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IDZ DISTRIBUTION: PROPERTIES AND APPLICATION
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
This paper introduces a novel two - parameter continuous distribution. This distribution is derived from the mixture of the Exponential, Weibull and Ailamujia distributions. The derived distribution is named as "Idz distribution". The probability density function of the Idz distribution is derived and some of its plots are presented. It can be observed that the Idz distribution can generate right tailed unimodal, non-monotonic decreasing and exponential shapes. Further, survival and hazard functions of the Idz distribution are derived. It reveals that the hazard function of the Idz distribution can accommodate three types of failure rate behaviors, namely, non-monotonic constant, right tailed unimodal and nonmonotonic decreasing. Moreover, some properties of Idz distribution such as moments, mean, variance, moment generating function, order statistics and maximum likelihood estimates are derived. In addition, the proposed distribution is applied into a Breast Cancer data and compare with the Exponentiated Generalized Inverse Rayleigh distribution, the Ailamujia Inverted Weibull distribution and the New Extended Exponentiated Weibull distribution. Result shows that the Idz distribution gives better estimates as compared with the said distributions for a given dataset.
Keywords: Weibull distribution, Exponential distribution, Ailamujia distribution.
1. Introduction
Non-negative continuous probability distribution is important in modelling real lifetime data, specifically, in the field of reliability, engineering and biomedical science. There are popular classical limetime distributions such as the exponential, log-normal, log-logistic, Weibull, Rayleigh and the Frechet distributions. But due to the complexity of the lifetime data, the classical distributions need to generalize or extend in order to cater the complex behaviour of the data. One method for extending the classical distribution is by using the generated family of distributions like the Exponentiated - G family of distributions [7], Marshall-Orkin - G family of distributions [11], Beta - G family of distributions [4] and other existing families of distributions.
Another method of facing the complex behaviour of the lifetime data is by using mixture distribution of two or more probability distribution functions. A random variable X is assumed to have a mixture of two or more probability distribution functions f1(x), f2(x), f3(x),...,fn (x) if its probability density function m(x) = YJi=i aifi(x) with ai <G [0,1] and YJi=i ai = 1. Years ago, several distributions have been derived from the mixing distributions, for example, the Aradhana distribution [13] which is a mixtures of the Gamma (2, 6), the Gamma (3, 6) and the Exponential (6) distributions with corresponding mixing proportions 92+25+2, 92+29+2 and 92+29+2. Other identified mixture distributions such as the Rama distribution [14], Darna distribution [15], Shanker distribution [12], Gharaibeh distribution [6], Alzoubi distribution [2] and Benrabia
distribution [3].
In this paper, the concept of mixture distribution is used to propose a two - parameter distribution named as Idz distribution which is a mixture of three distributions, namely, the Weibull (A, p) distribution [17], exponential (A) distribution [8] and the Ailamujia (A) distribution
[10] with mixing pr°p°rtions Aj62lAp+v Api and Api +1Ap+1, respectively. Other goals of the
paper are the following: (i) to derive some properties of Idz distribution such as its moments, moment generating function, mean, variance, order statistics and maximum likelihood estimates of the proposed distribution parameters; and (ii) to apply the proposed distribution into a real dataset and compare with the Exponentiated Generalized Inverse Rayleigh, the Ailamujia Inverted Weibull and the New Extended Exponentiated Weibull distributions.
This paper is arranged as follows: Idz distribution is introduced in section 2. In section 3, some properties of Idz distribution are derived. Order Statistics of the ID distribution is given in section 4 while the maximum likelihood estimates of the ID parameters is presented in section 5. In section 6, the application of Idz distribution is illustrated . Some concluding remarks is presented in section 7.
2. Idz Distribution
This section presents the definition of the Idz distribution and its special cases with the illustration of its pdf.
A random variable X is said to have an Idz distribution (ID) with parameters A and p if the probability density function of X is given by
( A p)_ A2e-Ax[p + 4xe-Ax + p3xp-1 eA(x-xp)]
f (x A p) _ Ap2 + Ap + 1 , (1)
where x > 0, A > 0 and p > 0. The corresponding cumulative distribution function of X is given
by
F(x, A, p)_ 1 - Ape-Ax + (2Ax2+ 1) e'2Ax + Ap2^. (2)
v Ap2 + Ap +1 w
(c) (d)
Figure 1: pdf plots of Idz distribution (ID) for different sets of values of the parameters: (a) A _ 2.5 and varying values of p; (b) A _ 0.5 and varying values of p; (c) p _ 0.5 and varying values of A; and (d) p _ 2.5 and varying values of A.
Figure 1 shows some possible density shapes of the ID distribution and it reveals that the pdf of the ID distribution can generate right tailed unimodal, non-monotonic decreasing and
exponential shapes.
Special Cases of Idz distribution
1. If ft = 1 then ID reduces to
f (x, A)
2. If A = 1 then ID reduces to
f (x, ft)
3. If ft = 2 then ID reduces to
2A2 e-Ax (1 + 2xe-Ax) 2A +1 .
e-x [ft + 4xe-x + ft3 xft-1 ex-xP ] ft2 + ft + 1
(4)
f (x, A)
2A2 e-Ax [1 + 2xe-Ax + 4xeA(x-x2 6A +1
(5)
We name the probability distribution functions (pdf) (3), (4) and (5) as the pdfs of the Edz distribution, Laks distribution and Aids distribution, respectively.
3. Statistical Properties
In this section, we derive some properties of Idz distribution such as its moments, mean, variance, moment generating function, survival fuction and hazard function.
3.1. Moments
Theorem 1. Let X be a random variable that follows an Idz distribution then the rth moment of X denoted by is given by
A1-
Aft2 + Aft + 1
r(r + 2) ft2 r( ? +
ftr(r +1) +
A2r
+
r +1
Aft-r
(6)
where r = 1,2,3,...,n and T(-) is a gamma function. Proof. The rth moment of X is defined by ^r = E[Xr ]
/TO
xrf(x)dx
-to
rTO rA2e-Ax[ft + 4xe-Ax + ft3xft-1 eA(x-xft)]
Aft2 + Aft + 1
dx
A2
Aft2 + Aft +1
A2
Aft2 + Aft +1
A1-r
Aft2 + Aft +1
/ /»TO /»TO /»TO ft \
( ft y xre-Axdx + 4 ^ xr+xe-2Axdx + ft3 J xrxft-1e-A^dx j ft (A+ ) r(r + 1) + 4 (
1 r+2
A r(r + 2) + ft^ ft»
ftfr) r( ft +1
ftr(r +1) +
r(r + 2) ft2r{ r + 1
A2r
+
Aft r
■
0
Corollary 1. Let X be a random variable with moment given in equation (6) then the mean p and variance a2 of X are, respectively, given by
F =
1
r(3)
ß-1 /1
ßr(2) + ^+ ß2A ß r - + 1
2A
ß
and
a2 = 1 I A-1
r(4) ß2r 2 + 1 ßr(3) + Vß
4A
Aß
2
ßr(2) + ^. +
r(3) , ß2nß +1
1-ß A ß
where c = Aft2 + Aft + 1.
Proof. The mean of X is derived when r = 1 in (6). Hence,
1
F
ßr(2) + ß2 A^ r( ß + 1
where c = Aft2 + Aft + 1. Next, the variance of X denoted by a2 can be computed as
a2 = f2 - (Pi)2.
Now, the 2nd raw moment of X is obtained by setting r = 2 in equation (6). It follows that
f2
A-1 c
r(4) ß2r ( ß + 1
ßr<3)+ 4A + a ß -2
Therefore, the variance a2 of X is
a
2 A-1
r(4) ß2r ß + 1
ßr(3) + rA1 + —rh-
Aß '
ßr(2) + ® + ß2 Ar( ß + 1
= C < A-1
r(4) ß2 r( ß + 1
ßr(3) + ~4T + 2
4A Aß
2
ßm + -a +
r(3) , ß2r(ß + 1
1-ß A ß
■
3.2. Moment Generating Function
Theorem 2. Let X be a random variable that follows an Idz distribution then the moment
generating function of X is given by
Mx (t) = E
~ tr A1-r
r(r + 2) ß2r( I + 1 ßr(r + 1) + ^^ + Vß
A2r
rt0 (Aft2 + Aft + 1) r! where t € R.
Proof. The moment generating function of X is defined by
/TO
etxfX (x)dx.
TO
Using equation (1), we have
p TO
MX (t) = y etxf (x, A, ft)dx.
Aß '
c
2
1
c
2
c
2
Recall that etx = E^=0 £ xr • Then,
r=0
/•œ^ tr tr rœ t'
Mx(t) = E hxrf (x, A, p)dx = E -J x'7(x, A, p)dx = E -, H'r.
r=0 r' r=0 r! J0 —n r!
Using equation (6) and hence,
Mx (t) = E
tr A1-1
r=0
(Ap2 + Ap + 1) r!
r(r + 2) , p r {~p
Pr(r + ^^^ +
p2 r( r + 1)
AP
where t G R.
■
r
3.3. Reliability Analysis
Let X be a random variable with cdf (2) and pdf (1) then the survival S(x, A, p) and hazard h( x, A, ) functions of X are respectively, given by
n/ Ape-Ax + (2A x + 1) e-2Ax + Ap2 e-Axp . „„ „
S(x,A,p) _ ^-(A^2 + Ap + 1 -,x > 0,A > 0,p > 0
and
A2 [p + 4xe-Ax + p3 xp-1 eA(x-xp)] h(x,A,p) _ r r
Ap + (2A x + 1) e-A x + Ap2 e-A(x-xP )
(c) (d)
Figure 2: hf plots of Idz distribution (ID) for different sets of values of the parameters: (a) A _ 2.5 and varying values of ; (b) A _ 0.5 and varying values of ; (c) _ 0.5 and varying values of A ; and (d) _ 2.5 and varying values of A.
Figure 2 presents some possible shapes of the hazard function of the ID distribution and it reveals that the hazard function of the ID distribution can accommodate non-monotonic constant, right tailed unimodal and non-monotonic decreasing behaviors.
4. Order Statistics
Let X(i), X(2),..., X(n) be the order statistics of a random sample X1, X2,..., Xn drawn from the continuous population with probability density function (pdf) fX (x) and cumulative distribution
function FX(x), then the pdf of rth order statistics X(r) is given by
n!
fX(r)(x)
(r - 1)!(n - r)!
fX(x) [Fx(x)]r-1 [1 - Fx(x)]
The pdf of rth order statistics X(r) of the ID distribution is derived by inserting (2) and (1) into (7) and is
fX(r)(x, A, ft)
;!A2 e-Ax [ft + 4xe-Ax + ft3 xft-1eA(x-xft)] (r - 1)!(n - r)! (Aft2 + Aft + 1)
-Ax i /oi-v i i \ „—2Ax i i a2„—Axftir 1
1
Aft e-Ax + (2Ax + 1) e-2Ax + Aft2 e-
Aft2 + Aft + 1
Aft e-Ax + (2 Ax + 1) e-2Ax + Aft 2e Aft2 + Aft + 1
2 Ax
(8)
The pdf of the smallest or 1st order statistics of the ID distribution is obtained by setting r = 1 in equation (8) and is
fX(1)(x, A, ft)
z!A2 e-Ax [ft + 4xe-Ax + ft3 xft-1 eA(x-xft)]
(n - 1)! (Aft2 + Aft + 1)n
Aft e + (2Ax + 1) e + Aft2 e
2 Ax
n—1
If r = n then the pdf of the nth or largest order statistics of ID distribution is given by
n!A2e-Ax [ft + 4xe-Ax + ft3 xft-1eA(x-xft)]
fx(n)(x, A, ft)
(n - 1)! (Aft2 + Aft + 1)
1
Aft e-Ax + (2Ax + 1) e-2Ax + Aft2 e Aft2 + Aft + 1
2 Ax
n—1
5. Maximum Likelihood Estimation
Let X1, X2,..., X n be a random sample of size n from Idz distribution (ID). Then the likelihood function of ID is given by
L
n
i=1
A2 e-Ax' [ft + 4xie-Ax, + ft3 xft 1 eA(xi-xft-
Aft2 + Aft + 1
Then, the log-likelihood function of ID is
i=1
i=1
^ = " 1 + ft3xft-1 [ft + f1 - Axft ) log(x!) dft k
A xi x
ft - 4xie-Axi + ft3xf e'
3x -1eA xi-xi
fi'\
nA(2ft +1) ; Aft2 + Aft + 1;
and
3 -1
31 2n " " ft xf
^ = A - k xi + k
xi- xi e
A xi-xft
Axi
nft (ft + 1)
Aft2 + Aft + 1'
(9)
nn
1 = 2n log(A) - A k x, + k log [ft - 4xie-Axi + ft3 xft eA(xi-xi)] - n log(Aft2 + Aft + 1). (10)
The partial derivatives of (10) with respect to parameters and A are presented as follow:
(11)
(12)
i=1 i=1 ft - 4x,e-Axi + ft3xft-1 e vx"-x<' J
The maximum likelihood estimates of the parameters and A of Idz distribution can be computed by setting equations (11) and (12) equal to zero. This can be done by using any numerical method like the Newton-Raphson iterative method.
nr
nr
n
e
6. Application
This section presents the application of Idz distribution to a medical dataset. In this application, we use breast cancer data from Lee [9]. This dataset is taken from a large hospital in a period from 1929 to 1938 and it represents the survival times of 121 patients with breast cancer. The observations are given as follow: 0.3, 0.3, 4.0, 5.0, 5.6, 6.2, 6.3, 6.6, 6.8, 7.4, 7.5, 8.4, 8.4, 10.3, 11.0,
11.8, 12.2, 12.3,13.5,14.4, 14.4,14.8,15.5,15.7, 16.2,16.3,16.5,16.8, 17.2,17.3,17.5, 17.9, 19.8, 20.4,
20.9, 21.0, 21.0, 21.1, 23.0, 23.4, 23.6, 24.0, 24.0, 27.9, 28.2, 29.1, 30.0, 31.0, 31.0, 32.0, 35.0, 35.0, 37.0, 37.0, 37.0, 38.0, 38.0, 38.0, 39.0, 39.0, 40.0, 40.0, 40.0, 41.0, 41.0, 41.0, 42.0, 43.0, 43.0, 43.0, 44.0, 45.0, 45.0, 46.0, 46.0, 47.0, 48.0, 49.0, 51.0, 51.0, 51.0, 52.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 60.0, 60.0, 61.0, 62.0, 65.0, 65.0, 67.0, 67.0, 68.0, 69.0, 78.0, 80.0, 83.0, 88.0, 89.0, 90.0, 93.0, 96.0, 103.0, 105.0, 109.0,109.0,111.0, 115.0, 117.0, 125.0,126.0,127.0,129.0,129.0, 139.0 and 154.0.
Fatima [5] used the above dataset for their proposed model named as the Exponentiated Generalized Inverse Rayleigh distribution (EGIR) and compared with the Exponentiated Inverse Rayleigh (EIR), the Generalized Inverse Rayleigh (GIR) and the Inverse Rayleigh (IR) distributions. They found that the EGIR had the best fit for the Breast Cancer dataset.
Here, we compare the proposed distribution with the EGIR, the Ailamujia Inverted Weibull distribution (AIW) [16] and the New Extended Exponentiated Weibull distribution (NEEW) [1]. The probability density functions of EGIR, AIW and NEEW are given as follow:
fEGIR (x) = 2 ^ (1 - e-^*)-2 y- [1 - (1 - e-^y} * > 0, a, A, 7 > 0;
7-1
fAIW(*) = 4a92x-2a-1 e-29x , x > 0,9, a > 0;
and
fNEEW (x)
aAxA-1 e-axA (1 - e-axA)
e9(i-eax )(2 + 9 - 9e-a*A) + 2
e9 +1
, x > 0, a, A, 9 > 0.
In this application, we use the following diagnostics statistics: (i) Akaike Information Criterion (AIC); (ii) Bayesian Information Criterion (BIC); (iii) Kolmogorov - Smirnov (K-S); (iv) Cramer - von Mises (W*); and (v) Anderson - Darling (A). Furthermore, a package "fitdistrplus" in R software is also used. In addition, the results are shown in the following tables. Table 1 presents the maximum likelihood estimates of the fitted models for Breast Cancer dataset while Table 2 indicates that Idz distribution gives better estimate for the given dataset since it has a smallest values of some diagnostics statistics as compared with the EGIR, AIW and NEEW distributions. Also, same result is noticed from Figure 3.
Table 1: ML estimates of the fitted models using different distributions
Distribution a 9 /1 7
ID 1.68335961 0.02058629
EGIR 0.3331558 5986.6441628 2239.7157204
AIW 0.5159137 4.8684926
NEEW 0.1099727 1.7984615 0.7582980
Table 2: Some diagnostic statistics of the fitted models using different distributions
Distribution AIC BIC K-S A W *
A
W *
ID 1164.159 1169.751 0.05341806 0.51366194 0.06178559
EGIR 1279.365 1287.753 0.2180867 9.7828973 1.5945699
AIW 1250.729 1256.321 0.1723414 7.0402261 1.1024897
NEEW 1167.178 1175.565 0.07776119 0.45898966 0.06532455
ID
EGIR AIW NEEW
50
100
150
Figure 3: Estimated pdf of the fitted models for the Breast Cancer dataset.
7. Concluding Remarks
This paper derives a novel two - parameter continuous distribution called as Idz distribution. Some properties of Idz distribution such as moments, mean, variance, moment generating function, survival function, hazard function and order statistics were derived. Maximum likelihood method was used to estimate the parameters of Idz distribution. The applicability of the proposed distribution was examined by applying into a breast cancer data and compared with the Exponentiated Generalized Inverse Rayleigh (EGIR), the Ailamujia Inverted Weibull (AIW) and the New Extended Exponentiated Weibull (NEEW) distributions. It was found that the Idz distribution provides better fit for the given dataset as compared with the said distributions.
[1] Arif, M., Khan, D. M., Aamir, M., Khalil, U., Bantan, R. A. R. and Elgarhy, M. (2022). Modeling COVID-19 Data with a Novel Extended Exponentiated Class of Distributions.
Journal of Mathematics, Article ID 1908161, https://doi.org/10.1155/2022/1908161.
[2] Benrabia, M. and AlZoubi, L. M. A. (2021). Alzoubi distribution: properties and applications. Electronic Journal ofStatistics Applications and Probability: An International Journal, 1-16.
[3] Benrabia, M. and AlZoubi, L. M. A. (2022). Benrabia distribution: properties and applications. Electronic Journal of Applied Statistical Analysis, 15(2):300-317.
[4] Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and Methods, 31:497-512.
[5] Fatima, K., Naqash, S. and Ahmad, S. P. (2018). Exponentiated Generalized Inverse Rayleigh Distribution with Application in Medical Sciences. Pakistan Journal ofStatistics, 34(5):425-439.
[6] Gharaibeh, M. (2021). Gharaibeh Distribution and Its Application. Journal of Statistics Applications and Probability, 10(2):1-12.
[7] Gupta, R. C., Gupta, P. L. and Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Communication Statistics Theory and Methods, 27:887-904.
References
[8] Kingman, J. (1982). The coalescent. Stochastic Processes and their Applications, 13(3):235-248.
[9] Lee, E. T. (1992). Statistical Methods for Survival Data Analysis. John Wiley: Newyork.
[10] Lv, H.Q., Gao, L.H. and Chen, C.L. (2002). Ailamujia distribution and its application in supportability data analysis. Journal of Academy of Armored Force Engineering, 16:28-52.
[11] Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84:641-652.
[12] Shanker, R. (2015). Shanker distribution and its Applications. International Journal of Statistics and Applications, 5(6): 338-348.
[13] Shanker, R. (2016). Aradhana Distribution and its Applications. International Journal of Statistics and Applications, 6(1):23-34.
[14] Shanker, R. (2017). Rama Distribution and its Applications. International Journal of Statistics and Applications, 7(1): 26-35.
[15] Shraa, D. and Al-Omari, A. (2019). Darna distribution: properties and application. Electronic Journal of Applied Statistical Analysis, 12(2):520-541.
[16] Smadi, M. M. and Ansari, S. I. (2022). Ailamujia Inverted Weibull Distribution with Application to Lifetime Data. Pakistan Journal of Statistics, 38:341-358.
[17] Weibull, W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech., 18:293-296.
