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0
ON MODELING OF BIOMEDICAL DATA WITH
EXPONENTIATED GOMPERTZ INVERSE RAYLEIGH
DISTRIBUTION
Sule Omeiza Bashiru1, Alaa Abdulrahman Khalaf2, Alhaji Modu Isa3 and Aishatu
Kaigama4
department of Mathematics and Statistics, Confluence University of Science and Technology,
Osara, Kogi State, Nigeria.
2Diyala Education Directorate, Diyala, Iraq.
3,4Department of Mathematics and Computer Science, Borno State University, Nigeria.
Email: 1bash0140@gmail.com; 2alaa.a.khalaf35510@st.tu.edu.iq; 3alhajimoduisa@bosu.edu.ng;
4a.kaigama@bosu.edu.ng
Abstract
This paper introduces and thoroughly examines the Exponentiated Gompertz Inverse
Rayleigh (EtGoIr) Distribution, a four-parameter extension of the Gompertz Inverse
Rayleigh distribution. The primary focus is on its application to biomedical datasets,
shedding light on its mathematical and statistical properties. Some properties of the
distribution that were derived include the quantile function, median, moments, incomplete
moments, Renyi entropy, and probability weighted moments. The model parameters were
estimated using the method of maximum likelihood. A simulation study was conducted to
investigate the consistency of the proposed model. The outcome of the investigation revealed
that the model demonstrates consistency, as evidenced by the reduction in both root mean
square error (RMSE) and bias as sample sizes increase. To showcase the practical relevance
of the EtGoIr distribution, the paper applies the model to three distinct biomedical datasets.
The results highlight its enhanced flexibility, demonstrating superior fit compared to its
counterpart.
Keywords: Exponentiated G, MLE, Moment, Renyi Enropy, Biomedical
I. Introduction
Statistical theory continually evolves to meet the demands of modeling complex natural phenomena
effectively. Traditional probability distributions have long served as foundational tools, yet the
complexities of modern biomedical datasets often necessitate the development of novel models to
extract deeper insights. This necessity is particularly pronounced in biomedical research, where
conventional distributions struggle to capture the intricacies of physiological measurements, disease
outcomes, and survival times across various medical conditions. Recent advancements in
distribution theory have underscored the importance of innovative models for accommodating the
skewness prevalent in the aforementioned datasets. This skewness poses a significant challenge to
conventional distributions, prompting researchers to explore extensions of established models to
better capture these complexities. Notable among these extensions are the works of [1] - [9].
In this study, we focus on extending the Gompertz inverse Rayleigh (GoIR) distribution,
introduced by [10], to create a more adaptable model. We investigate the exponentiated (Et) family
of distributions, as proposed by [11], to achieve this extension. By combining the GoIR distribution
with the Et family, our aim is to develop a versatile model capable of accurately fitting real-world
datasets, particularly in biomedical science applications.
The cumulative distribution function (cdf) and probability density function (pdf) of the Et family are
given respectively as:
F(x) = [G(x)]e ; (1)
fix) = 6g(x)[G(x)]
e-i
: в > 0
(2)
where G(x)and g(x) are the cdf and pdf of the baseline distribution.
The cdf and pdf of GoIR distribution taken as baseline are given as:
G(x) = 1 — e
and
g(x) = 2 Py2x~
(3)
>e xJ
1 — e~
-”-1 (F){1-
; в>0,р>0,у>0,а>0 (4)
The motivation for this research arises from the recognition that traditional distributions
often fall short in accommodating the complexities of biomedical datasets, especially those
exhibiting skewness. By extending the GoIR distribution, we seek to contribute to the development
of hybrid distributions that better reflect the intricacies of real-world data.
II. Methods
2.1 Derivation of Exponentiated Gompertz Inverse Rayleigh (EtGoIR)
Distribution
This section introduces a new model called the EtGoIR distribution. The cdf of the EtGoIR
distribution is derived by substituting equation (3) into equation (1), as follows:
в
2\—a
1-1 1-e x)
F(x) = 1 — e
{ }
On differentiating equation (5) with respect to x, we obtain pdf of EtGoIR distribution given as:
(5)
f(x) = 2вру2х 3e
1 — e
-a-1 ( )<1-\1-e x.
1 — e
1-1 1-e x)
в-1
The pdf plot of the EtGoIR distribution is given in Figure 1 below.
—a
2
Y
—a
2
Y
1-e x
2
2
Y
Y
P
—a
—a
2
2
Y
Y
P
2
2
Y
Y
a
e
X
X
Figure 1: pdf plot of EtGoIR distribution
2.2 Expansion of Density
Using the generalized binomial expansion given as
(1 - y)p-
= I
l=U
(~1)T(p) .
а г (p -i)y
Applying equation (7) on the last term in equation (6), we have
(7)
1 — e
{
Where
e-i
e
{-vyn}
-I
1=0
(—1 yv
■y
V]
and
e{-vyn}
Zvi
-y-
1=0 ■
Therefore,
y(§)
Iiry'
о t1
IM = lkiw
1=0
where
yl
2 — J
=I(
(—1)T(p) (§){i-(i-^(x)
О !Г(р — 0.
i=0
-|1 — (1 —■) }-I(H)c—«
fe=0
1 — e-©
-
Substituting back all the expansions into equation (6), we have
-a
-a
2
2
У
со
о
2
(8)
(9)
ГО ГО ГО ГО
= —тптй
(-1 )i+k+l (§) ( + 1) Г(-о(К + 1})r(0)
fix') = px 3
where
i=0 j=0 k=0 1=0
_(r2'm+1
e (x.
j! l\ Г(-а(К + 1) - т)Г(в - i)
ГОГОГОГО
(-1)i+fc+i (§V (У + 1) Г(-о(К + 1))r(0)
^ = 2^6^ZZZZ i! j! Z! Г(-ст(^ + 1) - т)Г(в - i)
i=0 j=0 k=0 f=0
—3
(10)
2.3 Properties of EtGoIR Distribution
This section derives some statistical properties of the EGILx distribution including moments,
survival function, hazard function, quantile functions, and order statistics.
2.3.1 Quantile function
The quantile function is the inverse of the cdf of a distribution and is used in simulation studies. It
is also applied as a measure of the spread of a distribution. The quantile function is obtained using:
Q(u)=F—1(u) (11)
Applying equation (13) to the cdf of the new model, we have the quantile function given as
x = у'
-log 1 - ( 1 - ■
[
dog( 1—uO I
\}
1
1
<x
p
(12)
2.3.2 Median
Median of EtGoIR distribution is obtained by setting u =0.5 in equation (12) and it is given as
^median У
-log 1 - ( 1 - ■
[
dog ( 1—0.50
(13)
\}
2.3.3 Moments
1
l
2
1
a
p
ГО
E(Xr) = J xrf(y)dx
0
ГО
E(Xr) = pJxr
—(Z)
e \xj
dx
On solving the integral part in equation (15), we have
E(Xr) = p
rrr(l-2)
,_L
.(k + 1)1 2
When r=1 in equation (16), we have mean of EtGoIR distribution
2
0
(14)
(15)
2.3.4 Incomplete Moments
The rth (r > 0) incomplete moments for the EtGoIR distributions follow from equation (10) as
lir(u) = J
*фхг 3e (m+14) dx
Let t
= c+«0‘»' = (!:fli
When x = 0 ^ t = 0 , and if x = и ^ t = (m + 1 ) (^)
Then
lir(u) =-------—-ту (l - r~,(m + 1 )y2)
2((ш+1)у2} 2
(17)
0
1
2.3.5 Renyi Entropy
Define the Renyi entropy of the EtGoIR distributions with the following formula [12]
tr(4) = Y^l03 J fn(x)dx , П>0,П*1
0
By equation (6) we find /(х)П:
f(x)7 = 20л pv y2v x~3v e
1 -e
^-(^+1)77 7
1 -e
By generalized binomial series and exponential expansion, we get
/ / 2\_ff\ П(в-1) / . 2\~°'
t Л/\ 2 \ \ „ . / / /Vs 2 \
1 — e
{
And
1-( 1-e \xJ
=Г(
г(П(е -1) + Q *
нгц(в — 1)) e
^/1-f1-e-©'
2\~a
1-( 1-e \xJ
(П+0
=z
jzpz(n + ty
z! az
1-(1-e-&)
Then
f(x)n = ^
г(П(б -1) + i)izPz(n + 0
i!rn(6-1))z! az
■(1-(1-e-e^)2)-ff)
Tie-1)
Again using a generalized binomial, we get
/(х)П = Wx-3ne-(^^(n+4)
~ 1 2бП£Пу2ПГ(П(0 - 1) + ОУ^ХП + i)z(-1)pr((ff + 1)П + op + qVz
Where W
= Г
i=0
t! ГП(^ - 1))z! azq! Г((а + 1)П + ffp)
0
By substituting equation (18) into the equation above, we get:
Т’я(П)
= -—— ZogJ Wx 3Пе (^) (П+ч)^х
1=0
Y
<г\ 2
о
е
Z=0
0
(18)
The last integral, we get
i l wr(fm-i)+i)
^(0)=—log
1-n V2((n+4 )y2)2(31+1),
(19)
2.3.6 Probability Weighted Moments
The probabilistic weighted moment (<p, n)thfor EtGoIR distributions can be expressed as follows:
Р(*Л) = E (xv(f*(X))] = j ** F*(x)f(x)dx
By equation (5), we can find Fn(x):
,А-Л вП
l{l-(l-e-®
F(x)n = 1 — e
{ }
By generalized binomial series:
1 — e
§(i-(i-e-©
-^ en
, , =W)
And using exponential expansion
e
.(en\ ^M1-e-®
2\-a'
iP
2\-a'
1-1 1-e \xJ
==Ш1—(1—‘-&)
Then
'M0 =I(—^C,n(i-(i-(9‘)
о - a\ r
r\ar V j
j=r=0
And using generalized binomial
2 —a\r “
i — (i—e—(X)2) ) =£(—i)sQ(i —e-®2)
2 —r a
=0
And (l—
Then
—(Г
e '-x.
2\ ra v-1 Г(т + w) _(7)
) 1 w\ Г(т) 6
F(x)n = Ke"
Where K
= I
w\ Г(т)
(—1)1+syr^rr(m + wbflm zr
r\nrw\r(m) V j
(20) into
— 3e — (w+m+1)(^)
(7)0
= r= = w=0
By substituting equation (20) into the equation above, we get:
РОЛ) = K^ j x^-3 e—(w+m+1)(x) dx
— O
Let u = (w + rn + 1) then
K^r(1
2((w+m+1)y2) 2
(20)
— o
2
oo
Y
co
r
a
r=0
o
2
w
w=0
2
Y
w
X
2
2.3.7 Survival function
S(x) = 1 - F(x)
(DMi-
S(x) = 1 - 1 - e
,ys2
-a
9
J
2.3.8 Hazard function
H(x) = ^
S(x)
2вру2х-3е
'1 2
1—e
£
a
l-(1-e
'T\2
-a
9-1
1-e
(22)
(23)
(24)
H(x) =
A
a,
l-(l-e
2V2
-а в
J
(25)
1- 1-e
{ J
2.3.9 Cumulative hazard function
C(x) = -log[5(x)]
A
a.
[ {
1-(1-e
'П2
-a
9
J ]
(26)
2.3.10 Reverse hazard function
R(x)=^>
F(x)
2Г 2-,-ff-l (§)|1-(1-e
у2 у 2 °
2epy2x-ie \x) 1-e \x) e
2\-a'
1-11-е X)
R(X) =
2\-ff
(28)
(29)
2.3.11 Order Statistics
The pdf of the rth order statistics of Xr:nis given as:
n—r
иЛ%) = g(r,n-r + l)Z(-1)'[F(x)]r+'—1/(x)
i=0
Inserting equation (5) and equation (6) into equation (30), we have
/r:n (^)
1
F(r,n—r+1)'
E
[(-1)' 1 - e
2\-ff
r+i—1
1—( 1—e x)
[{
} j
—p
e x.
^,—ff—1 (){1—(1—e x)
2\-ff
E
1 — e
24-ff
1—( 1—e x)
e—1
20^p2x 3e
1-
(30)
On bringing the like terms together, we have
„2 л-г
f (x) =
Г.Л
2врГ1
Б(г,л- г +1)
^(-iy'x-3e
L
3 {x
1 -e
-г-1 Щ1- 1-
1 -e
(е(г+1-у1)
Using the generalized binomial expansion on the last term in equation (31), we have
1 — e
1—( 1—e x)
-ст 0(r+i) — 1
=!(
(—1>r(6(r + 0) i§){1—(1—e (x)
e
{ }
Substituting back into equation (31), we have
. /!Г(6(Г + i) —/)
;=0 [
-ст 7
( ) = 2^r2 yn—гую —з xj
7r:nW B(r,n—r+1)^‘=0 ^'=0 7ir(e(r+i)—7) * e
.(_1^I+))x—зe—(^)2 [1 — e—(S
^,—ff—1 ( ){1—( 1—e \x)
-ст 7 + 1
(31)
(32)
Also, expanding the last term in equation (32), we have
1—( 1—e x)
=Z(—1)fc(/+1)1—(1—e—®)
2 — erfe
fe=0
0-1
-ст
У
p
a
1—e
0
У
P
*J11-l1-e
—e
0
У
2
У
x
У
У
е
i=0
2
2
у
Р
о
С7
2
У
е
2\-ст 7 + 1
У
£
оо
£7
fr:n(‘^')
2 в/ly2
n—r ГО ГО
'j +1
> > > ___________4 к ) _________x-3e KxJ
B(r, n — r + 1) Z-i Z-i Z-i j!F(e(r + i) — j)
' i=0 j=0 k=0 J J
(—1)i+J+1(J+k1)r(e(r + i)) 2
--------------------------x 3 e v^/ 1 — e
-ff(k+1)-i
(33)
L —(Г)2] —°(k+1)—1 ^ (—1)lr(—a(k + 1)) —(-)2]
1 — e (x) 4^ l! F(—a(k + 1) — l) e \xj
Putting all the expansions together, we have the rth order statistics of EtGoIR distribution given as:
fr:n (^')
2вру2
yn—r yro yro yro
B(r,n-r+1)^‘=0 Lj=° Lk=° Li=° j!i!Y(g(r+i)—j)Y(—G(k+1)—l) J
To obtain minimum order statistics for EtGoIR distribution, we set r=1 in equation (34) to get
(—1)i+J+k+1(l+r)r(—ff(fc+i))r(e (r+i))
—(l)
e
(34)
(—1)i+7+ft+1Vl+1/r(—ff(fc + 1))r(e(1+i))
hm(x) = ЗпвРу2^1=0 Zj=°Zb=0Zl=° j!l!T(e(1+i)—j)T(—a(k + 1)—i) X 3
— (Z)
e w
(35)
To obtain maximum order statistics for EtGoIR distribution, we set r=n in equation (34) to get
fn,n(x) = 2пвру2 Zf=oZro=oZT=o —rj7
(—1)j+k+i(j+1)r(—ff(fc+i))r(e(n))
e w
i+1
k=°L>l=° р.1!Т(в(п)—])Т(—а(к + 1)—1)
2.4 Maximum Likelihood Estimation (MLE)
(36)
2
1=0
1+1
2
3
1+1
2
2
Y
3
Given some observed data, a method known as maximum likelihood estimation (MLE) can be used
to estimate a probability distribution's parameters. This is accomplished by maximizing a likelihood
function to make the observed data as probable as possible given the assumed statistical model. The
log-likelihood function of EtGoIR is given as
logL = nlog(2) + nlog(d) + nlog(fi) + 2nlog(y) — 3 'Zt=1log(x) — y2 'Z'i=1 (1) — (a +
1) I.?=i log
1 — e—&' + M I3 1—[ \ — e—(Г —a-
G
+ (e — i)Zt
1 — e
\ -(Z)2 -a’
1-U—e (x)
(37)
l
The maximum likelihood estimate is the location in the parameter space where the
likelihood function is maximized. The maximum likelihood estimates of 0, p, у and a are the values
that maximize the likelihood function. We can find these values by taking the partial derivatives of
the likelihood function with respect to 0, p, y, a and setting them equal to zero. This gives us the
following equations:
dlogl
dG
= ^ + YH=ilog 1 — e
.a,
-() -a'
[1—[ 1—e (x)
1
= 0
(38)
[
dlogl
If
— _L — Vй
P + aLi= —
' ■ (П2' —a-
[1—[ 1 — e ("
— (e — 1)Zh
e
-(Г)2 -a
[1-[ 1-e \x)
(D (i)2 -a-
1-[ 1-e Ы
1—e
=0
(39)
dlogl __ 2n
ду = у
= f + 2 KZ?=1g) +(a + 1)Xl1I^i+ Р-ТЦ=12оУе~
1 — e
+ (в-
(Ё.) _m2 (a'
е^а'2&уе (x) e
f _(У)2 _o~
l_\l_e \x)
_(!)'
1-e (x)
1 ($) _(-)2 _G
i_ l_e (x)
1-e
= 0
(40)
dlogl _ p yn
= „Li=i
d& G
1—e-®
log
1 — e-®
I P ra
+ a2^i=1
1 — e-®
I.?=1 log
1 — e-®
+ (в
\ _(-) l_e (x) _G _(Ц2 _&) i_ \ _<r)2 l_e \x) _o~
e log 1-e (x) -e
1 ($) _(-)2 _G
i_ l_e (x)
1-e
= 0
(41)
2
У
— G—1
2
2
У
У
x
x
1-е
G_1
-о
о~\
2
2
2
2
1
Since equations (38), (39), (40) and (41) are non-linear in parameters, techniques such as Newton-
Raphson method in R-software can be used to accomplish the task of estimating the parameters from
equations (38), (39), (40) and (41).
III. Results
3.1 Simulation
In this section, we conduct a simulation study to assess the performance of the Maximum Likelihood
Estimation (MLE) for the EtGoIR distribution. We generate random numbers using the quantile
function (qf) of the distribution. Specifically, if U is a uniform random variable on the interval (0, 1),
then x follows the EtGoIR distribution. We generated a total of n = 10000 samples, with each sample
having sizes n=20, 50, 100, 250, 500, and 1000. These samples were drawn from the EtGoIR
distribution using its quantile function. Subsequently, we calculated the empirical means, biases,
and root mean squared errors (RMSE) of the MLE.
Table.1 MLEs, biases and RMSE for some values of parameters
(0.5,0.1,0.1,0.5) (2,1,3,2.5)
n Parameters Estimated Values Bias RMSE Estimated Values Bias RMSE
20 в 0.4548 -0.0452 0.1484 2.2647 0.2647 0.9692
P 0.1266 0.0266 0.0976 1.0825 0.0825 0.5743
r 0.1262 0.0262 0.0579 3.0253 0.0253 0.2659
a 0.5770 0.0770 0.1908 2.7354 0.2354 0.9156
50 в 0.4737 -0.0263 0.1151 2.1251 0.1251 0.6905
P 0.1075 0.0075 0.0503 1.0966 0.0966 0.4272
r 0.1110 0.0110 0.0310 3.0438 0.0438 0.1938
a 0.5366 0.0366 0.1216 2.5940 0.0940 0.6200
100 в P Y a 0.4890 0.1035 0.1054 0.5185 -0.0110 0.0035 0.0054 0.0185 0.0903 0.0338 0.0193 0.0889 2.0670 1.0951 3.0519 2.5425 0.0670 0.0951 0.0519 0.0425 0.4628 0.3017 0.1539 0.4413
250 в 0.4972 -0.0028 0.0665 2.0166 0.0166 0.2872
P 0.1006 0.0006 0.0227 1.0665 0.0665 0.2210
Y 0.1019 0.0019 0.0126 3.0435 0.0435 0.1115
a 0.5097 0.0097 0.0630 2.5241 0.0241 0.2873
500 в 0.5017 0.0017 0.0511 2.0052 0.0052 0.1923
P 0.1012 0.0012 0.0160 1.0511 0.0511 0.1606
Y 0.1006 0.0006 0.0091 3.0318 0.0318 0.0825
a 0.5012 0.0012 0.0415 2.5051 0.0051 0.1930
1000 в 0.5028 0.0028 0.0370 2.0010 0.0010 0.1367
P 0.1010 0.0010 0.0105 1.0434 0.0434 0.1208
Y 0.1002 0.0002 0.0064 3.0288 0.0288 0.0727
a 0.5000 0.0001 0.0289 2.5048 0.0048 0.1400
Table 1 presents the simulation outcomes corresponding to the EtGoIR distribution. It is observed
that as the sample size increases, the Root Mean Square Error (RMSE) and bias associated with the
parameter estimators consistently decreases. The outcome suggest that the model is consistent.
3.2 Applications
This section demonstrates the practical application of the EtGoIR distribution by utilizing it to model
biomedical datasets. We compare its performance in providing a robust parametric fit to the datasets
with that of the Gompertz Inverse Rayleigh (GoIR) distribution, the generalized Gompertz (GGo)
distribution, the exponentiated exponential (EtEx) distribution, and the inverse Rayleigh (IR)
distribution. Metrics such as the log likelihood, Akaike Information Criterion (AIC), and Bayesian
Information Criterion (BIC) are employed for this comparison. To discern the most suitable model,
computations of the log likelihood, AIC, and BIC values are carried out for both the proposed EtGoIR
model and the alternative models used for comparison. The model exhibiting the lowest log
likelihood, AIC, and BIC values is deemed the most appropriate match for the provided datasets.
For this analytical endeavor, the R software is employed, facilitating the necessary calculations and
comparisons.
Data set 1 has been utilized by [13] and [14]. The dataset comprises the summation of skinfold
measurements from 202 athletes at the Australian Institute of Sports. It consists of the following
values:
28.0, 98, 89.0, 68.9, 69.9, 109.0, 52.3, 52.8, 46.7, 82.7, 42.3, 109.1, 96.8, 98.3, 103.6, 110.2, 98.1, 57.0, 43.1,
71.1, 29.7, 96.3, 102.8, 80.3, 122.1, 71.3, 200.8, 80.6, 65.3, 78.0, 65.9, 38.9, 56.5, 104.6, 74.9, 90.4, 54.6,
131.9, 68.3, 52.0, 40.8, 34.3, 44.8, 105.7, 126.4, 83.0, 106.9, 88.2, 33.8, 47.6, 42.7, 41.5, 34.6, 30.9, 100.7,
80.3, 91.0, 156.6, 95.4, 43.5, 61.9, 35.2, 50.9, 31.8, 44.0, 56.8, 75.2, 76.2, 101.1, 47.5, 46.2, 38.2, 49.2, 49.6,
34.5, 37.5, 75.9, 87.2, 52.6, 126.4, 55.6, 73.9, 43.5, 61.8, 88.9, 31.0, 37.6, 52.8, 97.9, 111.1, 114.0, 62.9, 36.8,
56.8, 46.5, 48.3, 32.6, 31.7, 47.8, 75.1, 110.7, 70.0, 52.5, 67, 41.6, 34.8, 61.8, 31.5, 36.6, 76.0, 65.1, 74.7, 77.0,
62.6, 41.1, 58.9, 60.2, 43.0, 32.6, 48, 61.2, 171.1, 113.5, 148.9, 49.9, 59.4, 44.5, 48.1, 61.1, 31.0, 41.9, 75.6,
76.8, 99.8, 80.1, 57.9, 48.4, 41.8, 44.5, 43.8, 33.7, 30.9, 43.3, 117.8, 80.3, 156.6, 109.6, 50.0, 33.7, 54.0, 54.2,
30.3, 52.8, 49.5, 90.2, 109.5, 115.9, 98.5, 54.6, 50.9, 44.7, 41.8, 38.0, 43.2, 70.0, 97.2, 123.6, 181.7, 136.3,
42.3, 40.5, 64.9, 34.1, 55.7, 113.5, 75.7, 99.9, 91.2, 71.6, 103.6, 46.1, 51.2, 43.8, 30.5, 37.5, 96.9, 57.7, 125.9,
49.0, 143.5, 102.8, 46.3, 54.4, 58.3, 34.0, 112.5, 49.3, 67.2, 56.5, 47.6, 60.4, 34.9.
Data set 2, encompassing the remission times (in months) of a randomized collection of one
hundred and twenty-eight (128) bladder cancer patients, has been utilized by [15] and [14]. The
dataset comprises the following values:
0.08, 0.20, 0.40, 0.50, 0.51, 0.81, 0.90, 1.05, 1.19, 1.26, 1.35, 1.40, 1.46, 1.76, 2.02, 2.02, 2.07, 2.09, 2.23,
2.26, 2.46, 2.54, 2.62, 2.64, 2.69, 2.69, 2.75, 2.83, 2.87, 3.02, 3.25, 3.31, 3.36, 3.36, 3.48, 3.52, 3.57, 3.64,
3.70, 3.82, 3.88, 4.18, 4.23, 4.26, 4.33, 4.34, 4.40, 4.50, 4.51, 4.87, 4.98, 5.06, 5.09, 5.17, 5.32, 5.32, 5.34,
5.41, 5.41, 5.49, 5.62, 5.71, 5.85, 6.25, 6.54, 6.76, 6.93, 6.94, 7.09, 7.26, 7.28, 7.32, 7.39, 7.59, 7.62, 7.63,
7.66, 7.87, 7.93, 8.26, 8.37, 8.53, 8.65, 8.66, 9.02, 9.22, 9.47, 10.06, 10.34, 10.66, 10.75, 11.25, 11.64, 11.79,
11.98, 12.02, 12.03, 12.07, 12.63, 13.11, 13.29, 13.80, 14.24, 14.76, 14.77, 14.83, 14.83, 15.96, 16.62, 17.12,
17.14, 17.36, 17.36, 18.10, 19.13, 20.28, 21.73, 22.69, 23.63, 25.74, 25.82, 26.31, 32.15, 34.26, 36.66, 43.01,
46.12, 79.05.
Data set 3, representing the survival times of one hundred and twenty-one (121) patients with
breast cancer obtained from a large hospital during the period from 1929 to 1938, was obtained from
[17]. The dataset is outlined as follows:
0.3, 0.3, 1.0, 4.0, 5.0, 5.6, 6.2, 6.3, 6.6, 6.8, 7.4, 7.5, 8.4, 8.4, 10.3, 11.0, 11.8, 12.2, 12.3, 13.5, 14.4, 14.4, 14.8,
15.5, 15.7, 16.2, 16.3, 16.5, 16.8, 17.2, 17.3, 17.5, 17.9, 19.8, 20.4, 20.9, 21.0, 21.0, 21.1, 23.0, 23.4, 23.6,
24.0, 24.0, 27.9, 28.2, 29.1, 30.0, 31.0, 32.0, 35.0, 35.0, 37.0, 37.0, 37.0, 38.0, 38.0, 38.0, 39.0, 39.0, 40.0,
40.0, 40.0, 41.0, 41.0, 41.0, 42.0, 43.0, 43.0, 43.0, 44.0, 45.0, 45.0, 46.0, 46.0, 47.0, 48.0, 49.0, 51.0, 51.0,
51.0, 52.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 60.0, 60.0, 61.0, 62.0, 65.0, 65.0, 67.0, 67.0, 68.0, 69.0,
78.0, 80.0, 83.0, 88.0, 89.0, 90.0, 93.0, 96.0, 103.0, 105.0, 109.0, 109.0, 111.0, 115.0, 117.0, 125.0, 126.0,
127.0, 129.0, 129.0, 139.0, 154.0.
Table 2: Summary Statistics of data
N Min. Max. Q1 Q2 Mean Q3 Var. SD Ku Sk
Datal 202 28.00 200.80 43.85 58.60 69.02 90.35 1060.501 32.565 4.365 1.175
Data2 128 0.080 79.050 3.348 6.395 9.366 11.838 110.425 10.508 18.485 3.286
Data3 121 0.30 154.00 17.30 40.00 46.08 60.00 1259.567 35.490 3.372 1.029
Table 2 demonstrate that the three datasets exhibit a high degree of skewness.
Table 3: The models' MLEs and performance requirements based on data set 1
Models P 0 7 a ll AIC BIC
EtGoIR 0.1985 369.5184 0.3036 0.1799 -953.632 1915.2650 1928.4980
GoIR 0.0031 - 0.0000 0.8601 -987.520 1981.0410 1990.9660
GGo -0.0052 15.4031 - 0.0597 -956.086 1918.1730 1928.9200
EtEx 0.0406 8.5786 - - -958.006 1920.0130 1926.6300
IR 52.6054 - - - -966.462 1934.9250 1938.2330
Models P в 7 a ll AIC BIC
EtGoIR 0.0003 2.5796 0.0001 0.3400 -410.704 829.4088 834.1479
GoIR 0.0839 - 0.0041 0.5129 -413.575 833.1505 836.1377
GGo -0.0224 1.5034 - 0.1678 -413.183 832.3668 835.3539
EtEx 0.1213 1.2180 - - -413.077 830.1552 834.8592
IR 2.2612 - - - -774.341 1550.683 1553.535
Table 5: The models' MLEs and performance requirements based on data set 3.
Models P <9 y a ll AIC BIC
EtGoIR 0.0000 0.5664 0.4016 0.9033 -578.7145 1165.4290 1176.6120
GoIR 0.0933 - 0.0002 0.6341 -579.9791 1165.9580 1176.7450
GGo 0.0066 1.1485 - 0.0182 -579.9435 1165.9371 1176.7274
EtEx 0.0269 1.4244 - - -581.7091 1167.4182 1168.2120
IR 2.2612 - - - -1087.464 2176.9290 2179.7240
Figure 5: Density plots for data set 3.
Tables 3 to 5 showcase the superior ability of the proposed model to effectively fit the highly
skewed datasets compared to the competing models, as indicated by the evaluation metrics
employed. Figures 3 to 5 also showed that the proposed model fits the data set adequately.
IV. Discussion
This paper introduces a novel distribution termed the Exponentiated Gompertz Inverse Rayleigh
(EtGoIR) distribution, extending the framework of the Gompertz Inverse Rayleigh (GoIR)
distribution. The introduction of a new parameter enhances the distribution's adaptability in
capturing various nuances present in biomedical datasets. The paper extensively examines the
properties of the EtGoIR distribution, effectively demonstrating its practical applicability to real-life
scenarios through the implementation of Maximum Likelihood Estimation (MLE). The empirical
findings consistently substantiate that the proposed EtGoIR model outperforms the alternative
distribution models under consideration in accurately fitting the provided datasets.
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