ON GOMPERTZ EXPONENTIATED INVERSE RAYLEIGH DISRIBUTION Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

ON GOMPERTZ EXPONENTIATED INVERSE RAYLEIGH DISRIBUTION

Sule Omeiza Bashiru1 and Halid Omobolaji Yusuf2

1Prince Abubakar Audu University, Anyigba, Kogi State, Nigeria 2Ekiti State University, Ado-Ekiti, Ekiti State, Nigeria Email: 1bash0140@gmail.com, Email: 2omobolajihalid01@gmail.com

Abstract

In this paper, we proposed a four parameter Gompertz Exponentiated Inverse Rayleigh Distribution. The proposed distribution is an extention of the Exponentiated Inverse Rayleigh Distribution which was compounded with the Gompertz generated family of distribution. Several of its statistical and mathematical properties including quantiles, median, moments, skewness and kurtosis are derived. Also, the reliability and hazard rate functions are derived. To estimate the new model parameters, the maximum likelihood technique is used. To evaluate the effectiveness of the estimators in this model, a simulation study was carried out and the result of the simulation study indicated that the model is consistent since the value of the mean square error decrease as sample size increases. Finally, the usefulness of the proposed distribution is illustrated with two datasets and it is discovered that this model is more adaptable when compared to well-known models..

Keywords: Maximum likelhood estimation; Skewness; Kurtosis; Probability density function;

Cummulative probability distribution.

The classical distributions frequently do not offer an appropriate match to some real data sets in real-world circumstances. In order to create novel distributions, researchers devised numerous generators by inserting one or more parameters. The newly generated distributions are more adaptable than the classical distributions.

Gompertz[10] introduced a continuous probability distribution known as the Gompertz probability distribution (GD). The GD is employed to explore nature of human mortality by determining the value of life's unexpected events. Several branches of statistics have used the Gompertz distribution where survival time is necessary such as in demography Vaupel [18], Preston et al [16] and in actuary Willemse and Koppelaar[19); in gerontology, medicine, biology, and related sciences Economos [7], Brown and Forbes [6]. In this article, the Gompertz family of distributions is used to create a novel model. Some authors that have employed the Gompertz Family of distributions include : Halid and Sule [12], Alizadeh et al. [4], and Abdal-Hameed et al. [1] . Halid and Sule [12] defined the cummulative density function (CDF) of the Gompertz family of distribution as:

1. Introduction

(1)

and the corresponding PDF to (1) is given by

fX(x) = n(x)H - G(x)]-n-1 e(f»

e

(2)

where f and q are the extra shape parameters

The article is broken down into the following sections: In Section 2, the new distribution GEIR's derivation is described. In Section 3, the mathematical characteristics of the new distribution are explained and the Maximum likelihood estimation of the distribution is used to estimate the parameters. In Section 4, we presented and explored the new distribution's practical applicability. Finally, Section 5 displays the concluding remarks.

2. Derivation of Gompertz Exponentiated Inverse Rayleigh Distribution

In this section, we derived the Gompertz Exponentiated Inverse Rayleigh (GEIR) distribution. Rao and Mbwambo [17], introduced the CDF and PDF of Exponentiated Inverse Rayleigh (EIR) Distribution as

GX (x) = 1 - il - ex) ; x > 0,£ > 0, a > 0 (3)

The corresponding pdf is given as:

, , 2a£2 -(x" gX(x) = -3-e vxy

a-1

x > 0,£ > 0, a > 0

(4)

putting equation (3) into (1) we have the CDF of Gompertz Exponentiated Inverse Rayleigh (GEIR)

Fx(x) = 1 - [1-(1-exp(-(£/x)2)) "] x > 0,£ > 0,a > 0,q > 0, p > 0 (5)

Figure 1: CDF plot of GEIR distribution for different parameter values

Now, putting (3)and (4) into (2), we now obtained the PDF of the proposed GEIR distribution

given by

fx (x) = 2 p£2 x-3 e 1

1- e

- an-1 p { 1-

1—e (x)

21 -an

where q and a are shape parameters, p and £ are scale parameters.

2

2

£

e

2

2

£

£

e

PDF

0 1 2 3 4 5

Figure 2: PDF plot ofGEIR distribution for different parameter values

3. Properties of GEIR Distribution

3.1. Linear Mixture of GEIR

Given the CDF and PDF of GEIR distribution ( 5) and (6), the expressions

1- 1-е (x )

£ ( i -

к m! И

1- e

-ца

£ ц£ l Ï W1 -

к m! Vn

1- e

-ца

1

1- e

-qa

2ça<2x 3e

'r)2 œ œ I 1 \ k+m / 7.

}) ££ i-1 I k m!

km

m! m

f (x) = ££

(-1)k+m fk

km

m! m

2ça<r2x 3e

1 - e (

na(m+1)-1

—na(m+1)-1

1 - e

1 - e

-na(m+l)-l œ k œ

= ££ £

k m n

(-1) m!

k+ m+ n

k\ f-na(m + 1) - ^ e-n(r-' m M n

So therefore, the PDF of GEIR distribution can be expressed as

f (x) = wKm,n2 (n + 2) щ<2 x 3 e

n+1

where

,, = ££ £ (-1) fk\ f-Va(m + 1) - 1 ^k,m £ £ Îrm! (n + 2){ m

n

-na

k

2

i

n

e

k

2

i

k

2

(

2

2

?

?

2

2

?

2

г

and the CDF of GEIR distribution can be expressed as

F(x) = WKm,n^ x^ j

where gn+1(x) is the PDF of Gompertz Exponentiated Inverse Rayleigh distribution with shape parameter n+2

3.2. Survival Function

According to Ieren and Balogun [13], the survival function discribes the probability that a unit, or component, or individual will not fail at a given time". A survival function is generally expressed as

S(x) = 1 - F(x; q>, a, £) (7)

Therefore the survival function of GEIR distribution is derived by substituting (5) into (7) which resulted to

Sx (x) = e

1 x

1 _ e ( x )

(8)

Survival Function

0 1 2 3 4 5

x

Figure 3: GEIR distribution survival plot for various parameter values

3.3. Hazard Function

The hazard function is given as

HM = 1-|) (9)

The hazard function for GEIR distribution is derived by substituting (5) and (6) into (9) and it resulted into

2f?2x 3e

hX (x)

1 - e

-ay-1 n < 1- 1-e (x)

21 -na

nS

hX (x) = 2 f?2 x-3 e 1

1—e (x)

21 -na

1 e-

-ay-1

(10)

(11)

2

2

?

?

e

e

2

?

?

Hazard Function

Figure 4: Hazard plot of GEIR distribution for different parameter values

3.4. Cumulative Hazard Function

From this definition, the cumulative hazard function, Hx(x), of a continuous random variable, X, which follows the GEIR distribution is obtained.

Hx (x) = -log[SX (x)]

substituting equation (8) into (12), we obtain

Hx (x) = -log

1- 1-e ( x )

(12)

(13)

Hx (x) = -f< 1 -

1 e-

-ya

(14)

3.4.1 Reversed Hazard Function

The reversed hazard function can be obtained by applying the formula below:

T(x) =

№ F(x)

(15)

-na

f

n

e

2

?

Hence, we obtain the reversed hazard function by substituting (5) and (6) in (15)

2 x-3 e

-an

_ _ 1 CO I - - - I 4 )

)2

T(x) =

1- e

'П2

e

-ац-1 ф]1- 1-e (X)

l- Д[1-(1-exp(-(r/x)2))-na]

(16)

3.5. Quantile Function, Median, Skewness and Kurtosis

The pth quantile of the GEIR distribution is derived as

qx (p) = , ^ = (17)

- log (1 - [1 - ф log(1 - p)]-

we have the first three , Qi1 = Q(1/4) and Q3 = Q(3/4), that is by substituting value of p=0.25 and p=0.75 in Xp, respectively. Also Quantile is also used in finding the skewness and kurtosis of the distribution.

3.5.1 Median

Substitute p=0.5 in (17), we have

Me = Qx (0.5) = . Z _ (18)

- log (1 - [1 - ф log(0.5)]-

3.5.2 Skewness and Kurtosis

According to Galton [9] and Moors[15] we can obtain the skewness (Sk) and kurtosis (Ku) measures, respectively for GEIR distribution using the following expression

Sk = Q< 4 > + 0< 4 >- 2Q< 2 > (19)

Q(4> - Q<4>

and

3.5.3 Moment

Ku = Q(8) - Q(5) + Q(I) - Q( 1 ) (20)

Q(8) - Q(8)

In this section, we consider the moment of the GEIR distribution. Let X = (x\, x2,... ,xn) be a sample drawn from GEIR distribution with pdf, then the rth moment }ir can be written as

œ

ь-Г J

цr = xrf (x)dx

10

' Г œ 2 _3 ( -( <)2 An + 1

цг =J xr&k/m/n2 (n + 2) x 3I e vV I dx

th

after some mathematical derivations, we obtained the rth moment as

=uKm,n2 (n + 2) (n+1)(r/2-1)<K2r(1-2) r< 2

3.6. Maximum Likelihood Estimation

Due to its consistency, asymptotic efficiency, and invariance property, the Maximum Likelihood Estimation (MLE) method is frequently used to estimate unknown parameter(s). Let x2,..., xn be random sample of size n drawn from GEIR distribution, then the likelihood can be expressed as :

L(f, ?, y, a) = 2n f2?2n £ x-3e

'<=1 x2

n

n

i=1 L

1 e-

~,n f2?2n £ 3„ i=1

and the log-likelihood of expression (21) can be expressed as

-1-7« - £n=1

n | f ' ' 7

2\ -«7'

1- | 1-e x2

(21)

l = log L

i=1

^ f sr It ! t .ex-

n ln fa?2 - ya £ ln 1 - e-x2 - ?2 £ ^ +Z £ 1 - 1 -

i=1

i=1

na

-3 £ ln(x) - £ ln(1 - e-i=1 i=1

(22)

Differentiating (22) with respect to q>, a and q, if equated to zero, we obtain the following estimating equations

dL = n 1 £ 11 _

df = f 7 <£

1e-

na

(23)

dL_

2n n

? - 2na? £ ? i=1

-2? £ i=1

\

e x2

21

e x2

/

e x2

x2 1

ex

i=1

- 2? £ ^ + 2 fa? £

i=1

1 e-

?1\-na _£\

e x2

21

e x2

(24)

dy = -a £ inj1 - e x2

i=1

+ af £

2 -na

1 e-

ln 1 e-

-f SI1 -(1 -e x

2 -na

dl n » , -

da = a - 7 E^l1 - e i=1

+f£ i=1

2 -na

1 e-

ln 1 e-

(26)

The maximum likelihood estimator d = (q>,?, q, a) of d = (q>,?, q, a) is obtained by solving the nonlinear system of equations (23) - (26). In this study, we used the Newton Raphson technique, a nonlinear optimization procedure, to numerically optimize the log-likelihood function shown in (22). The asymptotic distribution of the element of the 4 x 4 observed information matrix of GEIR distribution can be expressed as

/n(d - d) - N4(0, E-1 )

(27)

where E is the expected information matrix. Thus, the expected information matrix is expressed as

E-

-E

r d2l d2l d2l d21 "

df2 dfd? dfdy dfda

d2l d2l d2l d21

dfd? d? 2 dyd? dad?

d2l d2l dH d21

dfdy dyd? dy2 dyda

d2l d2l d2l d21

_dfda dad? dyda da2 .

(28)

(25)

n

e

n

2

n

?

2

x

2

?

2

?

x

x

x

2

?

n

2

n

?

2

2

x

x

2

2

n

c

s

2

2

2

x

x

x

1

The solutions to be obtained by solving (28) will yield the asymptotic variance and covariances of the parameters f, <j , a and fj. Using (28), the approximate 100(1 — A)% confidence intervals for p, a and n can be expressed as

p ± Z A>/£u,£j ± Z a VÊ22, n ± z x VË33, a ± z xVtu

2 2 2 2

where Z a is the upper Xth percentile of the standard normal distribution. where

dL = k

dn2 k

2ф 1- 1-e

2 \ -ца ^

(29)

dL d ф2

dL

da2

Ф2

(30)

(31)

dL dt2

t2 k ^ i=1

2yae

21

e x

+ k

i=1

4ца£2e

9

e x2

_2n_ k

X2 k X i=1

-k

i=1

2 ч -ца

4ф 1 - e

x4 ( 1 - e x ^2 N 2 1

Ц02£2 e

x2

x4 1 - e x2 n

+ k

i=1

4ца£2 e

2N 2

41

e x2

+ k

i=1

2 \ -ца

2ф 1 - e

nae

x2 ( 1 - e x2 j ц

2 \ -ца

4 ф 1 - e

nat2e

9 -tl 2e x2

41

-k

i=1

2e-

x2 1

ex

e x2 ц

+ k

i=1

-k

i=1

2 \ -ца

4 ф 1 - e

ца£2 e

x4 ( 1 - e x2 J ц

4t e

2e-Ï2

x4 1

ex

+ k

i=1

4n£2 e

2

41

e x2

(32)

2

x

3

ц

n

n

2

а

2

?

n

n

2

x

22

s

x

С

2

2

x

x

x

n

22

2

2

2

?

2

2

2

x

x

x

n

n

2

22

?

С

2

2

?

n

n

2

x

22

?

Э2 dtdç

k

i=1

2n 1 - e x2

2 \ -ца

e

x2 1

e x2 ц

д2 дадф

д2 дцдф

-k

i=1

2 \ -ца >

1- 1 - e-

Э2 дцда

(33)

(34)

(35)

(36)

2

с

2

x

n

2

?

0

2

x

n

ц

0

dqd?

- E

i=1

2 \ -qa

2n f 1 - e

qa? e

x2 ( 1-e-x2 1 n2

JL

dad?

(37)

(38)

2

?

2

2

x

x

n

0

4. Data Analysis

4.1. Simulation Studies

In this section, we simulated data set for sizes n = 30,100,200 and 500 that follows Gom-pertz Exponentiated Inverse Rayleigh distribution using different parameter values for the four parameters f, q, a and ? using the quantile function (inverse transformation method of simulation). We considered the following combinations for the parameters (f, q, a, ?) = ((0.5,1,1,0.8), (1.5,0.6,1.2,1.5), (0.5,0.5,0.5,0.5 ) and (1,0.5,0.5,1))at different sample sizes n = 30, 100, 200, and 500. The results presented in Table 1 displayed the true values of (f, q, a, ?) and estimated values of (f, q, a, ?) with the standard errors. The results are replicated 10,000 times and the average result were presented in the Table 1.

Table 1: The MLE estimates and their MSEfor different parameter values

f q a ? MSEf MSEq MSEK MSE?

30 f = 0.5 0.4164 0.6713 1.0835 0.6710 5.9423 9.5783 15.4494 0.1593

100 q = 1 0.5453 1.4859 0.6865 0.7407 2.7936 7.6042 3.5094 0.1109

200 a = 1 0.4947 0.9795 1.0074 0.7908 2.6984 5.3368 5.4865 0.0784

500 ? = 0.8 0.4410 1.3019 0.8647 0.7593 1.2450 3.6662 2.4340 0.0556

30 f = 1.5 1.1616 0.7094 0.8432 1.2341 16.8549 10.3031 12.2263 0.2242

100 q = 0.6 1.0910 1.0760 0.8447 1.3581 8.7396 8.6236 6.7624 0.1484

200 a = 1.2 1.0830 0.6560 1.2001 1.4536 5.4328 3.2944 6.0152 0.1018

500 ? = 1.5 0.8998 0.7517 1.2599 1.4447 5.4193 4.5324 7.5899 0.0701

30 f = 0.5 1.8939 1.3197 0.1331 0.4055 14.6817 10.2453 1.0298 0.0964

100 q = 0.5 0.4297 0.5878 0.4620 0.4533 3.1636 4.3259 3.3989 0.0672

200 a = 0.5 0.4996 0.5109 0.4939 0.4895 4.1287 4.2278 4.0834 0.0474

500 ? = 0.5 0.5095 0.7248 0.4092 0.4760 1.5357 2.1832 1.2322 0.0317

30 f = 1 1.3023 0.5704 0.3324 0.8097 27.7198 12.1465 7.0727 0.1621

100 q = 0.5 0.9354 0.7846 0.3940 0.9009 4.7474 3.9884 1.9979 0.1091

200 a = 0.5 0.9282 0.5241 0.5117 0.9699 4.5882 2.5960 2.5287 0.0763

500 ? = 1 0.9232 0.7251 0.4500 0.9601 5.0199 3.9469 2.4472 0.0514

4.2. Data Description

The strength data was originally reported by Badar and Priest [5] where the strength is measured in GPA for single carbon fibers and impregnated 1000-carbon fiber tows at gauge lengths of 20 mm. These data set were fitted to GEIR distribution, the Half- Logistics Inverse Rayleigh (HLIR) distribution by Almarashi et al [3] and the Type II Topp-Leone Inverse Rayleigh (T2TLIR) distribution by Mohammed and Yahia [14]. Other distributions that have been fitted to these same data are the Transmuted Inverse Rayleigh distribution (TIR) by Ahmad et al [2], the Odd Frechet Inverse Rayleigh (OFIR) distribution by Elgarhy and Alrajhi [8], one parameter Inverse Rayleigh (IR) by Trayer [20].

го

о

s ^ %

13* <ъ

Л) t-o сл

П ся

I

о* rt-

<-t- fD

a œ ? £

& 3

2 3

3 tu

75 i-(

tli ^

Я о fî rt-

№ S-S

fi ff.

CL n

7)

0 О

CL rt-tu ЦГ

5Г m ^ О с M

3 " Crq Д

g E

I' -

s ^

es ?d

** " -, n H

as % &

1 s й-

§ й • CL

OQ С

tTI

3

a =

n

О

Pi.

CD

g S

n'

QJ

О СГ QJ СГ

CD

Ci)

CDF

0.0 0.4 0.8

J_I_I_I_I_L

[4 CD

CD

Empirical probabilities 0.0 0.4 0.8

о J_I_I_I_I_L

CD ГО

CD CO

cr С

m

3 ■g

—:

n fu

U 13 CL

fl> O

n> rt-

Ci'

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Tl

to

a

j_

■a o

tD O

O □J

-a

o3

(D (fl

Density

0.0 0.2 0.4 0.6 0.8

U>

L

_L

_L

_L

J

ho

CD

1-0 CD

CD

Empirical quantiles 1.5 2.5 3.5

I I I I I

u-l ho

ho

in

1-0 CD

CO

in

m 3

T3

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n> Q.

O -<

(D

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(I>

ifl

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ö a\ ö

1£) 00 M Ü1

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|sj NO OJ ON ii

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GJ Ol 00 Ol

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go

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to on

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M M M \fi

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hf^ ON

no oj no o vi

ho

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On

ho

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Table 2: Goodness-of-fit measures based on AIC, BIC, HQIC, K-S values for the Strength (20mm) data set (data 1)

Models AIC BIC HQIC K-S Value P-value

GEIR( çz 7]z a 1 ) 104.43 107.367 109.112 0.021 0.9321

HLIR(h, /1) 105.003 109.472 106.776 0.0:96 0.966S

T2TLIRO, B) 108.137 112.605 109.91 0.0776 0.7993

TTR(Q, A) 145.879 148.113 146.765 0.2:4 0.0002

OFIR(0, a) 147.423 151.891 149.196 0.1S01 0.0227

[R(ft) 178.826 1S1.06 179.713 0.3549 0

Data II: Patients receiving an analgesic dataset

The data set is taken from Gross and Clark [11] which consists of 20 observations of patients receiving an analgesic! 1.1, 1.4,1.3, 1.7,1.9, 1.8,1.6, 2.2,1.7, 2.7, 4.1, 1.8,1.5, 1.2,1.4, 3.0,1.7, 2.3, 1.6, 2.0

Figure 6: Empirical and theoretical plot for Patients receiving an analgesic

Table 3: Estimates and Goodness-of-fit measures based on AIC, BIC, HQIC, K-S values for

Patients receiving an analgesic

Distribution

Parameters

AIC CAIC BIC HQIC Pvalue

39.356 40.212 43.338 40.1332 0.4493

46.365 47.0709 48.3564 46.7537 0.1435

48.5149 50.0149 51.5021 49.098 0.4597

40.805 41.5109 42.7965 41.1938 0.463

GEIR 2.3362134 -0.3288 2.4S1S 2.2056 EIR 0.8714 3.16S6

WR 11.8552 1.2364 0.0545

GR 3.2748 0.6926

5. Conclusion

In this study, a proposed four parameter distributions are added to Gompterz family of distribution called Gompterz exponentiated Inverse Rayleigh (GEIR). Some structural mathematical properties; Moment, Order Statistic, Skewness and kurtosis of the derived model are obtained. A simulation study is carried out to estimate the behaviour of the shape and scale parameters, also maximum likelihood estimation method was employed to estimate the parameters of the distribution and simulation studies were performed to assess the flexibility of the proposed distribution. For the simulated dataset, the result presented in Table (1), from the result, we observed that the estimated values gotten are close to the predefined parameters and that as n increases the MSE reduces which confirms to the law of large numbers.

However, application of two real-life data set shows that the GEIR has strong and better fit than other competing models i.e., the data sets were fitted to the Half- Logistics Inverse Rayleigh (HLIR) distribution and the Type II Topp-Leone Inverse Rayleigh (T2TLIR). Other distributions that have been fitted to these same data are the Transmuted Inverse Rayleigh distribution (TIR), the Odd Frechet Inverse Rayleigh (OFIR) distribution, exponentiated inverse Rayleigh distribution (EIR), Weibul Rayleigh (WR), Gamma Rayleigh (GR), one parameter Inverse Rayleigh (IR) distributions using goodness of fit and information criterion.

References

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[2] Ahmad, A, Ahmad S.P.,and Ahmed A. (2014) Transmuted Inverse Rayleigh distribution: A generalization of the inverse rayleigh distribution. Math. Theory Model, 4(7), 111-131

[3] Almarashi A.M., Majdah M.B., Elgarhy M., Jamal F. and Chesneau C. (2020). Statistical Inference of the Half-Logistic Inverse Rayleigh Distribution. Entropy, 2020,22, 449

[4] Alizadeh, M., Cordeiro, G. M., Pinho, L.G.B. and Ghosh, I. (2017). The Gompertz-G family of distributions. Journal of Statistical Theory and Practice, 11(1), 179-207.

[5] Bader, M. and Priest, A. (1982) Statistical Aspect of Fiber and Bundle Strength in Hybrid Composites. In: Hayashi, T., Kawata, S. and Umekawa, S., Eds., Progress in Science and Engineering Composites, ICCM-IV, Tokyo,, 1129-1136

[6] Brown, K. and Forbes, W.A. (1974). Mathematical model of aging processes. Journal of Gerontology, 29(1), 46-51.

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