x) y
= log
f 3 2 ^
в (в2 + Ä+ a2 )
~"3 2 V в2 (в1 +Ä + a1)
+ log
2
в^а x + 2^ x + 2в^ 2
x + 2А x + 2в V 2 2 2 2 J
Therefore f
d dx
log
fX ( x )
fY ( x)
(авА —авА) x2+^вв (a — a ) x+4(ä$2—Ä2$I )
(ва x2+2 А x+2в ) (
в2а2 x + 2 А x+2в
)
(в1 —в2
(в1 — в2 )
Thus, if А = , а1 = а^ and > #2 or в^ = в2, а1 = а2 and А < А
or
вц =$2, А = Ä2 and а1 < а2,
d dx
log
fX ( x ) fY ( x)
< 0 . This means that X Y and hence
X <hrY, X <W Y and X <stY .
"mrl
5. Deviation from Mean and Median
The amount of dispersion in a population is an evidently measured to some extent by the totality of deviations from the mean and median. These are known as the mean deviation about the mean and mean deviation about median and are defined by
S (x) = J | x - n | f (x)dx and 8~ (x) = J | x - M | f (x)dx respectively, 1 0 2 0
where ^ = E(X) andM = Median(X) .
The measures Si (x) and S2 (x) can be calculated using the following relationships
U
S, (x) = 2/j.F(u) — 2 J xf (x)dx
0
and
S (x) = —/ + 2 J xf (x)dx 2 M
Thus, the mean deviation about the mean Sj (x), and the mean deviation about the median §2 (x) of ATPSD are obtained as
s:( x) =
ae2u2 + 261/ + 4a6u + 262 + 41 + 6a
2
S(x) =
6(6 + 1 + a)
ae3M3 + e2 (21 + 3a) M2 + (263 + 461 + 6a6)M + 262 + 41 + 6a
26(62 +1 + a)
-6M
■u
(7)
(8)
x
6. Bonferroni and Lorenz Curves and Indices
The Bonferroni and Lorenz curves by Bonferroni [9] and Bonferroni and Gini indices have wide applications in economics to study income and poverty, but it also used in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as
1 q 1 B(p) =-J xf (x)dx =-
PU0 PU
x x
J xf (x)dx - J xf (x)dx 0 q
1
PU
U - J xf (x)dx
q
1q 1 and L(p) = — J xf (x)dx = — U 0 u
J x f (x) dx — J x f (x) dx 0 q
U — J x f (x) dx
respectively.
The Bonferroni and Gini indices are obtained as
1 1 B = 1 - J B(p) dp and G = 1 - 2 J L(p) dp, respectively. 0 0
Using pdf of ATPSD and little algebraic simplification, we get 1
B( p) =
1 -
a63q3 +62(21 + 3a)q2 + (263 + 461 + 6a6)q+ 262 + 41+ 6a
-6q
2(6 + 21 + 3a)
L( P) = 1 —
a e3q3 +62(21 + 3a)q2 + (263 + 461 + 6a6)q+ 262 + 41 + 6a
-¡2
(9)
(10)
2(6 + 2A + 3a)
Now using equations and after some simple algebraic simplifications, the Bonferroni and Gini indices of ATPSD are obtained as
B = 1 —
a63q3 +62(21 + 3a)q2 + (263 + 461+ 6a6)q+ 262 + 41 + 6 a
G =
2(6 + 21 + 3a) ae3q3 + e2 (21 + 3a) q2 + (263 + 461 + 6a6) q + 262 + 41 + 6a
2(62 + 21 + 3a)
— 1 .
(11) (12)
X
X
2
7. Maximum Likelihood Estimation
Let (x,, x^,...,xn) be a random sample of size n from ATPSD(6, A, a) . Then the likelihood function is given by
L =
^2 (в2 + А + а)
n i 2\ -nвx —
П (2в + 2Аx + ва x )e , where x is the sample mean.
The log-likelihood function is thus obtained as
log L = n
21og6 - log2 - log(d2 +A + a)J+ E log(26 + 2Ax( + 6ax(2)-n6x . (13)
The maximum likelihood estimates (6, A, a) of parameters (6, A, a) are the solution of the following log-likelihood equations
д log L 2n 2пв
+ Е-
2 + ax^
дв в в2 + А + а i=12в + 2Äxi + 0axi 2
- nx = 0
д log L
-n n
+ Е
2x-
дА в2 + А + а i=1 2 в + 2 Ax +вax,2
- nx = 0
д log L
- n
в x,
i i 2
n
= + е_l-_
да в2 +А + а i=1 2в + 2 Axf +вaxj 2
= 0
These three log-likelihood equations do not seem to be solved directly. We have to use Fisher's scoring method for solving these three log-likelihood equations. We have
дв
2
д log L 2
дА2
2
д log L да2
2
д log L
двдА
2
д log L
дв да
(в2 - А - а) n (2 + ax^2) (в2 +А + а) i=1 ( 2в + 2Ax■ + в ax-2)
д2 logL_ -2n 2п($ -А-а) n
X" = "ТХ + ^ X - iE1
- Е
4 V
(в2 +А+а) 1 1 (
|в + А + а 2п$ (в2 +А + а 2п$
у- Е 2 i=1,
2 в + 2 А x + в ax
Л 4
в xi
2 2
(в2 + А + а)2 2=1 (
n
- Е
2в + 2AXj +ва x,
4 x + 2ax-~
i2)2
(в2 + А + a) 2 1 (
2в + 2 А^- + в a x■2
2AX:'
у + Е 2 i=1,
(в2 + А + а) ' (
2 в + 2 А x + в a x.
.2 Г
2
д log L
дв да
-12п6 n
-Т - е
( л2 Л2 i=V л 3V
(а в + 6 ) (а + в x )
2
д log L
да дв
n
n
n
n
2
8 log L
818 a
n 29x; - X_-
(d2 + 1 + a) l~l (29 + 21x; +dax2
2
8 log L
8a 81
The following equations can be solved for MLEs (0, A,a) of (0, A, a) for ATPSD
82 log L 82 log L 82 log L
2 89 8981 898a
82 log L 82 log L 82 log L
8189 2 812 818 a
82 log L 82 log L 82 log L
8a89 8 a 81 8a 2
70
/1 -1
0
a - a
1=1
a=a
8 log L
89 8 log L
81 8 log L 8a
9=90 1=1
a=a,
0
(14)
where (0q,Aq,®q)are the initial values of (0,A, a) respectively. These equations are solved iteratively till sufficiently close values of (0, A,a) are obtained.
8. Applications to Lifetime Data
The following real lifetime datasets have been considered for testing the goodness of fit of ATPSD over other one parameter and two-parameter lifetime distributions.
Data Set 1: The real data discussed by Almongy et al [10] that represents a COVID 19 mortality rate data belongs to Mexico of 108 days that is recorded from 4 March to 20 July 2020. This data formed of rough mortality rate. The data are as follows:
8.826, 6.105, 10.383, 7.267, 13.220, 6.015, 10.855, 6.122, 10.685, 10.035, 5.242, 7.630, 14.604, 7.903, 6.327, 9.391, 14.962, 4.730, 3.215, 16.498, 11.665, 9.284, 12.878, 6.656, 3.440, 5.854, 8.813, 10.043, 7.260, 5.985, 4.424, 4.344, 5.143, 9.935, 7.840, 9.550, 6.968, 6.370, 3.537, 3.286, 10.158,8.108,6.697, 7.151, 6.560, 2.988, 3.336, 6.814, 8.325, 7.854, 8.551, 3.228, 3.499, 3.751, 7.486, 6.625, 6.140, 4.909, 4.661, 1.867, 2.838, 5.392, 12.042, 8.696, 6.412, 3.395, 1.815, 3.327, 5.406, 6.182,4.949, 4.089, 3.359, 2.070, 3.298, 5.317, 5.442, 4.557, 4.292, 2.500, 6.535, 4.648, 4.697, 5.459, 4.120, 3.922, 3.219, 1.402, 2.438, 3.257, 3.632, 3.233, 3.027, 2.352, 1.205, 2.077, 3.778, 3.218, 2.926, 2.601, 2.065, 1.041, 1.800, 3.029, 2.058, 2.326, 2.506, 1.923.
Data set-2: The following bi-modal dataset, discussed by Ghitany et al. [11], is obtained from the banking sector discusses the waiting time (in minutes) before the customer received service in a bank. The values are:
0.8, 0.8, 1.3, 1.5, 1.8, 1.9, 1.9, 2.1, 2.6, 2.7, 2.9, 3.1, 3.2, 3.3, 3.5, 3.6, 4.0, 4.1, 4.2, 4.2, 4.3,4.3, 4.4, 4.4, 4.6, 4.7, 4.7,4.8, 4.9, 4.9, 5.0, 5.3, 5.5, 5.7, 5.7, 6.1, 6.2, 6.2, 6.2, 6.3, 6.7, 6.9,7.1, 7.1, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.2, 8.6, 8.6, 8.6, 8.8, 8.8, 8.9, 8.9, 9.5, 9.6, 9.7, 9.8, 10.7,10.9, 11.0, 11.0, 11.1 ,11.2, 11.2, 11.5,11.9, 12.4, 12.5, 12.9, 13.0, 13.1,13.3, 13.6, 13.7,13.9,14.1, 15.4, 15.4, 17.3, 17.3, 18.1, 18.2, 18.4, 18.9, 19.0, 19.9, 20.6, 21.3, 21.4, 21.9,23.0,27.0, 31.6, 33.1, 38.5.
In order to compare lifetime distributions, values of -2 log L, AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), K-S Statistics (Kolmogorov-Smirnov Statistics)
Hosenur Rahman Prodhani and Rama Shanker
ON SOME STATISTICAL PROPERTIES AND APPLICATIONS OF RL&A, N°.3 (74) THREE PARAMETER SUJATHA DISTRIBUTION_Volume 18, September 2023
and the corresponding probability value (p-value) for the above data set has been computed. The
formulae for computing AIC, AICC and K-S Statistics are as follows:
AIC = -2logL + 2k , AICC = AIC + 2k(k + 1 , D = Sup | Fn (x) - FQ (x) |
n - k - 1 x
where k = number of parameter, n = sample size
The distribution corresponding to the lower values of -2 log L, AIC, AICC, and K-S Statistics is the best fit distribution. These statistical values for the two datasets have been computed and presented in tables 2 and 3 respectively. It is obvious from the goodness of fit of distributions given in tables 2 and 3 that ATPSD gives much closure fit as compared to other one parameter and two-parameter distributions and hence it can be considered as a suitable model for the given dataset.
Table 2: ML estimates, -2logL, AIC, AICC, K-S value and p-value of the distribution for the data set-1
MLE
Distributions в a А -2logL AIC AICC K-S p- value
ATPSD 0.5209 2277.6180 0.1000 533.29 539.29 539.52 0.0584 0.9246
TPSD 0.4867 0.0100 536.45 540.45 540.56 0.0682 0.8029
NTPSD 0.4842 0.0100 542.75 546.75 546.86 0.0869 0.5153
ANTPSD 0.4869 931.2583 536.37 540.37 540.48 0.0684 0.7975
QSD 0.4825 0.1000 537.97 541.97 542.08 0.0671 0.8205
NQSD 0.4868 137.8985 536.39 540.39 540.50 0.0626 0.8758
WSD 0.9828 3.7285 510.77 514.77 515.47 0.0845 0.5561
PSD 0.3491 1.1648 537.06 541.06 541.76 0.0937 0.4454
Sujatha 0.4631 543.36 545.36 545.39 0.0950 0.3961
Table 3: ML estimates, -2logL, AIC, AICC, K-S value and p-value of the distribution for the data set-2
Distributions MLE -2logL AIC AICC K-S p- value
в a А
ATPSD 0.2025 0.1270 108.8797 634.60 640.6 640.85 0.0564 0.9539
TPSD 0.2769 2.4379 639.25 643.25 644.05 0.0764 0.7146
NTPSD 0.2316 20.3400 635.03 639.03 639.15 0.0699 0.8086
ANTPSD 0.2769 0.4102 639.25 643.25 644.05 0.0750 0.7361
QSD 0.2769 0.6752 639.25 643.25 644.05 0.0897 0.5133
NQSD 0.2769 0.1136 639.25 643.25 644.05 0.0908 0.4959
WSD 0.1958 0.0100 602.44 606.44 607.24 0.1079 0.2627
PSD 0.3571 0.9012 636.48 640.48 641.28 0.0682 0.8206
Sujatha 0.2846 639.63 641.63 641.88 0.0949 0.4447
The fitted plot of the considered distributions of the data set-1 and data set-2 are presented in figure 6.
Figure 6: Fitted plot of distributions of the data set-1 and data set-2.
9. Conclusion
Some important and useful statistical properties of a three-parameter Sujatha distribution (ATPSD) including descriptive measures based on moments, reliability properties, mean deviations, stochastic ordering and Bonferroni and Lorenz curves have been derived and discussed. Maximum likelihood estimation has been discussed for estimating the parameters. Applications and goodness of fit of the ATPSD have been demonstrated with two real lifetime datasets and it shows better fit over several one parameter and two-parameter lifetime distributions.
10. Conflict of Interest
The Authors declare that there is no conflict of interest.
References
[1] Sharma, V., Shanker, R. and Shanker, R. (2019). On some one parameter Lifetime distributions and Their Applications, Annals of Biostatistics & Biometrics Application, 1(2):1 - 6.
[2] Nwikpe, B.J. and Iwok, I. A. (2020). A three-parameter Sujatha distribution with application, European Journal of Statistics and Probability, 8(3):22 - 34.
[3] Shanker, R. (2016). Sujatha Distribution and Its Applications, Statistics in Transition-New Series, 17(3):391- 410.
[4] Lindley, D.V. (1958). Fiducial distributions and Bayes' theorem, Journal of the Royal Statistical Society, Series B, 20(1):102 - 107.
[5] Shanker, R. (2015). Akash Distribution and Its Applications, International Journal of Probability and Statistics, 4 (3):65 - 75.
[6] Shanker, R. (2015). Shanker Distribution and Its Applications, International Journal of Statistics and Applications, 5 (6):338 - 348.
Hosenur Rahman Prodhani and Rama Shanker
ON SOME STATISTICAL PROPERTIES AND APPLICATIONS OF RT&A, N°.3 (74) THREE PARAMETER SUJATHA DISTRIBUTION_Volume 18, September 2023
[7] Shanker, R. and Mishra, A. (2013). A quasi Lindley distribution, African Journal of Mathematics and Computer Science Research (AJMCSR), 6 (4):64 - 71.
[8] Shaked, M. and Shanthikumar, J. (1994). Stochastic Orders and Their Applications. Academic Press, New York.
[9] Bonferroni, C.E. (1936). Teoria statistica delle classi e calcolo delle probability [Statistical class theory and calculation of probability], Publicazioni del R Istituto Supriore di Scienze Economiche e Commerciali diFirenze, 8:3-62.
[10] Almongy, H.M., Almetwally, E.M., Aljohani, H.M., Alghamdi, A.S. and Hafez, E.H. (2021). A new extended Rayleigh distribution with applications of COVID-19 data, Results Physics. 23:1-8.
[11] Ghitany, M.E., Atieh, B. and Nadarajah, S. (2008). Lindley distribution and its application, Mathematics and Computers in Simulation, 78:493-506.
