BAYESIAN AND CLASSICAL ESTIMATIONS OF TRANSMUTED INVERSE GOMPERTZ DISTRIBUTION Текст научной статьи по специальности «Математика»

BAYESIAN AND CLASSICAL ESTIMATIONS OF TRANSMUTED INVERSE GOMPERTZ DISTRIBUTION

+1T. M. Adegoke, 2K.O. Obisesan, 3O. M. Oladoja & 4G.K. Adegoke

*1,2,3Department of Mathematics and Statistics, First Technical University, Ibadan, Nigeria.

In this article, the use of the transmuted inverse Gompertz distribution in modeling lifetime data is investigated particularly in cases where standard probability distributions are not able to properly handle complex datasets. The quadratic rank transmutation map scheme is utilized to obtain the distribution. The study explores several characteristics of the transmuted inverse Gompertz model, including the estimation of parameters through classical approaches such as maximum likelihood estimation, least-squares estimation, Crammer-Von Misses estimation, and maximum product spacing estimation. Additionally, the Bayesian techniques is used under different loss functions, such as the Linex loss function, square error loss function, and general entropy loss function. The estimates obtained from both classical and Bayesian techniques are evaluated using simulation. To illustrate the potential benefits of the transmuted inverse Gompertz model, a dataset on the strength ofaircraft window glass is employed. The results obtained from the application of the new distribution to the real-life dataset demonstrate that it yields superior fits in comparison to other well-known distributions. The study's findings suggest that the transmuted inverse Gompertz model can provide a useful alternative for modeling lifetime data. The research offers valuable insights into the distribution's properties and estimation techniques, as well as its superiority over other commonly used distributions. The new model can contribute to the development of more accurate and efficient models for analyzing lifetime dataset. Overall, the study highlights the importance of exploring new statistical models and techniques to improve data analysis and decision-making

Keywords: Classical method, Bayesian method, Posterior distribution, Loss functions,

Transmuted inverse Gompertz distribution.

The Gompertz distribution with two parameters is an expansion of the exponential distribution and was proposed by Gompertz [1]. It is widely used in survival analysis to construct accurate actuarial and human mortality tables. Additionally, it is a valuable tool for modeling survival distributions with increasing hazard rates and for describing the distribution of adult lifespans by demographers and actuaries (Willemse and Koppelaar [2]).

The inverse distribution was developed to model actuarial surveys biological and demography (El-Bassiouny et al. [3]). The inverse Gompertz distribution was proposed by Eliwa et al. [4] and it is useful in modeling lifetime observations. The cumulative probability density (CDF) and probability density function (PDF) are expressed as

4

Department of Computer Science, Nasarawa State University,Keffi, Nigeria. *[email protected], [email protected], [email protected], [email protected]

Abstract

1. Introduction

7 - i

g(x-, e, i) = -Je p

ei-1 + -

9 > 0, 7 > 0, x > 0

(2)

where the scale and shape parameters are presented as 9 and 7.

Recently in literature the development of new family of lifetime distributions using transmuted method has recently been attempted to estimate model parameters efficiently for the subject model (see Aryal and Tsokos [5] and Khan et al. [6]). Using the techniques known as quadratic rank transmutation map, proposed by Shaw and Buckley [7], we are able to propose a new three parameters transmuted inverse Gompertz (TR-IG) distribution.

If a random variable X has a transmuted distribution, then its cumulative probability density function (CDF) and probability density function (PDF) satisfy the following relationship:

F(x) = (1 - A)G(x) - AG2(x), |A f (x)= g(x)((1 - A) - 2AG(x))

1

(3)

(4)

where G(x) is the CDF of the baseline model, g(x) and f(x) are the corresponding PDF associated with G(x) and g(x), respectively.

The motivation of this work is to proposed a model that can be used to model complex dataset which the standard probability distributions model can't handle properly. Also we examine the potential use of the TR-IG distribution and determine a number of the mathematical features.

2. Transmuted inverse Gompertz distribution

Consider a positive value of x with three parameters TR-IG distribution having a shape parameter 7 and scale parameter 9 and the transmuted parameter |A| < 1 , the CDF can be derived by substituting Equation (1) into Equation (3)

F(x; 7,9, A) = (1 + A)e

-!( ef -1

A

7 > 0,9 > 0,x > 0, |A| < 1 (5)

2

1

ex

e

0 1 2 3 4 5

Figure 1: CDF Plot of TR-IG distribution for different parameter values .

Figure 1 shows the CDF plots of TR-IG distribution and we deduced that as x increases the CDF increase and remains constant as it tends to 1.

Substituting Equations (1) and (2) into Equation (4) produces the PDF of TR-IG distribution

f (x; 7,9, A) = 7e-9

1 + A - 2Ae

-91 et-1

7 > 0,9 > 0, x > 0, |A| < 1 (6)

Figure 2: PDF plot ofTR-IG distribution for different parameter values

Figure 2 shows that the PDF of TR-IG distribution is positively skewed. When |A| = 0, the TR-IG distribution CDF and PDF (5) and (6) becomes inverse Gompertz distribution with CDF (1) and PDF (2).

3. Statistical Properties

This section looks into some of the statistical characteristics of TR — IG(y, ft, X) such survival function, reversed hazard function, odds function, hazard function and moments.

3.1. Survival Function

A survival function of the TR-IG distribution can be expressed as

S(x) = 1 - (1 + A)e

-01ex-1

-A

-01ex-1

Y > 0,0 > 0, |A| < 1, x > 0

(7)

3.2. Hazard Function

The hazard function for the TR-IG distribution can be expressed as

Y e

x2

-U ei-1 ) + x

h(x) =

1 + A - 2Ae

- H ei-1

1 - (1 + A)e

-Y e* -1

A

- V ex-1

1 2

Y > 0,0 > 0, |A| < 1, x > 0

(8)

2

e

3.3. Reversed Hazard Function (RHF)

The reversed hazard function for the TR-IG distribution can be expressed as

1 e

- Ц ex-1 ) + 9

Ф)

1 + Я - 2Ae

- Ц ex-1

(1 + Я)е

- Ü e -1

Я

-Ü ei-1

1 2

Y > 0,9 > 0, |Я| < 1, x > 0 (9)

3.4. Odd Function

The odd function for the TR-IG distribution can be expressed as

(1 + Я)e

-91eX-1

Я

O(x)

-Y ex-1

1 - (1 + Я)e

- V e -1

Я

-Y ex-1

2

Y > 0,9 > 0, |Я| < 1,x > 0 (10)

3.5. Cumulative Hazard Function

The cumulative hazard function for the TR-IG distribution can be expressed as

/

H(x) = - log

- 9 eX-1 1 - (1 + Я)е У J - Я

\

-91 ex-1

Y > 0,9 > 0, |Я| < 1, x > 0 (11)

3.6. The Quantile Function

The expression for the quantile function is Q(u) = F-1 (u). Consequently, the TR-IG distribution quantile function can be written as

Q(u)

9

log 1 - 9 log

(1+Я)-у/ 1+(2-4и)Я+Я2 2Я

(12)

3.7. Moments

Moments can be used to analyze a variety of distributional features, including tendency, skewness, dispersion, and kurtosis. If X ~ TR — IG(y,ß, A), then the rth moment expression of TR-IG distribution can be expressed as

r<X.

xrf(x)dx (13)

pœ

>,r=j0 "

after performing some mathematical expressions, we obtained the moment generating function for the TR-IG distribution

ц'г = Mx(t)= E lCj(-1)l+jT]

i,j,r=0

1 (-t)r r(1 - r)

il r! [9(j - i - 1)]1-r

2

e

e

2

e

œ

4. Estimations of Transmuted Inverse Gompertz Distribution

4.1. Maximum Likelihood Estimation (MLE)

Let x1,x2,...xn be random variables (rvs) drawn from TR — IG(7,d,X). The log-likelihood function is defined as follows:

L( y, 9, A\x)

n " 1 - j L=1 ex-1j +9 LUX*

y L xe J n

1 + A - 2Ae

-¥ ^-1

i=1A i=1 taking the logarithm of Equation (15), we obtained the log-likelihood function

(15)

i = n iogy - log(x) - i l - ^ + 9 L 1 + L log

9 }

i=1

1

i=1

i=1

1 + A - 2Ae

- Ü -1

(16)

Maximizing logL(y,9, A) with respect to y, 9 and A, the system with non-linear equations is obtained as:

dL_ = n_ L dY = Y L

d[ = L d9 L

Y (e x - 1

92

e x — 1

-L

i=1

+ L

i=1

2Ap [ex - 1 9w

Yex ~9x

n1 + L x - L

i=1

i=1

2AQ^

where q

y( ex -1

dL = L

dA = L

i=1

! 9

Y e x -1

(1 - 2u)

(

(17)

(18) (19)

Ye x

~9x~

p = e

, ( = (1 + A - 2Ap)

To obtain the estimates of Jmle, @mle and xmle, we equate expressions (17) - (19) to zero and solve the system of nonlinear equation. It was observed that solution of this system can not be obtained analytically so we employed a numerical approach known as Newton Raphson method.

4.2. Least-squares Method (OLS)

Let the related order statistics (or) of rvs x\,x2,...,xn from the TR-IG distribution sorted in ascending order are denoted as X1:n, X2:n,..., Xn:n. According to Swain et al. [8], the OLS estimates of 7oLs, &oLs and AoLs can be obtained by minimizing Equation 20 with respect to y, 9 and A and equate the non-linear equations to zero.

S(y , 9, A\x) = L

i=1

(1 + A)e

ex — 1

A

/a \ 2 1

- K e1 1 1 i

e V J > n + 1

(20)

4.3. Cramér-von Mises (CVM)

Let the related os of rvs x\, x2,..., xn from the TR-IG distribution sorted in ascending order are denoted as X\.n, X2:n,..., Xn:n. To obtain the CVM estimates of 7cvm, 9 cvm and Acvm, Equation 21 is minimized with respect to , 9 and A and equate the non-linear equations to zero.

1n

C( Y,9 A) = 12n + £

i=1

(1 + A)e

-i ex-1

A

/ ñ \ ' 21 1

-if e -A e1 2i - 1

>--

I 2n

-1 2

(21)

n

n

n

n

9

n

9

2

4.4. Maximum Product Spacing Method (MPS)

Let the related os of rvs x\, X2,..., xn from the TR-IG distribution sorted in ascending order are denoted as X1:n, X2:n,..., Xn:n. According to Cheng and Amin [9], the MPS estimates of Jmps, QMPS and XMPS, can be obtained by minimizing Equation 22 with respect to 7, Q and X and equate the non-linear equations to zero.

D = (1 + A)e

-91 ex< -1

A

-91 ex< -1

(1 + A)e

-91e9- -1

A

-91e*-1 -1

(22)

4.5. Bayesian Analysis

Let x = (x1, x2,... xn) be a rv with parameters 7, Q and X having a size n. The posterior probability density function of the parameters 7, Q and X given x can be expressed as

Pr( 7,9, A\x)

n( 7,9, A)l( 7,9, A)

fff n(7,9, A)l(7,9, A)d(7,9, A)

(23)

where l(7,9, A) is the likelihood function expressed in Equation (15) and n(7,9, A) is the prior probability distribution, which is expressed in Equation (24).

n( 7,9, A)

jAT(a)b'

-9a-1e-9

7,9, A, a, b > 0

(24)

where: 7 ~ Uniform(0,7); X ~ Uniform(0, X); Q ~ Gamma(a, b) substituting Equations (i5) and (24), we obtained the posterior probability distribution

Pr( 7,9, A\x)

1 aß-1- ll - 9(e9-l) + X A{a)ba9 e b*x2e V 7 1" -9 (e9 -1)] 1 + A - 2Ae 9 \ J

r r r 1 an 1 -9 Y -9(eX-1)+X HI YA T{aW9a-1e X2e V ; 1 + A - 2Ae 9 V 7 d( 7,9, A)

(25)

2

2

9

e

1

4.6. Loss Functions

The Bayesian method was employed for the estimation of TR-IG distribution parameters, utilizing three different types of loss function. The first loss function considered was the LINEX loss function (LLF), also referred to as the linear-exponential loss function, initially proposed by Varian [11]. LLF is an asymmetric loss function that rises exponentially on one side of zero and linearly on the other, as described by Preda and Panaitescu [i2].

The second loss function used was the General Entropy loss function (GELF), first introduced by Calabria and Pulcini [13]. GELF is also an asymmetric loss function that has been utilized by several authors, such as Dey and Liu [14], Sule and Adegoke [15], and Ogunsanya et al. [17], who used GELF in its original form by setting c to be equal to 1.

The third and final loss function considered was the Squared Error loss function (SELF), which is a common loss function in statistics and machine learning. SELF is also known as the L2 loss function and is used to measure the difference between the estimated and true values squared.

If 9 is an estimator required to estimate the parameter 9, then the square error loss function can be defined as follows:

Lself a (9 — 9)2 (26)

The LLF can be expressed as

LLLF a K(ec(9—9) — c(9 — 9) — 1) k > 0, c = 0 (27)

where c and k are the scale and shape parameters of the LLF. In this study, we assume that k= 1. Provided that E9 [e—c9] exits, Bayes estimator of the LLF is the value 9 that minimizes Equation (27) (Zeller [18]).

— 1 ln (e9

e—c9

(28)

The GELF is defined as

/~\c \ c 1

/ 9 \ / 9 \ / 9 \

(29)

lgelf 9 a

9 — c loM 9 — 1

where c > 0. The minimum occurred at 9 = 9. The GELF Bayes estimator is the value 9 that minimizes Equation (29) and can be expressed as

9 = [E (9—c)]—c (30)

4.7. Lindley Approximation Method

To estimate Equation (25), we will employ an iterative techniques known as Lindley's approximation to estimate the parameter of interest for 7,9 and X. According to Lindley [19], if n is large enough, any ratio of the integral of the form

I(x) = Elu(y 9 X)] = ///u(7,9,X)e1^7,9,X)+p(y,9,X)Hi,9,X) (31)

i (x)= elu(7,9, X)\= J, J, J e1(Y,9,X)+p(Y,9,X) d(7r 9, X) (31)

where u(7,9, X) is a function of 7, 9 and X only, l(7,9, X) is the log-likelihood and p(7,9, X) is the log of prior distribution n(7,9, X). Equation (31) can be evaluated as

1

I(x) = u(7, 9, X) + (u1 «1 + U2«2 + U3 + «3 + «4 + «5) + ^ A(u\&11 + U2012 + U3C13)

11 + -B(u10'21 + u2^22 + u3023) + ~ 031 + u2032 + u3033) (32)

«i = P1 0i1 + P20i2 + P3°i3

«4 = u12 O12 + u13 O13 + u23023

«5 = 2 (uhOU + u22022 + ^3033)

A = 0n Lm + 2°12 L121 + 2°13 L131 + 2°23 L231 + 022 L221 + 033 L331

B = 011L112 + 2°i2 L122 + 2013 L132 + 2023 L232 + 022 L222 + 033 L332

C = 011L113 + 2012 L123 + 2013 L133 + 2023 L233 + 022 L223 + 033 L333

(33)

(34)

(35)

(36)

(37)

(38)

dp.

du(91,92,93) _ d2 u(91,92,93)

d9i

d9i d9j

Li,j,k

d3l(91,92,93) 39; 39; 39k

(39)

where 91 = 7, 92 = 9 and 93 = X. 0i,j is the (i, j) element of the matrix's inverse Li;j all evaluated at the MLE estimation and i, j, k = 1,2,3

P

u

d2l =_ E

dY2 = Y2 ¿Í

m =

dв2 = E

-E

2 e 2ЛYtf2 ц

+ E

2Л_2 ц

2Ye x l2^

-E

i=Í

-E

4Л2_2Ц2

в2w2

(40)

Yex

в x2

-E

2Лфц

W

W

-E

4Л2Y tf2^1

fl dЛ2

d2l дYдв

д21 дYдЛ

nY2

d21 двдЛ

-E

W2

2Y ф>ц

W

+ E

2ЛYtfцy

W

E Ы E

e x dx

+E

2ЛeYtfЦ

вw

E

д31 = 2n E д y3 Y'3

2_ц 2Л_3 ц

-E

i=

n

+ En

-E

i

n

+E

i=

2Лeцy 12пЛ2 e3ц2

в3 W2

2Л_ц

!pw

4Л2_ц2 ytf

+ E

2Лe x ц в xw

вw2

d3i = e \ 2Y3

дЛ3 = E W3

+ E

16пЛ3 e3ц3 в3 w3

(4Í)

(42)

(43)

(44)

(45)

(46)

m дв3

E

i=

-E

i=

д31 дY 2дв

n

+E

i=

д31

6 e [ в4 J n - En i= 6Yex в3 x n + E i= г„ в 1 3yne x в2 x2 n - En i= - в -Yex в x3 n\ - E i= 2Лтц w n\ - E i= 6ЛфYtfЦ w

12Л2фц2Ytf

W2

-E

2ЛYtfъЦ

W

-E

12Л2 y Ц>Ъц2

W2

дY 2 дЛ

E

i=

8Л2_2Ц2 в3 w2

= - E

4Л_2Ц

HW

-E

4Лeцex

в2 wx

-E

2Л_2 Ytfц

-E

8Л2

ex

в2 w2 x

-E

в2 w

16Л3 e^3Y tf в2^3

-E

i=

n

- En

16Л3 Ytf3Ц3

W3

(47)

12л2_2ц^ tf в2^2

2e2 ц в^ n + E i= 2Л_2 цY в2w2 _ n - En i= 8Л_2Ц2' в2 w2 _ n + E i= 8Л2e2^Y в2w3 _

(48)

(49)

д31 дв2 дЛ

-E

i= n

+E

2фц w

+ E

2Лфцy

W2

-E

2ЛYtf2 цy

W

-E

i= 22

2nYtf2Ц

8ЛYtf2Ц

W

n

+ E

8Л^ tf2 ц2 Y

W

(50)

д31 дв2дY

-E

i= n

+ E

i= n

+ E

n + E i= Го 2e x в2 x n - En i= Г в e x в x2 n\ - En i= 2Лфц w n\ + E i= 2ЛФ_Ц вw

2e

¥

4пЛ2ФЦ2e

w2в

12Л2 y tf^^

w2в

-E

i=

-E

4ЛYtfЦYtf

W

8Л2YtfЦ2Ytf

+ E

2ЛYtf2 _Ц

вw

W2

+E

16Л^ tf2 ц3 e

W3^

(5Í)

n

n

n

д31 дА2ду

- Е

i=

д31

+Е

4Y2 Яец

дШд<у

-Е

i= n

+Е

w3Q

n г27фц

w

2Я7фц\

+ Е

д3г

дЯ2дд ~ n r2nyфец

Е

i=

+ Е

4Yy фц

w2

-Е

4Y2 Я7фц

w3

(52)

w

2

-Е

Qw

n г2Яу$еЦ*

Qw2

87 фц2 Яе w2 Q

8 Я2

-Е

w3Q

(53)

where:

Y = (1 - 2ц) , Ф

e x -1 _ ex 1 ф

Q2 Q x I , Г

n- - ex ), ф = (--

27 ( e x-1

Ж

+

27e x 7e x

- "ei2"

Ф

2 e x-1

Q3

+ iâX _ I r

+ Q2x Qx2 I ' r

67 e x -1

_ 67ex + _ 7ei I е = fe x _ -Л

Q3 x + Q2 x2 Qx3 ,е = x 1

From the prior distribution in expression (24),

p = log(n(7,Q,Я)) = (a - 1) lnQ - b - ln7 - ln Я

p1

д_ д7

p2

д_ дв

a -1 -9~

p3

д_ дЯ

(54)

substituting Equations ((40)-(54)) into Equation (31) reduces the Lindley integral, therefore the Bayes estimates using SELF are thus

i. If u(7, Q, Я) = 7

7BS = 7 - ^011 +

ii. If u(7, Q, Я) = Q then

a-1-bê

Q

012 - 1013 + 2 ( A011 + B012 + C013 )

9bs = 9 — 1021 + 9 022 — 1023 + 1 (A012 + B022 + C032)

iii. If u(y, 9, X) = X then

XBS = X — 1031 + 032 — 1033 + 2 (A013 + B023 + C033) Also, the Bayes estimate using LINEX are thus

i. If u(y, 9, X) = e—c^ then

7bl = 7 + log (1 — c (—1011 + —9012 — 1013 — 2011 + 1 (A0U + B012 + C013)

ii. If u(y, 9, X) = e—c9 then

9bl = 9 + log (1 — c (— 4021 + —9022 — 1023 — 2022 + 1 (A012 + B022 + C032)

iii. If u(y, 9, X) = e—cX then

XBL = X + log (1 — c (— ±031 + —9032 — 1033 — 2033 + 1 (A013 + B023 + C033) Finally, the Bayes estimate using the GELF are thus i. If u(y, 9, X) = 7—c then

7 BG

7 -

1 - c (- 1011 + 012 - 1013 - ^ + 2 (A011 + B012 + C013))

ii. If u(7, Q, Я) = Q-c then

BG =

1 - Q (-1021 + a-QM022 - 1023 - c+1 + 2 (A012 + B022 + C032))

iii. If u(7, Q, Я) = Я-c then

Я BG =

7 -

1 - Я (-1031 + a-TM032 - 1033 - С2Я + 1 (A013 + B023 + C033)

n

c

c

c

5. Results

5.1. Simulation Techniques

In this section, we consider a monte carlo simulation study to evaluate the performance of all the estimators using the MSE and biases with respect to different sample sizes n =(20, 50, 100,200, 500) for different parameters TR - IG(y, 6, A) = [(1,1,1), (0.7,1 0.5), (0.5,1, -0.5) and (0.5, 0.5 ,-0.2)] respectively. The results obtained from the analysis are displayed in Tables (1 - 7) and the results shown that the estimates using both the classical techniques and Bayesian methods performed excellently in estimating model parameters of the TR-IG distribution since the estimated results are closed to the true parameter values with small MSE and bias as the sample sizes increases for all the estimation techniques considered in this study.

Table 1: The MLE estimates, Bias and MSE for different parameter values.

n Values 7 6 A 7 Bias 6Bias A Bias 7 MSE 6 MSE A MSE

20 1.3933 1.7840 -2.170 0.3933 0.7840 -3.1707 0.1547 0.6146 0.6146

50 7 = 1 1.0849 1.0267 2.0412 0.0849 0.0267 1.0412 0.0072 0.0007 0.0007

100 6 = 1 1.2748 1.4741 1.5233 0.2748 0.4741 0.5233 0.0755 0.2248 0.2248

200 A = 1 1.0842 0.9210 1.0594 0.0842 -0.0789 0.0594 0.0071 0.0062 0.0062

500 1.0406 0.9230 1.0752 0.0406 -0.0769 0.0752 0.0016 0.0059 0.0059

20 0.8490 0.9426 0.9628 0.1490 -0.0573 0.4628 0.0222 0.0032 0.0032

50 7 = 0.7 0.8012 0.4173 2.1917 0.1012 -0.5826 1.6917 0.0102 0.3395 0.3395

100 6 = 1 0.9900 0.6209 1.5020 0.2900 -0.3790 1.0020 0.0841 0.1436 0.1436

200 A = 0.5 0.8170 1.0175 0.5633 0.1170 0.01757 0.0633 0.0137 0.0003 0.0003

500 0.7696 0.9641 0.5986 0.0696 -0.0358 0.0986 0.0048 0.0012 0.0012

20 0.6706 1.2368 -1.7885 0.1706 0.2368 -1.2885 0.0291 0.0561 0.0561

50 7 = 0.5 0.4520 0.5452 -0.3392 -0.0479 -0.4547 0.1607 0.0023 0.2068 0.2068

100 6 =1 0.5724 0.9028 -0.5957 0.0724 -0.0971 -0.0957 0.0052 0.0094 0.0094

200 A = -0.5 0.6274 0.9089 -0.3868 0.1274 -0.0910 0.1131 0.0162 0.0082 0.0082

500 0.5189 0.9966 -0.4976 0.0189 -0.0033 0.0023 0.0003 1.1E-05 1.1E-05

20 0.0679 -0.0188 -0.1817 -0.4320 -0.5188 0.0182 0.1866 0.2691 0.2691

50 7 = 0.5 0.2330 0.0920 0.2068 -0.2667 -0.4079 0.4068 0.0712 0.1664 0.1664

100 6 = 0.5 0.3784 0.3681 -0.2509 -0.1215 -0.1318 -0.0509 0.0147 0.0173 0.0173

200 A = -0.2 0.4593 0.4362 -0.1983 -0.0406 -0.0637 0.0016 0.0016 0.0040 0.0040

500 0.4179 0.5230 -0.3623 -0.0820 0.0230 -0.1623 0.0067 0.0005 0.0005

Table 2: The Bayesian estimate using GELF, Bias and MSE for different parameter values.

n Values 7 6 A 7 Bias 6Bias ABias 7MSE 6MSE AMSE

20 „ 1.5700 1.9749 0.2172 -0.5700 -0.9749 0.7828 0.3249 0.9504 0.9504

Y = 1

50 6 = 1 1.0552 0.9894 1.6972 -0.0552 0.0106 0.6972 0.0030 0.0001 0.0001

100 A6 == 11 1.2573 2.1781 1.8270 -0.2573 -1.1781 -0.8270 0.0662 1.3878 1.3878

200 A = 1 1.0766 0.9160 1.0860 -0.0766 0.0840 -0.0860 0.0059 0.0071 0.0071 c = 1

500 1.0377 0.9208 1.0854 -0.0377 0.0792 -0.0854 0.0014 0.0063 0.0063

20 0.7795 0.8400 -2.7292 -0.0795 0.1600 1.2292 0.0063 0.0256 0.0256 = 0.7

50 6 = 1 0.8106 0.4566 -0.8339 -0.1106 0.5434 1.3339 0.0122 0.2953 0.2953

100 A_0 5 0.9805 0.6114 8.3239 -0.2805 0.3886 -7.8239 0.0787 0.1510 0.1510

200 A = 0.5 0.8090 1.0061 0.6476 -0.1090 -0.0061 -0.1476 0.0119 0.0000 0.0000 c =1

500 0.7667 0.9596 0.6238 -0.0667 0.0404 -0.1238 0.0044 0.0016 0.0016

20 = 0.5 0.6499 1.1989 -1.7308 -0.1499 -0.1989 1.2308 0.0225 0.0395 0.0395

50 Vi 0.4390 0.5267 -0.2967 0.0610 0.4733 -0.2033 0.0037 0.2240 0.2240

100 A6==-01.5 0.5669 0.8942 -0.5799 -0.0669 0.1058 0.0799 0.0045 0.0112 0.0112

200 A = -0.5 0.6246 0.9049 -0.3776 -0.1246 0.0951 -0.1224 0.0155 0.0090 0.0090 c = -1

500 0.5181 0.9953 -0.4949 -0.0181 0.0047 -0.0051 0.0003 0.0000 0.0000

20 = 0.5 0.0362 -0.0674 -0.0715 0.4638 0.5674 -0.1285 0.2151 0.3219 0.3219

50 6 = 0.5 0.2153 0.0708 0.2833 0.2847 0.4292 -0.4833 0.0811 0.1842 0.1842

100 A I°052 0.3703 0.3577 -0.2202 0.1297 0.1423 0.0202 0.0168 0.0202 0.0202

200 A = -0.2 0.4553 0.4314 -0.1808 0.0447 0.0686 -0.0192 0.0020 0.0047 0.0047 c = -1

500 0.4167 0.5214 -0.3567 0.0833 -0.0214 0.1567 0.0069 0.0005 0.0005

Table 3: The Bayesian estimate using LINEX, Bias and MSE for different parameter values

n Values 7 6 A 7 Bias 6Bias ABias 7MSE 6MSE AMSE

20 = . 0.3809 0.4126 2.2994 0.6191 0.5874 -1.2994 0.3833 0.3451 0.3451

50 = 1 1.2102 0.6444 -2.5123 -0.2102 0.3556 3.5123 0.0442 0.1264 0.1264

Û _ 1

100 A = 1 1.1813 0.1028 0.6599 -0.1813 0.8972 0.3401 0.0329 0.8050 0.8050

200 A = 1 1.1288 1.0870 0.9989 -0.1288 -0.0870 0.0011 0.0166 0.0076 0.0076 c =1

500 1.0672 1.0210 1.0269 -0.0672 -0.0210 -0.0269 0.0045 0.0004 0.0004

20 0.9130 1.0132 0.2642 -0.2130 -0.0132 0.2358 0.0454 0.0002 0.0002 = 0.7

50 6 = 1 1.2158 0.9125 -0.3400 -0.5158 0.0875 0.8400 0.2661 0.0077 0.1685

100 A_0 5 1.1905 0.8456 -0.2682 -0.4905 0.1544 0.7682 0.2406 0.0239 0.0239

200 A = 0.5 0.7941 0.9342 0.4847 -0.0941 0.0658 0.0153 0.0089 0.0043 0.0043 c =1

500 0.7636 0.9384 0.5599 -0.0636 0.0616 -0.0599 0.0040 0.0038 0.0038

20 = 0.5 0.6660 1.2612 -0.7200 -0.1660 -0.2612 -0.2200 0.0276 0.0682 0.4852

50 V-1 0.4446 0.5059 -0.3653 0.0554 0.4941 -0.1347 0.0031 0.2442 0.2442

100 A6==-01.5 0.5743 0.9084 -0.6158 -0.0743 0.0916 0.1158 0.0055 0.0084 0.0084

200 A =_'"0 0.6285 0.9109 -0.3955 -0.1285 0.0891 -0.1045 0.0165 0.0079 0.0079

500 c=-1 0.5193 0.9980 -0.5008 -0.0193 0.0020 0.0008 0.0004 0.0000 0.0000

20 0.0007 -0.3024 -0.2992 0.4993 0.8024 0.0992 0.2493 0.6438 0.6438 = 0.5

50 6 = 0.5 0.2015 0.0137 0.0638 0.2985 0.4863 -0.2638 0.0891 0.2365 0.2365

100 A6 == -00..52 0.3741 0.3568 -0.2791 0.1259 0.1432 0.0791 0.0159 0.0205 0.0205

200 A =_''0. 0.4584 0.4332 -0.2103 0.0416 0.0668 0.0103 0.0017 0.0045 0.0045

500 c=-1 0.4181 0.5233 -0.3653 0.0819 -0.0233 0.1653 0.0067 0.0005 0.0005

Table 4: The B«yesi«n estim«te using SELF, Bi«s «nd MSEfor different p«r«meter v«lues

n 7 9 X 7 Bi«s 9Bi«s X Bi«s 7mse 9mse XMSE

20 0.9476 1.2083 0.9019 0.0524 -0.2083 0.0981 0.0027 0.0434 0.0434

50 = 1 1.1452 0.8396 1.0749 -0.1452 0.1604 -0.0749 0.0211 0.0257 0.0257

100 9 = 1 1.2268 0.9133 1.1424 -0.2268 0.0867 -0.1424 0.0514 0.0075 0.0075

200 X = 1 1.1058 1.0044 1.0302 -0.1058 -0.0044 -0.0302 0.0112 0.0000 0.0000

500 1.0537 0.9721 1.0515 -0.0537 0.0279 -0.0515 0.0029 0.0008 0.0008

20 0.8749 0.9668 0.6708 -0.1749 0.0332 -0.1708 0.0306 0.0011 0.0011

50 7 = 0.7 1.0160 0.6759 0.8044 -0.3160 0.3241 -0.3044 0.0998 0.1051 0.1051

100 Q = 1 1.0910 0.7341 0.8397 -0.3910 0.2659 -0.3397 0.1529 0.0707 0.0707

200 Я = 0.5 0.8050 0.9751 0.5277 -0.1050 0.0249 -0.0277 0.0110 0.0006 0.0006

500 0.7664 0.9509 0.5806 -0.0664 0.0491 -0.0806 0.0044 0.0024 0.0024

20 0.5120 1.0705 -0.6552 -0.0120 -0.0705 0.1552 0.0001 0.0050 0.0050

50 = 0.5 0.6015 0.8504 -0.4043 -0.1015 0.1496 -0.0957 0.0103 0.0224 0.0224

100 Q = 1 0.5988 0.9835 -0.4885 -0.0988 0.0165 -0.0115 0.0098 0.0003 0.0003

200 Я = -0.5 0.6511 0.9742 -0.3346 -0.1511 0.0258 -0.1654 0.0228 0.0007 0.0007

500 0.5214 1.0081 -0.4678 -0.0214 -0.0081 -0.0322 0.0005 0.0001 0.0001

20 0.4064 0.5934 -0.6487 0.0936 -0.0934 0.4487 0.0088 0.0087 0.0087

50 = 0.5 0.4748 0.4553 -0.3847 0.0252 0.0447 0.1847 0.0006 0.0020 0.0020

100 Q = 0.5 0.4836 0.5361 -0.4501 0.0164 -0.0361 0.2501 0.0003 0.0013 0.0013

200 Я = -0.2 0.5183 0.5333 -0.3103 -0.0183 -0.0333 0.1103 0.0003 0.0011 0.0011

500 0.4336 0.5511 -0.3879 0.0664 -0.0511 0.1879 0.0044 0.0026 0.0026

Table 5: OLS estim«tes, Bi«s «nd MSEfor the simul«ted v«lues

n Values 7 Q Я 7 Bias QBias XBias 7MSE (¡USE XMSE

20 1.0444 0.7986 1.1393 -0.0444 0.2014 -0.1393 0.0020 0.0405 0.0405

50 = 1 0.9979 1.0074 1.0142 0.0021 -0.0074 -0.0142 0.0000 0.0001 0.0001

100 Q = 1 1.0355 1.0912 0.9270 -0.0355 -0.0912 0.0730 0.0013 0.0083 0.0083

200 Я = 1 1.0684 1.1656 0.8362 -0.0684 -0.1656 0.1638 0.0047 0.0274 0.0274

500 1.0309 1.0772 0.9979 -0.0309 -0.0772 0.0021 0.0010 0.0060 0.0060

20 0.5445 1.0569 0.6307 0.1555 -0.0569 -0.1307 0.0242 0.0032 0.0032

50 = 0.7 0.7043 1.0029 0.5329 -0.0043 -0.0029 -0.0329 0.0000 0.0000 0.0000

100 Q = 1 0.7956 1.0147 0.4648 -0.0956 -0.0147 0.0352 0.0091 0.0002 0.0002

200 Я = 0.5 0.8361 1.0624 0.3252 -0.1361 -0.0624 0.1748 0.0185 0.0039 0.0039

500 0.7327 1.0585 0.4770 -0.0327 -0.0585 0.0230 0.0011 0.0034 0.0034

20 0.4300 0.9754 -0.3656 0.0700 0.0246 -0.1344 0.0049 0.0006 0.0006

50 = 0.5 0.5046 1.0022 -0.4892 -0.0046 -0.0022 -0.0108 0.0000 0.0000 0.0000

100 Q = 1 0.5923 1.0111 -0.5143 -0.0923 -0.0111 0.0143 0.0085 0.0001 0.0001

200 Я = -0.5 0.6208 1.0654 -0.5974 -0.1208 -0.0654 0.0974 0.0146 0.0043 0.0043

500 0.5374 1.0534 -0.5178 -0.0374 -0.0534 0.0178 0.0014 0.0029 0.0029

20 0.5315 0.3582 -0.1056 -0.0315 0.1418 -0.0944 0.0010 0.0201 0.0201

50 = 0.5 0.5138 0.4900 -0.1926 -0.0138 0.0100 -0.0074 0.0002 0.0001 0.0001

100 Q = 0.5 0.5237 0.5591 -0.2425 -0.0237 -0.0591 0.0425 0.0006 0.0035 0.0035

200 Я = -0.2 0.5523 0.5970 -0.3090 -0.0523 -0.0970 0.1090 0.0027 0.0094 0.0094

500 0.5197 0.5442 -0.2383 -0.0197 -0.0442 0.0383 0.0004 0.0020 0.0020

Table 6: MPS estim«tes, Bi«s «nd MSEfor the simul«ted v«lues

n Values 7 Q Я 7 Bias QBias %-Bias 7MSE (¡USE XmSE

20 1.0230 0.9286 0.7827 -0.0230 0.0714 0.2173 0.0005 0.0051 0.0051

50 = 1 1.0807 0.8271 0.9403 -0.0807 0.1729 0.0597 0.0065 0.0299 0.0299

100 Q = 1 1.1303 0.9557 1.0101 -0.1303 0.0443 -0.0101 0.0170 0.0020 0.0020

200 Я = 1 1.0660 1.0193 0.9686 -0.0660 -0.0193 0.0314 0.0044 0.0004 0.0004

500 1.0181 0.9949 1.0038 -0.0181 0.0051 -0.0038 0.0003 0.0000 0.0000

20 0.9815 0.6772 0.6045 -0.2815 0.3228 -0.1045 0.0792 0.1042 0.1042

50 = 0.7 1.0213 0.6075 0.7410 -0.3213 0.3925 -0.2410 0.1032 0.1540 0.1540

100 Q = 1 1.0915 0.6921 0.7970 -0.3915 0.3079 -0.2970 0.1532 0.0948 0.0948

200 Я = 0.5 0.8248 0.9325 0.5298 -0.1248 0.0675 -0.0298 0.0156 0.0046 0.0046

500 0.7810 0.9256 0.5875 -0.0810 0.0744 -0.0875 0.0066 0.0055 0.0055

20 1.6379 0.1379 0.5668 -1.1379 0.8621 -1.0668 1.2947 0.7432 0.7432

50 = 0.5 1.5484 0.1713 0.6225 -1.0484 0.8287 -1.1225 1.0991 0.6868 0.6868

100 Q = 1 1.6676 0.2513 0.6615 -1.1676 0.7487 -1.1615 1.3633 0.5606 0.5606

200 Я = -0.5 0.6542 0.9326 -0.3554 -0.1542 0.0674 -0.1446 0.0238 0.0045 0.0045

500 0.9890 0.6552 0.2423 -0.4890 0.3448 -0.7423 0.2391 0.1189 0.1189

20 0.8555 1.1375 0.4301 -0.3555 -0.6375 -0.6301 0.1264 0.4065 0.4065

50 = 0.5 1.1167 0.0400 0.5593 -0.6167 0.4600 -0.7593 0.3803 0.2116 0.2116

100 Q = 0.5 1.2114 0.0889 0.5987 -0.7114 0.4111 -0.7987 0.5062 0.1690 0.1690

200 Я = -0.2 0.5270 0.5020 -0.3156 -0.0270 -0.0020 0.1156 0.0007 0.0000 0.0000

500 1.1111 0.0000 0.6778 -0.6111 0.5000 -0.8778 0.3735 0.2500 0.2500

Table 7: The CVM estim«tes, Bi«s «nd MSEfor the simul«ted values

n Values 7 Q Я 7 Bias QBias X-Bias 7MSE (¡USE XmSE

20 1.0209 0.9660 1.1022 -0.0209 0.0340 -0.1022 0.0004 0.0012 0.0012

50 = 1 1.0513 1.0769 0.9379 -0.0513 -0.0769 0.0621 0.0026 0.0059 0.0059

100 Q = 1 1.0598 1.1720 0.8364 -0.0598 -0.1720 0.1636 0.0036 0.0296 0.0296

200 Я = 1 1.1118 1.2235 0.7814 -0.1118 -0.2235 0.2186 0.0125 0.0500 0.0500

500 1.0926 1.0991 0.8669 -0.0926 -0.0991 0.1331 0.0086 0.0098 0.0098

20 0.6705 1.0208 0.5809 0.0295 -0.0208 -0.0809 0.0009 0.0004 0.0004

50 = 0.7 0.7955 1.0136 0.4675 -0.0955 -0.0136 0.0325 0.0091 0.0002 0.0002

100 Q = 1 0.8104 1.0894 0.3628 -0.1104 -0.0894 0.1372 0.0122 0.0080 0.0080

200 Я = 0.5 0.8899 1.1010 0.2976 -0.1899 -0.1010 0.2024 0.0361 0.0102 0.0102

500 0.7668 1.0906 0.4191 -0.0668 -0.0906 0.0809 0.0045 0.0082 0.0082

20 0.4851 1.0183 -0.4424 0.0149 -0.0183 -0.0576 0.0002 0.0003 0.0003

50 = 0.5 0.5593 1.0397 -0.5468 -0.0593 -0.0397 0.0468 0.0035 0.0016 0.0016

100 Q = 1 0.6335 1.0521 -0.5911 -0.1335 -0.0521 0.0911 0.0178 0.0027 0.0027

200 Я = -0.5 0.6754 1.0881 -0.6524 -0.1754 -0.0881 0.1524 0.0308 0.0078 0.0078

500 0.5966 1.0494 -0.5740 -0.0966 -0.0494 0.0740 0.0093 0.0024 0.0024

20 0.5134 0.4739 -0.1671 -0.0134 0.0261 -0.0329 0.0002 0.0007 0.0007

50 = 0.5 0.5278 0.5544 -0.2396 -0.0278 -0.0544 0.0396 0.0008 0.0030 0.0030

100 Q = 0.5 0.5601 0.5904 -0.3026 -0.0601 -0.0904 0.1026 0.0036 0.0082 0.0082

200 Я = -0.2 0.6012 0.6143 -0.3492 -0.1012 -0.1143 0.1492 0.0102 0.0131 0.0131

500 0.5453 0.5665 -0.2794 -0.0453 -0.0665 0.0794 0.0021 0.0044 0.0044

5.2. Application

In this section, we analyzed the strength of glass of aircraft window datasets adopted by Fuller et al. [20] (whose dataset is displayed in Table ( 8) to ascertain that the TR-IG distribution is a good lifetime model, when compared with three known distribution like Inverse Gompertz (IGD), inverse Rayleigh (IR) and inverse Exponential distribution (IE). To assess the TR-IG distribution's goodness-of-fit with these distributions, some criteria (such as the Kolmogorov-Smirnov test statistic (KS), log-likelihood (L) values, Akaike information criterion (AIC), Bayesian information criterion (BIC), and Cramer-von Mises statistic (W*) and Anderson-Darling statistic (A*)) were used to fit all of the above-mentioned distributions.

Figure 3 displayed the estimated PDFs, estimated CDFs of all tested distributions and the empirical and theoretical CDF. From the results displayed in Table 9, we deduced that the TR-IG distribution fits the data better than the other four model. Table 10, shows the classical and Bayesian estimates for the strength of glass of aircraft windows.

Table 8: The strength of glass of aircraft window.

18.83 20.8 21.657 23.03 23.23 24.05 24.321 25.5 25.52

26.77 26.78 27.05 27.67 29.9 31.11 33.2 33.73 33.76

35.75 35.91 36.98 37.08 37.09 39.58 44.045 45.29 45.381

Table 9: The estimates and goodness-of-fit measurements for glass strength data set.

Statistics Model

IE IR IG TR-IG

7 29.215 810.504 1.249 0.6563

e A KS - -0.6411 119.762 126.5584

0.477 0.325 0.139 0.1349

-L 137.262 118.201 107.884 94.5072

AIC 241.2363 208.4237 195.9174 193.0145

BIC 242.5321 209.7196 198.5090 196.902

A* 7.2108 3.5316 0.6034 0.5349

W* 1.5051 0.6627 0.0841 0.0748

Histogram and theoretical densities Empirical and theoretical CDFs

20 25 30 35 40 45 20 25 30 35 40 45

data data

Figure 3: The Histogram, empirical and theoretical densities for Glass strength dataset.

Table 10: Estimated values of the strength of glass for aircraft window under different estimation techniques

Estimation Techniques 7 6 A

CVM 0.4574 124.1316 -0.7821

MPS 0.9403 112.6889 -0.6803

SELF 0.5428 121.5032 -0.8790

LINEX (c =-1) 0.5467 129.2076 -0.9585

OLS 0.5663 122.5142 -0.5257

GELF (c =1) 0.7698 129.7899 -0.5152

6. Conclusion

We introduced a novel model called the TR-IG distribution, which expands upon the inverse Gompertz distribution for analyzing data with a real support. One clear motivation for extending a standard distribution is to increase the flexibility in modeling complex dataset. We derived some properties such as hazard function, survival function and etc. The parameters were estimated using both the classical and Bayesian estimation techniques.The utilization of the TR-IG distribution on actual data demonstrates that this new distribution can be employed with great effectiveness to yield superior fits in comparison IE, IR and IG distributions.

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