ON MODELING OF BIOMEDICAL DATA WITH EXPONENTIATED GOMPERTZ INVERSE RAYLEIGH DISTRIBUTION Текст научной статьи по специальности «Математика»

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: [email protected]; [email protected]; [email protected];

[email protected]

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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