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30
A Novel Extended Version of the Ailamujia Inverted
Weibull Distribution
Idzhar A. Lakibul •
Department of Mathematics and Statistics Mindanao State University - Iligan Institute of Technology Iligan City, Philippines idzhar.lakibul@g.msuiit.edu.ph
Abstract
Statististical distributions with support on the set of non-negative real numbers are important in modelling and describing the behaviour of lifetime data. Ailamujia distribution is one of the non-negative continuous distribution that has an application in lifetime data. In this paper, a new three-parameter non-negative continuous distribution which is an extension of the Ailamujia Inverted Weibull distribution is introduced. This extended distribution is labeled as the Cubic Transmuted Ailamujia Inverted Weibull distribution. The proposed distribution is derived from the cubic transmuted family of distributions by specifying Ailamujia Inverted Weibull distribution as a baseline distribution. The probability density function of the proposed distribution is derived and some of its plots are presented. It can be observed that the proposed distribution can model the data which are exponentially and skewed unimodal right tailed data. In addition, survival and hazard functions of the proposed distribution are derived. It reveals that the hazard function of the proposed distribution can model both monotonic and non-monotonic decreasing failure rate behaviour of the data. Some properties of the proposed distribution such as its moments, moment generating function, mean, variance are derived. The Maximum Likelihood approach is used to estimate the proposed distribution parameters. Furthermore, parameter estimates as well as the performance of the proposed distribution is investigated by utilizing two sets of lifetime data. For point of comparison, this paper uses the following criteria: Akaike Information Criterion, Bayesian Information Criterion, Kolmogorov - Smirnov statistics, Anderson-Darling and Cramer-von Mises. Results show that for both sets of data, the proposed distribution produce better estimate as compared to the Quasi Suja and the Weibull-Lindley distributions. So, the proposed distribution consider as the best model for modelling the given two real datasets.
Keywords: Ailamujia distribution, Ailamujia Inverted Weibull distribution, Cubic transmuted
family of distributions
1. Introduction
Ailamujia distribution is one of the newly lifetime distributions that has many application in engineering [5] . Due to its flexibility in modelling lifetime data then it becomes an area of interest for some researchers.
Different extensions were considered in literature. Uzma [11] introduced the weighted analogue of Ailamujia distribution and derived some of its properties. Other identified extensions such as the area biased distribution [3] , the size biased Ailamujia distribution [8] and the inverse analogue of Ailamujia distribution [1].
Moreover, Smadi [10] proposed an extended version of the Ailamujia distribution called as Ailamujia inverted Weibull distribution (AIWD) and various properties such as reliability function, hazard function, moments, moment generating function, order statistics, mean residual function and Shannon's entropy were studied.
In this paper, a novel extension of the Ailamujia Inverted Weibull distribution called Cubic Transmuted Ailamujia Inverted Weibull distribution (CTAIWD) is derived. Some of its properties such as moments, moment generating function, mean and variance are studied. Maximum Likelihood approach is also discussed to estimate the proposed distribution parameters. Two real data sets are analyzed to evaluate the performance of the proposed distribution.
The rest of paper is organized as follows: The Cubic Transmuted Ailamujia Inverted Weibull (CTAIW) distribution is introduced in section 2. In section 3, some statistical properties of CTAIW distribution are presented. Maximum Likelihood Estimation of the proposed distribution is discussed in section 4. In section 5, the application of proposed distribution is illustrated. Some concluding remarks is presented in section 6.
2. Cubic Transmuted Ailamujia Inverted Weibull distribution
This section presents the derivation of Cubic Transmuted Ailamujia Inverted Weibull (CTAIW) distribution. The cumulative distribution function of the Ailamujia Inverted Weibull distribution is given by
F(x,6,a) = (1 + 26x-a)e-26x-a,x > 0,a > 0,6 > 0 (1)
with corresponding probability density function
f (x, 0, a) = 4a02 x-2a-1 e-20x-a.
Rahman [7] proposed a cubic transmuted family of distributions for extending any continuous distribution. The cumulative distribution function of the cubic transmuted family of distributions is given by
F(x) = (1 - A)G(x) + 3AG2(x) - 2AG3(x),x e R (2)
where A e [-1,1]. The cumulative distribution function of the Cubic Transmuted Ailamujia Inverted Weibull distribution is obtain by inserting (1) into (2) then, we have
,3,-60 x-
F(x, 0, a, A) = (1 - A)ze x + 3Az2e-40x - 2Az3e where z = 1 + 20x-a. The corresponding probability density function is
f (x,0,a, A) = 4a02x-2a-1 e-20x-a(1 - A + 6Aze-20x-a - 6Az2e-40x-a).
(3)
(4)
Note that if A = 0 then Cubic Transmuted Ailamujia Inverted Weibull distribution reduces to Ailamujia Inverted Weibull distribution.
Figure 1: pdf p/ois of CTAIW distribution for a = 1.5, 0 = 1.5 andfor varying va/wes of A
Figure 2: pdf plots of CTAIW distribution for a = 0.2, 6 = 0.5 and for varying values of A
Figures 1 and 2 present some possible shapes of the pdf of CTAIW distribution. It can be observed that the pdf of proposed distribution can generate a skewed unimodal and exponential shapes which are important in modelling lifetime data.
3. Statistical Properties
In this section, some statistical properties of the Cubic Transmuted Ailamujia Inverted Weibull distribution (CTAIW) such as moments, moment generating function, mean, variance, survival function and hazard function are derived.
3.1. Moments Theorem 1. The rth moment of CTAIW distribution with density (4) is
1
yi!r = A(20) a [(1 - 2a-13 + 3a-12 - -)r(-^ + 2) - (2a-23 - 3a-24)r(a + 3) + 3a-32r(a + 4)]
r
ft
■ r
"a
r -3".r(r
cc
The mean and variance are respectively, given by
(5)
H = A (20)a [(1 - 2a3-1 + 312-1 - A)r(-1 + 2) - (21-23 - 31 -24)r( 1 + 3) + 312-3r( 1 + 4)]
2-3W1
a2 =A(20)2 [(1 - 22-13 + 3a-12 - 1 )r(-2 + 2) - (22-23 - 32-24)r(2 + 3) + 3a-32r(2 + 4)]
A a a a
- (A(20)1 [(1 - 21-13 + 31 -12 - 1 )r(-- + 2) - (21 -23 - 31 -24)r(- + 3)
A
1
a:
'ft
+ 31 -32r(- + 4)])2 ft
Proof. The rth moment is defined by
V'r = E[Xr ]
xrf(x)dx
/•TO
= xr4a02x-2a-1 e-20x-a (1 - A + 6A(1 + 20x-a)e-20x-a - 6A(1 + 20x-a)2e-Jo
a\ 2„-40x-
')dx
TO
TO
/•TO _ /»TO _
= 4(1 - A)02«y xrx-2a-1 e-20x-dx + 24A02«J xrx-2a-1 (1 + 20x-")e-40x-dx
/»TO _
- 24A02« / xrx-2a-1 (1 + 20x-")2e-60x-«dx Jo
= (1 - A)[-(20)rГ(-Г + 2)] - 6A[^(Г(-a + 2) + 2Г(-Г + 3))] 1 r r 2 r 1 r
+6A[-9(60) r (r(-r + 2) + 3 r(-r + 3) + -r(-r + 4))]
= A(20)1 [(1 - 213-1 + 3a2-1 - A)Г(-- + 2) - (2«-23 - 31 -24)Г(1 + 3) + 312-3Г(- + 4)]
A
The mean of CTAIW distribution is obtained by using r = 1 in (5) and is
ц = A(20)1 [(1 - 2«3-1 + 3«2-1 - A)Г(-1 + 2) - (2«-23 - 3«-24)Г( 1 + 3) + 3«2-3Г( 1 + 4)] The 2nd raw moment of CTAIW distribution is obtained by using r = 2 in (5) and is ц2 = A(20)«[(1 - 2«-13 + 32-12 - A)Г(-2 + 2) - (2«-23 - 32-24)Г(« + 3) + 3«-32Г(« + 4)] The variance a2 of CTAIW distribution obtained as
a2 = ц2 - (ц1)2
22 212 2 2 2 2 2 = A(20)« [(1 - 22-13 + 32-12 - -)Г(— + 2) - (22-23 - 3«-24)Г(- + 3) + 3«-32Г(- + 4)]
A
- (A(20)1 [(1 - 21 -13 + 3«-12 - A)Г(-- + 2) - (21 -23 - 3«-24)Г(- + 3)
A
+ 3«-32Г(1 + 4)])2. a.
3.2. Moment Generating Function
Theorem 2. Let X follows the CTAIW distribution then the moment generating function Mx (t)
is given by
Mx(t) = £ tA?0)i[аГ(-- + 2) - ЬГ( r + 3) + сГ(- + 4)] n r! « « «
r=0
where a = 1 - 2r-13 + 3s-12 - A, b = 2r-23 - 3r-24, c = 3r-32 and t G R.
Proof. By definition of moment generating function and equation (5), we have
f TO
MX(t) = E(etX) = y etxf(x,6,a,A)dx. Recall that etx = ETO=0 rFT xr then we have
TO TO jr to tr TO TO jr
MX(t) = E ^xrf(x, 6,a, A)dx = E ryxr / f (x, 6,a A)dx = E ry^
70 r=0 r" r=0 r 70 r=0 f'
Thus,
Mx(t) = £ [ar(-r + 2) - br(- + 3) + сГ(- + 4)]
r!
r=0
where a = 1 - 2r-13 + 3Z-12 - A, b = 2r-23 - 3r-24 and c = 3r-32.
3.3. Reliability Analysis
Let X be a random variable with cdf (3) and pdf (4) then the survival S(x) and hazard h(x) functions of CTAIW distribution are respectively, given as follows:
-26 x-
— 3Az2e—46x-a + 2Az3e—66x—a;
S(x, 6, a, A) = 1 — (1 — A)ze—
4a62 x—2a—1 e—26x—a (1 — A + 6Aze—26x—a — 6Az2 e—46x—a) h(x 6 a, A) = 1 — (1 — A)ze—26 x—a — 3Az2 e—46 x—a + 2Az3 e—66x—a '
where z = 1 + 26x—a, x > 0,a, 6 > 0 and A e [—1,1].
Figure 3: hf plots of CTAIW distribution for a = 1.5, 6 = 1.5 and for varying values of A
0 0 0.5 1 0 1.5 2.0
X
Figure 4: hf plots of CTAIW distribution for a = 0.2, 6 = 0.5 and for varying values of A
Figures 3 and 4 present some possible shapes of hazard rate function of CTAIW distribution. It reveals that the hazard rate function of the proposed distribution can model both monotonic and non-monotonic decreasing failure rate behavior of the data which are common in the reliability data.
4. Maximum Likelihood Estimation
In this section, the maximum likelihood approach is used to estimate the CTAIW distribution parameters.
Let X1, X2,..., X n be a random sample of size n from CTAIW distribution, then the likelihood function is
n
L = n[4a62 x—2a—1 e—26 x— (1 — A + 6Aze—226x— — 6Az2 e—46 x—)] i=1
where z = 1 + 29X- a. The log-likelihood function is
n
l = £ log[4a92xr2»-1 e-29x- (1 - A + 6Aze-29x-a - 6Az2e-49x-a)]. (6)
i=1
Taking the deivative of (6) with respect to the parameters a, 9 and A then, we have the following:
dl n " , . , „ " a, . , „ . n log(xi)x-2ae-29x-a (1 - 2ze-29x-a) m
- = - - 2 £ log(xi) + 29 £ x-alog(xi) + 24A9 £ '-±-f (7)
a £ i) ^ i i) £1 - a - 6Aze-29x-a - 6Az2e-49x-a K!
dl = 2n 2 yx-« 24A9 y x-2ae-29x-a (1 - 2ze-29x-)
d9 = 9 - 2 £ x- - 24A9 £1 - AA - 6Aze-29xlCi - 6AZ2e-49x-a (8)
dl = _ " 1 - 6ze-29x-a + 6z2e-49x-a
dA i=11 - A - 6Aze-29x-a - 6Az2e-49x-a ( )
Equating equations (7), (8) and (9) to 0, respectively, one can get numerical maximum likelihood estimates of the CTAIW distribution parameters.
5. Application
In this section, the proposed distribution is applied to two real data sets and compared with the following distributions:
- Quasi Suja (QS) distribution [9]
94
f (x, a, 9) = —;—— (a + 9x4)e-9x, x > 0,9 > 0, a > 0. JX ' a93 + 24 '
- Weibull-Lindley (WL) distribution [2]
f (x,a,p) = a(fixP-1(1 + x)ex)e-a((1+x)eX^-1),x > 0,a,p > 0.
A data analysis is performed using a package "fitdistrplus" in R. Moreover, AKaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Kolmogorov-Smirnov (K-S), AndersonDarling (A) and Cramer-von Mises (W*) statistics are used for comparison in this analysis. In addition, the two sets of data used in the analysis are given as follows:
Data Set 1. This data set is from Murthy [6]. It is a set of failure times of 50 electronic components. The data set is given as follows: 0.036, 0.058, 0.061, 0.074, 0.078, 0.086, 0.102, 1.103, 0.114, 0.116, 0.148, 0.183, 0.192, 0.254, 0.262, 0.379, 0.381, 0.538, 0.570, 0.574, 0.590, 0.618, 0.645, 0.961, 1.228, 1.600, 2.006, 2.054, 2.804, 3.058, 3.076, 3.147, 3.625, 3.704, 3.931, 4.073, 4.393, 4.534, 4.893, 6.274, 6.816, 7.896, 7.904, 8.022, 9.337, 10.940,11.020, 13.880,14.730 and 15.080.
Data Set 2. This dataset is from Khan [4] and it is an ICU data set which assess the intensive care unit (ICU) patients agitation-sedation (A-S) status. The data set is given as follows: 9, 3, 27, 8, 4, 3, 4, 3, 23, 3, 3, 4, 28, 18, 19, 6, 3, 26, 3, 12, 6, 9, 43, 4, 4, 3,5, 12, 4, 36, 6, 8, 6,5, 3, 3 and 33.
Tables 1 and 3 list the MLEs of the CTAIW, QS and WL distributions fitted to first and second sets of data. Tables 2 and 4 indicate that the proposed distribution has a lower values of the AIC, BIC, K-S, A and W* compared to the QS and WL distributions for both sets of data. So, the propsed distribution fit well the said datasets. Furthermore, the estimated pdf of the fitted models to both sets of data are presented in figure 5 and 6, respectively.
Table 1: MLEs of the fitted models for first set of data.
Distribution a P 0 X
CTAIW 0.5116727 0.6782907 -0.9999998
WL 0.07258343 0.12084431
QS 211.8090902 0.6493223
Table 2: Numerical values of AIC, BIC, K-S, A and W* of the fitted models for first set of data.
Distribution AIC BIC K - S A W *
CTAIW WL QS 209.0186 228.7096 213.2069 214.7547 232.5337 217.031 0.1531139 0.2500551 0.2179699 1.3078260 3.3786131 3.6483026 0.2494361 0.5503281 0.4827523
Table 3: MLEs of the fitted models for second set of data.
Distribution a P 0 X
CTAIW 1.1988356 8.5119091 -0.9999997
WL 0.02157018 0.11712539
QS 20023.91 0.1678021
Table 4: Numerical values of AIC, BIC, K-S, A and W* of the fitted models for second set of data.
Distribution AIC BIC K - S A W*
CTAIW WL QS 232.7614 256.3175 253.2686 237.5941 259.5393 256.4904 0.1724946 0.2192403 0.3155350 1.4724752 2.3094837 2.5730059 0.2293943 0.4026443 0.4309077
0 5 10 15
Figure 5: Estimated pdf of the fitted models for the first set of data.
6. Concluding Remarks
In this study, an extended version of the Ailamujia Inverted Weibull distribution called Cubic Transmuted Ailamujia Inverted Weibull distribution has been introduced. Some properties of the proposed distribution such as moment, moment generating function, mean and variance were derived. The maximum likelihood approach was used to estimate the proposed distribution parameters. Two real datasets were used to examine the flexibility of proposed distribution and compared to Quasi Suja and Weibull - Lindley distributions. It was found that the proposed distribution fit well the given data sets compared to said distributions.
References
[1] Aijaz, A., Ahmad, A. and Tripathi, R. (2020). Inverse analogue of Ailamujia distribution with statistical properties and applications. Asian Research Journal of Mathematics, 16:36-46.
[2] Chacko, V. M., S, D. K, Thomas, B. and C, R. (2018) Weibull-Lindley Distribution: A Bathtub Shaped Failure Rate Model. Reliability: Theory and Application, 13:9-20.
[3] Jayakumar B. and Elangovan, R. A. (2019). New generalization of Ailamujia distribution with applications in bladder cancer data. IJSRMSS, 6:61-68.
[4] Khan, M. S., King, R. and Hudson, I. L. (2017). Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data. Communication Statistics : Simulation and Computation, 46:4377-4398.
[5] Kotz, S. L. and Pensky, M. The stress strength model and its generalizations, Theory and Applications, World Scientific, Singapore, 2003.
[6] Murthy, D. N. P., Xie, M., Jiang, R. Weibull Models, John Wiley and sons Inc, Hoboken, 2004.
[7] Rahman, M. M., Al-Zahrani, B. and Shahbaz, S. H. (2019). Cubic Transmuted Uniform Distribution: An Alternative to Beta and Kumaraswamy Distributions. European Journal of Pure and Applied Mathematics, 12:1106-1121.
[8] Rather, A., Subramanian, C., Shafi, S., Malik, K. A., Ahmad, P. J., Para, B. A. and Jan, T. A. (2018). New size biased distribution with Applications in Engineering and Medical Science. IJSRMSS, 5:75-85.
[9] Shanker, R., Upadhyay, R. and Shukla, K. K. (2022). A Quasi Suja Distribution. Reliablilty: Theory and Application,17:162-178.
[10] Smadi, M. M. and Ansari, S. I. (2022). Ailamujia Inverted Weibull Distribution with Application to Lifetime Data. Pakistan Journal of Statistics, 38:341-358.
[11] Uzma, J., Kawsar, F. and Ahmad, S. P. (2017). On weighted Ailamujia distribution and its applications to lifetime data. Journal of Statistics Applications and Probability an International Journal, 6:619-633.
