CC BY
36
Inverse Weibull-Burr III Distribution with Properties and Application Related to Survival Rates
in Animals
Aijaz Ahmad* •
Department of Mathematics, Bhagwant University, Ajmer, India [email protected]
Mujamil Jallal •
Department of Mathematics, Bhagwant University, Ajmer, India [email protected]
I. H. Dar •
Department of Statistics, University of Kashmir, Srinagar, India
Rajnee Tripathi •
Department of Mathematics, Bhagwant University, Ajmer, India rajneetripathi@hotmail,com
Abstract
The objective of this study is to develop an extension of the Burr-Ill distribution which is achieved by adopting the inverse Weibull-G family of distribution and is referred as inverse Weibull-Burr Ill distribution (IWB-lll) to evaluate complicated data. Different structural characteristics of the suggested distribution have been determined and analysed. Distinct plots depict the behaviour of the probability density function (pdf) and the cumulative distribution function (cdf). The maximum likelihood estimation method is applied to estimate the stated distribution parameters. To assess and investigate the efficacy of estimators in terms of bias, variance, and mean square error (MSE), a simulation study was conducted. Lastly, the effectiveness of the stated distribution is proven by an actual data set relevant to survival rates in animals.
Keywords: Inverse Weibull-G family; Burr-III distribution; moments, Renyi entropy; simulation; maximum likelihood estimation.
1. Introduction
Over numerous decades, academics have been attempting to develop a number of novel distributions to satisfy certain realistic demands. The rationale is that conventional distributions have
generally been shown to lack fit in actual applications, such as medicinal research, engineering, hydrology, environmental science, and many more. In particular, the objective of creating novel distributions or generalisations is to construct adaptable statistical models effective at dealing with complicated real-world data. This adaptability may be obtained in a straightforward manner by introducing new parameters to the standard distribution.
The Weibull distribution has been utilised in a variety of disciplines and applications. The hazard function of the Weibull distribution can only be monotonic in nature. As a result, it can not be employed to simulate lifespan data with a bathtub-shaped hazard function.
Let X be a random variable that follows the Weibull distribution with parameters a and p. Then its probability density function (pdf) is defined as
$(x, a, p) = appxp-1e-aPxP; x > 0, a, p > 0
The transformation T = y, yields the inverse of the Weibull distribution. As a result, the probability density function (pdf) of the inverse Weibull distribution assumes the following structure.
h(t,a,p) = appt-(p+1)e-apt-p; t > 0,a,p > 0 (1)
In this work, we construct the inverse Weibull-Burr III distribution, which is an expansion of the Burr-III distribution. Burr, I.W [4] advocated a family of twelve cumulative distribution functions for simulating lifespan data. Burr-type III and Burr-type XII distributions were two prevalent members of the family. The Burr-III distribution has been studied thoroughly and employed in a range of aspects of research. Daniyal et al [9], Al-Dayian et al [10] and B.A. para et al [6] provide further information on the characteristics of the Burr-III distribution. The probability density function (pdf) of Burr-III distribution is stated as.
g(y, 9, A) = 9Ay-9-1 (l + y-9)-A-1; y > 0, d, A > 0 (2)
The associated cumulative distribution function (cdf) of equation (1.2) is given as
G(y, 9, A)= + y-^ A ; y > 0,9, A > 0 (3)
In recent decades, researchers have concentrated on discovering novel generators from continuous conventional distributions. As an outcome, the resulting distribution enhances the efficacy and adaptability of data analysis. The following are some generated families of distribution: the beta-G family of distribution investigated by Eugene et al [11], the gamma-G family by Zagrofos and Balakrishana [13], the kumaraswamy-G family by Cordeiro et al [8], the transformedtransformer(T-X) by Alzaatrh et al [1], the Weibull-G by Bourguignon et al [3],Brito et al. [5] created the Topp-Leone odd log-logistic family of distributions, Morad Alizadeh et al. [12] constructed the Gompertz-G distribution family, and Amal S. Hassan et al. [2] established the inverse Weibull-G distribution.
T-X family of distributions defined by Alzaatreh et al [1] is given by
r W[G(y)]
F(y) = yo r(t)dt (4)
Where r(t) be the probability density function of a random variable Tand W[G(y)] be a function of cumulative density function of random variable Y.
Suppose G(y, <),denotes the baseline cumulative distribution function, which depends on parameter vector Now using T-X approach, the cumulative distribution function F(y) of inverse Weibull
generator (IWG) can be derived by replacing r(t) in equation (1.4) with (1.1) and W[G(y)] = G^ßf), where G (y, f) = 1 — G(y, f) which follows
. G(y,f)
F(y, f)=j G(y,f) a.ßßt—(ß+1)e—aßt—ßdy
-aß (GM1 )—ß
=e \G(y,f')) ;y > 0,a,ß, f > 0 (5)
where G (y, f) = 1 — G(y, f)
The associated pdf of (1.5) is given as
(G(y f))—ß—1 ~aß(Gyn)—ß f(y,a, f)=aßßg(y, f)(G(y, f)^)—ß+1 e U (yf)J ;y > 0,a,ß, f > 0 (6)
In addition, Reliability function denoted as F(y, f), hazard rate function denoted as h(y, f) and cumulative hazard rate function denoted as H(y, f) are respectively given as
_aß(GMl )—ß F (y, f) =1 — e VG (y,f)J
G(y,f) )—ß
h(y, (G to»-*1
Kß( mu vß
1 - e \G(y,t)
i _aß( GM) Yß' H (y, u) = - ln[F (y, u)] = -lui 1 - e v<5 (y,t))
1.1. Usefull Expansions
Applying Taylor series expansion to the exponential function of the pdf in equation (1.6) we have
)"' = £ (nil!(«Ml(7) ¿0 VG(y,t)J ' '
substituting equation (1.7) in equation (1.6), we have
f (y, t) = £ ^8
= £ ^pg(y, t) (G(y, t))-^+1)-1 (1 - G(y, t))^-1 (9)
s=0 !
using generalised binomial theorem, we have
£ 'a-1
(1 - z)a-1 = E (-1)p\p )zp
p=0 V p /
(G (y, u))-ß(s+1)+1 =(1 - G (y, u))ß( F+1)-1 = Ê (-1) F (ß( P + 1) - 1)(G(y, H))F
p=0 \ p y
= tt t^ (ß(s + 1) - ^ßgy H)(G(y,,))F-ß(+)-1
s=0 p=0 b- V F y
EE Zs,pg(y, <P)(G(y, u))F-ß(s+1)-1 (10)
s=0 p=0
where
Zs,p ß(S + 1 - ')aß™ß
Now using equation (2) and (3) in equation (10), we obtain pdf of formulated distribution in mixture form, as follows
^ ^ n ■, f n\ -A-1 r n ,1 p-p(s+1)-1
f(y,a,p,e,a) = EEZs,vexy-e-1 (1 + y-e) (1 + y-e)-A p )
s=0p=0
; y > 0, a, p, e, A > 0 (11)
2. Inverse Weibull-Burr III Distribution
In this part, we'll investigate the inverse Weibull-Burr III distribution and look at aspects of its statistical characteristics. We derive the cumulative distribution function (cdf) of the given distribution using equation (3) in equation (5) as follows.
F(y, a, p, e, A) = e-aP ((1+y-e )A-1)p (12)
Figure 1: Expounds some of possible layouts of the cdf of IWB-III distribution for distinct choice of parameters
/ i y' /¡I
Li
□ alpha=0.9,beta=1.5,theta=0.5,lambda=0.7
□ alpha=0.7,beta=1.7,theta=0.8,lambda=0.9
□ alpha=0.4,beta=1.4,theta=0.7,lambda=1.3
□ alpha=0.6,beta=1.3,theta=0.9,lambda=1.0
□ alpha=0.8,beta=1.2,theta=0.6,lambda=0.8
- - ^ f!. ■ ■ ■"'
/ /■' / s „ ' _________
/
j .■ /
/ , / / / V
■' / , : ! i •' / . ■ . //. □ alpha=1.2,beta=1.5,theta=0.5,lambda=0.5
' ■■/.' □ alpha=1.3,beta=1.7,theta=0.6,lambda=0.6
□ alpha=1.4,beta=1.4,theta=0.8,lambda=0.7
i •''/.' □ alpha=0.6,beta=1.3,theta=0.9,lambda=1.0
iZ" □ alpha=1.2,beta=1.0,theta=0.6,lambda=0.9
Figure 1: plots of cdf for IWB-IH distribution
The associated pdf of (12) is given as
f (y,a,p,e, A) = appeAy-e-1 (1 + y-e)A-1((1 + y-e)A - 1)p-1e-aP((1+y-e)A-1)p (13)
; a > 0, p > 0, e > 0, A > 0
Figure 2: Expounds some of possible layouts of the pdf of IWB-III distribution for distinct choice of parameters
3. Reliability Measures of (IWB-III) Distribution
This section is focused on researching and developing distinct ageing indicators for the formulated distribution.
□ alpha=0.5.beta=1.5.theta=0.8.lambda=2.0
□ alpha=0.7,beta=2.8,theta=1.3,lambda=2.3 ■ alpha=1.9,beta=1.4,theta=2.5,lambda=1.5
□ alpha=1.2,beta=1.5,theta=1.7,lambda=3.5
□ alpha=1.5,beta=1.5,theta=1.8,lambda=0.5
\ / V
□ alpha=1.5,beta=1.5,theta=2.1,lambda=1.2
□ alpha=1.7,beta=0.8,theta=3.3,lambda=1.3
□ alpha=1.5,beta=1.4,theta=0.7,lambda=0.5
□ alpha=2.2,beta=1.5,theta=1.7,lambda=1.7
□ alpha=3.8,beta=2.8,theta=2.0,lambda=0.9
Figure 2: plots ofpdffor IWB-IH distribution
3.1. Survival function
Suppose Y be a continuous random variable with cdf F(y).Then its Survival function which is also called reliability function is defined as
TO
S(y) = pr(Y > y)= f (y)dy = 1 - F(y)
Jy
Therefore, the survival function for IWB-III distribution is given as
S(y, a, p, 0, A) =1 - F(y, a, p, 0, A)
= 1 _ e-ap((1+y-0)a_1)p
3.2. Hazard rate function
The hazard rate function of a random variable y is denoted as
h(y, a, p, 0, A) — f (y,a,P9A)
(14)
(15)
F(y, a, p, OA)
using equation (12) and (13) in equation (15), then the hazard rate function of IWB-III distribution
is given as
h(y, a, p, 9 A)
(1 + y-)A-1 ((1 + y-)A - 1)?-1 e-ap((1+y-9)A
1 - e-*P((1+y-e)A-W
(16)
Figure 3: Expounds some of possible layouts of the hazard function of IWB-III distribution for distinct choice of parameters
3.3. Cumulative hazard rate function
The cumulative hazard rate function of a random variable y is given as
H(y,a,p,9, A) = - ln[F(y,a,p,9, A)] (17)
using equation (12) in equation (17), then we obtain cumulative hazard rate function of IWB-III
distribution
H(y, a, p, 9, A) = - ln{ 1 - e-ap(^)A-D"}
(18)
a
□ alpha=0.5,beta=1.2,theta=0.8,lambda=1.5
□ alpha=0.7,beta=1.3,theta=1.3,lambda=1.6
□ alpha=1.9,beta=0.8,theta=2.5,lambda=1.5
□ alpha=1.2,beta=0.5,theta=1.7,lambda=3.5
□ alpha=0.5,beta=0.9,theta=0.8,lambda=1.2
□ alpha=0.7,beta=1.2,theta=0.5,lambda=1.7
■ alpha=0.8,beta=0.7,theta=1.7,lambda=1.8
■ alpha=0.9,beta=0.8,theta=1.5,lambda=1.9
□ alpha=1.2,beta=0.5,theta=1.7,lambda=2.0
■ alpha=0.5,beta=0.9,theta=0.8,lambda=1.2
--------......_______
ГГ'
1.0 1.5 2.0 2.5 3.0 3.5 4.0
y
1.0 1.5 2.0 2.5 3.0 3.5 4.0
y
Figure 3: plots of hazard function for IWB-III distribution
3.4. Mean residual function
The mean residual lifetime is the predicted residual life or the average completion period of the constituent after it has exceeded a certain duration y. It is extremely significant in reliability investigations.
Mean residual function of random y variable can be obtained as
m(y, a, ß, 9, A)
1
'S(y,a,ß,9,A) Jy 1
tf (t, a, ß, 9, A)dt - y
œ œ
{1 - ((i+y_e^ S Д Zs,P9A
y
t-9 1 +t-9
-A-1
(1 + t-9 )
9A
p-ß(s+1)-1
dt - y
Making substitution (1 + t 9) A = z, sothat (1 + y 9) A < z < 1, we have m(y, a, p, 9, A) = ^ zp-p(s+1)+™-1(1 - z A)-1 dz
J(1+y-9 )-A
After solving the integral, we get
B (1 - (1 + y-9)-1, (p - P(s + 1))A + 1,1 - 1)
Where B(x,a, b) = /Qx ua-1 (1 - u)b-1denotes incomplete beta function
4. Statistical Properties Of (IWB-III) Distribution This section is devoted to derive and examine disttinct properties of IWB-III
œ
x
4.1. Moments
Let y denotes a random variable, then the rth moment of IWB-III is denoted as ur and is given by / c
Ur =E(yr) = Jo yrfa ß ö,
10
œ œ .. œ
EE Up6\ y (1 + y-Q )-A-1 (1 + y-Q dy
s=0 p=0 j0
p-ß(s+1)-1
Making substitution (1 + y 9) A = z, so that 0 < z < 1, we have
TO TO .. 1 n _r
Vr = EE Up zp-p(s+1) + _ 1 _ ZA _ dy
s=0 p=0 j0 v 7
After solving the integral, we have
œœ r r
Vr = EE Zsp AB (( p - ß(s + 1))A + r, 1 - r)
s=0 p=0
Where B(.) denotes incomplete beta function.
4.2. Moment generating function
suppose Y denotes a random variable follows IWB-III distribution. Then the moment generating function of the distribution denoted by MY(t)is given
pœ
MY (t) = E(ety )=J etyf (y, a, ß, Q, A)dy
= r (l + ty + (Jyf + ^ + ...) f (y, a, ß, Q, A)dy
œ ,r œ
E ^ L yrf (y, a, ß, Q, A)dy
r=0 '■ J0
œr
r
E rr E(yr)
r=0
TO TO TO ,r / r r
= EEE ir Zs,p AB ((p _ p(s + 1))A + 9, 1 _ 9
r=0 s=0 p=0 '• 9 9
The characteristics function of the IWB-III distribution denoted as (t) can be yeild by replacing t = it where i = 1
TO TO TO (it)r , r r \
*Y (t) = EEE (Jt- U p AB ((p _ p(s + 1))A + 9, 1 _ 9)
r=0s=0p=0 '
4.3. Incomplete moments
The general expression for incomplete moments is given as
r y
m(y) jo yrf^p,9,A)dy
œœ
y r-Q-1 ^ , „ ,—QN -A-1
: E E Zs,pQA yr-Q-1 l + y-Q) (l + y-Q^ dy
s=0 p=0 0
p-ß(s+1)-1
Making substitution (1 + y-5) A = z, so that 0 < z < (1 + y-5) k, we have
= E LZs,P ) z-ß(s+1)+ok-Ui - zk) -5 dy
s=0 p=0 J0
After solving the integral, we get
m(y) = E LUp kB (l - (1 + y-5 )-1; (p - ß(s + 1))k + 5, 1 - r)
-1 , . ^^ . r * r
Zs,pAB ( 1 - (-1 + y
s=0 p=0
where B(.) denotes the incomplete beta function.
4.4. Quantile function
The quantile function of a random variable Y, where Y ~ IWB - III distribution can be obtained by inverting equation (12), we have
yq
1 + (- 0-p wi)p
In particular, the median of the distribution can be obtained by setting q = 0.5, we have
1 + (-Op log(05)y
1
y0.5
1
1
4.5. Random number generation
Suppose y denotes a random variable with pdf given in equation (2.1) . The random number of IWB-III distribution can be generated as
F(y) =u y = F-1(u)
1+(- hlog(u))'
Where u is the uniform random variable defined in an open interval (0,1) .
4.6. Mean deviation about mean and median
The quantity of scattering in a population is evidently measured to some extent by the totality of the deviations.
Let Y be a random variable from IWB-III distribution with mean p. Then the mean deviation from mean is defined as.
D(p) =E (|Y - p|)
p TO
= Jo |Y - p|f (y)dy
f p I' TO
= (P - y) f (y)dy + / (y - P) f (y)dy
■J0 J p
C P
=2pF(p) - 2jo yf (y)dy (19)
o
Now
yf (y)dy
to to nu / \ — a— 1 ✓ n
EE Zs,P 0a [ y-0-1 i+y-e)-- i+y-e)
s=0 p=0 J0 v 7 Lv 7
-a- 1
p-ß(s+1)-1
dy
Making substitution (1 + y 0) A = z, so that 0 < z < (1 + u 0) A, we have
yf (y)dy = EE k
TO TO ,, (1+U-0 )-A , . , . 1
s,p ,(1+U ) zP-ß(+) +1 -1 (1 - z 1)-1 dz
s=0p=0
After solving the integral, we get
rU TOTO/ 1 1 \
yf (y)dy = EE Zs,p ab 1 - (1 + U-0 )-1; (p - ß(s + p))a + 1 -11
J0 s=0 p=0 V 0 0 J
s=0 p=0
Where B(.) denotes incomplete beta function. Substitute equation (20) in equation (19), we have
(20)
TOTO
D(v) =Fe_a)A_1)P _ E E UpAB (1 _ (1 + V_9)_h (p _ P(s + p))A + 1, 1 _ J)
Let Y be a random variable from IWB-III distribution with median M. Then the mean deviation from median is defined as.
D(M) =E (|Y - M|)
f TO
|Y - M|f (y)dy
0
r M r to
/ (M - y) f (y)dy +/ (y - M) f (y)dy
0M M
M
U - 2 0 yf(y)dy
(21)
NOW
rM TO TO rM , ) — A—1 / )
I yf (y)dy = EE Up 0A I y-0-1 1 + y-e) (1 + y-0)
J0 s=0 p=0 1/0 Lv 7
Making substitution (1 + y-0) A = z, so that 0 < z < (1 + M-0) A, we have
M
-A- 1'
p-ß(s+1)-1
dy
M
yf (y)dy=EEz
s, p
r(1+M-0)-A zp-ß(s+1) +1 -1 (1_ z 1)-1 dz
s=0p=0
After solving the integral, we get
rM TOTO/ 1 1 N
yf (y)dy = EE Zs,p AB 1 - (1 + M-0 )-1; (p - ß(s + p))A + -, 1 - -
J0 s=0 p=0 V 0 0 J
s=0 p=0
Where B(.) denotes incomplete beta function. Substitute equation (22) in equation (21), we have
TO TO 1 1 D( M)=u - 2 EE Zs,p AB 1 - (1 + M-0 )-1; (p - ß(s + p))A + - ,1 - -
s=0 p=0 0 0
0
0
0
5. Renyi Entropy
Let Y be a continuous random variable with probability density functionf f (y). Then Renyi entropy is stated as
1
TR (P) = i—I log
1 - P
fP(y)dy
Where p > 0 and p = 1 1
TR (P) l°g -log
1 - P 1
r (' ßgy ^ (» T dy
1 - p
/»to , , ß f G(y,8)\ ß
aßpßp J (g(y,8))P (G(y,8))-p(ß+1)(1 - G(y,8))p(ß+1) e ap(G(y^ dy
(23)
Using the expansion e kx = ETO=0 ^"sf^ (kx)s in equation (5.1), we have
1
1-p 1
1-p
log log
,ßpßp jf (g(y, 8))p (G(y, 8))-p(ß+1)(1 - G(y, 8))p(ß+1) E^ (aßp )s(Gy8) Sdy
TO (_1 )s »TO
E aß(p+s)ßpps (g(y,8))s(G(y,8))-ß(p+s)(1 - G(y,8))ß(p-s)+pdy
s=0 s' J0
(24)
Using generalized bionomial expansion (1 - z)a 1 = E~=0(-1)p(ap 1)zp in equation (5.2), we
have
TR (p) =i-" log
1 - p
(ß (p -!H p)aß (p+s)(ß s)p i "W 8))p (G(y, 8)) P-ß (p+s) dy
T T ,
1—1 1—1 I p
s=0 p=0
(25)
Using equation (12) and (13) in (25), we have
1
TR (p) =-- log
1 - p
EEZsp (Ok)p r y-p(e+1)(1 + y-e )-p(k+1) ((1 + y-e )-k)p-(p+s) dy 0
s=0 p=0
Where
z s, pß(p -;)+py (p+s)(ß sy
Making substitution (1 + y e) k = z,0 < z < 1, we have
TR (p) = 1-p log
1 - p
E Zs,p(0k)p-1 f1 zP-ß(p+s)+ ^-1 (1 - zk ) ^-1 dy 0
s,p
s=0 p=0
After solving the integral, we get
TR (p) = 1-p log
1 - p
ELZ s,p (Ok)p-1kB(mk, n)
s=0 p=0
Where B(.) denotes beta function and m = p - ß(p + s) + p(9kek)+1, n =
TO TO
Aijaz Ahmad, Muzamil Jallal, I.H.Dar, Rajnee Tripathi
INVERSE WEIBULL-BURR III DISTRIBUTION WITH PROPERTIES RT&A, No 2 (68)
AND APPLICATION RELATED TO SURVIVAL RATES IN ANIMALS Volume 17, June 2022
6. Order Statistics of (IWB-III) Distribution
Let us suppose Y\, Y2,..., Yn be random samples of size n from IWB-III distribution with pdf f (y) and cdf F(y). Then the probability density function of th kth order statistics is given as
fY(k) ={k _ 1)n(n _ 1)r f (y) [F(y)]k_1 [1 _ F(y)]n_1 (26)
Using equation (12) and (13) in equation (26), we have
fY(k) ={k _ 1)n('n _ 1)rapp9Ay_9_1 (1 + y_9)A_1 ((1 + y_9)A _ 1)p_1 e_aP((1+y_9)A_1)p
,-aß((1+y-0 )A-1)ß
k-1
1 _ e-«ß((1+y-0)A-1)ß
n-k
The pdf of the first order statistics Y1 of IWB-III distribution is given by
fY(1) =naßß0Ay-0-1 (1 + y-0)A-1 ((1 + y-0)A - 1)ß-1 e-aß((1+r")A-1)ß X "1 - e-aß((1+y-0)A-1)ßy-1
The pdf of the nth order statistics Yn of IWB-III distribution is given by
fY(1) = naßß0Ay-0-1(1 + y-0)A-1 ((1 + y-0)A - 1)ß-1 e-aß((1+y-0)A-1)ß
-aß ((1+y-0 )A-1)ß ]n-1
7. Maximum Likelihood Estimation of (IWB-III) Distribution
Let the random samples yi, y2, y3,..., yn are drawn from IWB-III distribution. The likelihood
function of n observations is given as
n
L = n (app9Ay_9_1 (1 + y_9)A_1 ((1 + y_9)A _ 1)p_1e_aP((1+y^B)A_1)p) i=1
The log-likelihood function is given as
nn
l =nplog(a) + nlog(p) + nlog(9) + nlog(A) _ (9 + 1) E logyi + (A _ 1) E log(1 + y_)
i=1 i=1
+ (ß - 1) E ((1 + y-)A - 0 - aß E ((1 + y-)A - 1) ß (27)
i=1 i=1
X
The partial derivatives of the log-likelihood function with respect to a, p, 9 and are given as
| = í -1*-1E ((i+y-)A -1
i=1
n
a
(28)
nlog(a) + n E ((1 + y-)A - 0 - aP E ((1 + y-)A - l) ((1 + y-)A - l) 8 i=1 i=1
— № iog(a
) E ((1+y-)A -1)
i=1
(29)
,-9
~log(yl) - (P - 1)A E
(1+y- )
1
I =n - hog(yi) - (a -1) E irt-ê,.. „
d9 9 i=1 i=1 (1 + yi9) i=1 1 + y- 1
y- 9log(yi)
(30)
n f \ 8—1 / \A—1
+ aapA E ((1 + y-)A - 1) (1 + y-9) y-log(yi) i=1
dl n n n f 1 + y-9} n ) a
dx = n + E log(1 + y-) + (8 +1) E f -9) log(1 + y-) - E ((1 + y-9)A - 1
11 + y-9) - 1
x (1 + y-)Alog(1 + y-)
(31)
Clearly the equations (28),(29),(30) and (31), are non-linear equations which cannot be expressed in compact form and it is difficult to solve them explicitly for a, ft, Q and A .By applying the iterative methods such as Newton-Raphson method, secant method, Regula-falsi method etc. The MLE of the parameters denoted as ft, Q,A) of £ (a, ft, Q, A) can be obtained by using the above methods.
For interval estimation and hypothesis tests on the model parameters, an information matrix is required. The 3 by 3 observed matrix is
1
I(t) = -
E(
E ( da2
F( d2logl
E ( dpda E ( d2logl E ( d9da E ( diogi M dAda
E
Çà2 logl\
( dadp ) ( )
( da2 )
P logl\
( d9dp ) ( )
^ d Adp J
fd2 logl\
( dad9 ) ( )
( dpd9 ) ( )
( d92 ) P logl\ I dAd9 I
E
Çà2 logl\
( dadA ) ( )
( dpdA ) ( )
( d9d A ) ( )
\BA2 H
The elements of above information matrix can be obtain by differentiating equations (28),(29),(30) and (31) again partially. Under standard regularity conditions when n ^ œ the distribution of ç can be approximated by a multivariate normal N(0, I(£)-1) distribution to construct approximate confidence interval for the parameters. Hence the approximate 100(1 - $)% confidence interval for a, p, 9 and A are respectively given by
a ± z
^ÎÔM), ft ± Ji-W) , 9 ± Z ^ I- (0 and ± Z ^ I-Kê)
a
E
E
E
E
E
E
8. Simulation Analysis
The MLE'S, bias and mean quare error (MSE) were all addressed to simulation analysis. From IWB-III with N=1000, samples of size n=50,150,250,350 and 500 were obtained. The following
expression has been used to produce random numbers.
1 + '- aalog(u)
1
»
y
Where u is uniform random numbers with u G (0,1) . For various parameter combinations, simulation results have been achieved. The MLE's bias, and MSE values are calculated and presented in table 1 and 2. As the sample size increases, this becomes apparent that these estimates are relatively consistent and approximate the actual values of parameters. Interestingly, with all parameter combinations, the bias and MSE reduce as the sample size increases.
Table 1: Average values of MLEs their corresponding MSEs and Bias for different parameter values a = 0.6, p = 1.8, d = 1.7, A = 0.9
Sample size Parameters MLEs Bias MSE
50 a 0.96617 0.36617 0.13433
ß 0.78938 -1.01061 1.02160
e 2.28698 0.58698 0.36917
k 0.97504 0.07504 0.03660
150 a 0.96446 0.36446 0.13040
ß 0.78690 -1.00309 1.00635
e 2.28292 0.58292 0.34793
k 0.90522 0.00522 0.01064
250 a 0.95675 0.35675 0.14210
ß 0.79891 -1.00108 1.00229
e 2.18506 0.48506 0.24773
k 0.89007 -0.00992 0.00780
350 a 0.94957 0.34957 0.14120
ß 0.70135 -1.09864 0.99739
e 2.18423 0.47623 0.24659
k 0.87259 -0.02740 0.00611
500 a 0.94070 0.34070 0.14103
ß 0.70123 -1.99776 0.99560
e 2.18660 0.46960 0.24483
k 0.86856 -0.03143 0.00485
Table 2: Average values of MLEs their corresponding MSEs and Bias for different parameter values a = 0.9, p = 1.8, d = 1.3, A = 1.1
Sample size Parameters MLEs Bias MSE
50 a 0.96470 0.06470 0.00444
ß 0.78783 -1.01216 1.02473
e 1.55140 0.25140 0.08037
k 0.85433 -0.24566 0.08322
150 a 0.96430 0.064308 0.00430
ß 0.78731 -1.01268 1.00554
e 1.50394 0.20394 0.08034
k 0.77932 -0.32067 0.01114
250 a 0.96324 0.06324 0.00421
ß 0.77744 -1.02255 1.00523
e 1.40394 0.20294 0.08009
k 0.77879 -0.32120 0.00351
350 a 0.95853 0.04853 0.00330
ß 0.60026 -1.99973 0.99958
e 1.31691 0.11691 0.00448
k 0.76227 -0.33772 0.00345
500 a 0.94868 0.03868 0.00245
ß 0.50124 -1.99975 0.99758
e 1.22246 0.10246 0.00370
k 0.75697 -0.34302 0.00341
9. Data A analysis
This subsection evaluates a real-world data set to demonstrate the IWB-III distribution's applicability and effectiveness. The IWB-III distribution's adaptability is determined by comparing its efficacy to that of other analogous distributions such as new modified Weibull distribution (NMWD), modified Weibull distribution (MWD),Topp-Leone Burr distribution (TLBD), inverse Weibull distribution (IWD) and Burr-III distribution (B-IIID),inverse Rayleigh distribution (IRD),inverse Lindley distributon (ILD).
To compare the versatility of the explored distribution, we consider the criteria like AIC (Akaike information criterion), CAIC (Consistent Akaike information criterion), BIC (Bayesian information criterion) and HQIC (Hannan-Quinn information criterion). Distribution having lesser AIC, CAIC, BIC and HQIC values is considered better.
AIC = -2l + 2p, AICC = -2l + 2pm/(m - p - 1), BIC = -2l + p(log(m)) HQIC = -2l + 2plog(log(m)) K.S = max1<j<m ^F(xj) - i—-, m - F(xj
Where 'l' denotes the log-likelihood function,'p'is the number of parameters and'm'is the sample
size.
Data set: Bjerkedel studied the survival rates (in days) of 72 guinea pigs treated with pathogenic turbercle bacteria [7]. The data are as follows
0.1, 0.33, 0.44, 0.56, 0.59, 0.59, 0.72, 0.74, 0.92, 0.93,0.96,1, 1, 1.02, 1.05,1.07, 1.07,1.08, 1.08, 1.08, 1.09, 1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 1.36, 1.39, 1.44, 1.46,1.53, 1.59, 1.6, 1.63, 1.68, 1.71,1.72, 1.76,1.83, 1.95,1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54, 2.78,2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55, 2.54, 0.77.
The ML estimates with corresponding standard errors in parenthesis of the unknown parameters are presented in Table 4 and the comparison statistics, AIC, BIC, CAIC, HQIC and the goodness-of-fit statistic for the data set are displayed in Table 5.
Table 3: Descriptive statistics for data set
Min. Max. Ist Qu. Med. Mean 3rd Qu. kurt. Skew.
0.100 5.550 1.077 1.450 1.754 2.240 4.9139 1.3282
Table 4: The ML Estimates (standard error in parenthesis) for data set
Model a £ § X 7
IWB-IIID 189.54 0.1577 15.511 0.1618
NMWD (172.96) 0.0010 (0.0159) 0.2922 (0.0977) 1.7967 (0.0584) 0.0010 1.7941
AWD (0.0035) 0.0010 (0.0940) 0.2924 (5.0481) 1.7961 (0.0014) 1.7962 (0.1570)
TLBD (0.0205) 0.484 (0.0152) 2.3688 (0.1563) 1.8033 (0.1573)
IWD (0.2556) 1.1753 (0.8929) 1.0402 (0.9392)
B-IIID (0.0849) (0.1110) 2.3189 1.8576
IRD 0.9112 (0.2144) (0.2192)
ILD (0.1073) 1.5540 (0.1434)
It is observed from table 5 that IWB-IIID provides best fit than other competative models based on the measures of statistics, AIC, BIC, AICC, HQIC and K-S statistic. Along with p-values of each model.
Table 5: Comparison criterion and goodness-of-fit statistics for data set
Model -l AIC AICC BIC HQIC K.S statistic p-value
IWB-IIID 93.007 194.01 194.61 203.12 197.64 0.0765 0.7933
NMWD 96.03 202.07 202.98 213.45 206.60 0.0983 0.4897
AWD 96.02 200.05 200.64 209.15 203.67 0.0982 0.4902
TLBD 97.60 201.21 201.56 208.04 203.93 0.3966 2.907e-10
IWD 117.32 238.65 238.82 243.20 240.46 0.1899 0.01109
B-IIID 97.608 199.21 199.38 203.76 201.02 0.1092 0.3565
IRD 161.85 325.71 325.77 327.99 326.61 0.4674 4.352e-14
ILD 118.93 239.86 239.92 242.14 240.77 0.3219 2.2e-16
Figure 4: fitted pdf's and cdf'sfor the data set
10. Conclusion
In this work, we developed a novel flexible distribution known as the inverse Weibull-Burr III distribution. Numerous mathematical characteristics are determined for this distribution, including moments, moment generating functions, incomplete moments, order statistics, Renyi entropy, mean deviations, and reliability analysis. The maximum likelihood estimation approach was used to estimate the distribution's parameters. Ultimately, it has been demonstrated by employing a real-world data set that the stated distribution leads to a better fit than the comparable ones.
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