CC BY
45
The New Mixed Erlang Distribution: A Flexible Distribution for Modeling Lifetime Data
Therrar Kadri1, Souad Kadri2, Seifedine Kadry3*, and Khaled Smaili4
•
Department of Education, Lebanese University, Beirut, Lebanon1 Department of Mathematics and Physics, Lebanese International University, Khyara, Lebanon1,2
Department of Applied Data Science, Noroff University College, Norway3 Department of Applied Mathematics, Faculty of Sciences, Lebanese University, Zahle, Lebanon4
*Corresponding author
Abstract
We introduce a new mixed distribution of the Erlang distribution that is generated from the convolution of the Extension Exponential distribution denoted by the Mixed Erlang distribution (ME). We derive an exact closed expression of the probability density function which is used to obtain closed expressions of the cumulative function, reliability function, hazard function, moment generating function and kth moment. The method of maximum likelihood and method of moments is used for estimating the model parameters. Two applications to real data sets are given to illustrate the potentiality of this distribution.
Keywords: Erlang Distribution, Extension Exponential Distribution, Probability Density Function, Maximum likelihood estimation, Moments, Akaike Information Criterion
1. Introduction
Numerous classical distributions have been extensively used over the past decades for modeling data in many applied areas such as lifetime analysis, finance and insurance, as the Exponential distribution and its alternatives the Erlang and Gamma distribution, see [1] and [2]. There is a clear need for extended forms of these distributions. In recent statistical literature modified extensions of the Exponential distributions have been proposed to give more flexibility to model real data. For example, Gupta and Kundu [6] introduced an extension of the Exponential distribution typically called the generalized exponential (GE) distribution and Mudholkar et al.
[9] introduced the exponentiated Weibull (EW) distribution as another extension. Gómez et al. [5] introduced a new extension of the Exponential distribution denoted as the Extended Exponential (EE) distribution of two positive parameters.
On the other hand, the sum of independent random variables, the convolution of random variables, also plays a significant role in modeling many events in most domains of science, as communications, computer science, and teletraffic engineering (Trivedi [12]; Jasiulewicz and Kordecki [7]), Markov process, reliability and performance evaluation (Kadri et al. [8]; Smaili et al.
[10]). A comprehensive study of these distributions is needed for modeling and the importance of providing closed and exact forms of probability density function (PDF), cumulative distribution function (CDF), reliability and hazard functions, moment generating function (MGF), and kth moments...
The main aim of this paper is to study a new distribution generated from the convolution of the Extension Exponential distribution which is observed to have the form of a mixed distribution of the Erlang distribution. We denote this distribution by the Mixed Erlang Distribution (ME). We provide a comprehensive account of the mathematical properties of this new distribution
by deriving closed and exact forms of PDF, CDF, reliability and hazard functions, MGF, and kth moments. Moreover, we propose that our distribution is quite flexible, evidence by its closed and simple expressions. Also, this distribution can be used quite effectively in analyzing positive data in place of proposed distribution in the literature, which indicates that the ME distribution is a serious competitor to the others. Thus, we preform a parameter estimation of the model by the method of maximum likelihood and the method of moment. Next, we fit the new distribution to two real data sets to examine the performance of the new model and compare it to lately new distributions proposed in the statistical literature.
2. Some Preliminaries
2.1. Erlang Distribution
Erlang distribution is a two-parameter continuous probability distribution with shape integer parameter n and scale parameter a > 0. It is considered as the sum of n independent identical Exponential distributions of parameter a, so for n = 1 the Erlang distribution is simplified to the Exponential distribution. Erlang distribution like Exponential is widely used in life time analysis, see [4].
Let Y ~ Erl(n, a). The PDF of Y is given as:
. , , (at)" 1 ae-at
fY(t)=( (n - 1)! ,t > 0 (1)
and CDF of Y is
F (t) =. r(nat) (2)
Fy (t) =1 - (2)
where r(-, ■) is the incomplete Gamma function. Also the MGF of Y is
<Py (o = ( a^Y,(<a (3)
and the moment of order k of Y is
E[Yk ] = r(kn + k) (4)
L J akr(n) v '
2.2. Extension Exponential Distribution
Gomez et al. [5], introduced a new extension of the Exponential distribution denoted as the Extended Exponential (EE) distribution of two positive parameters, denoted by EE (a, ft). They characterized this distribution having X ~ EE (a,ft) with PDF
fx(t) = ^^^ ^t > 0
where a is a scale parameter and ft is a shape parameter.
The EE distribution is considered as a mixed distribution of the Exponential distribution, E(a), and Erlang distribution Erl(2, a), i.e.
fx (t) = ar/E(")(t) + aar/Eri(2,*)(t) (5)
For further use, we have the Laplace transform of fX(t) as
rSf (tu a2(t + a + ft) (6)
L{fx (t)} = (a + ft)(t + a)2. (6)
Other properties of EE distribution can be found in [5].
3. Mixed Erlang Distribution Let Xj, j = 1,2,..., n be n identical independent (iid) random variables that follow Extension
n
Exponential distribution i.e. Xj — EE (a, ft) and let Sn = £ Xj. We denote Sn — ME (a, ft, n) to
j=1
be the Mixed Erlang distribution for a > 0, ft > 0 and n E N*. This name is derived from the obtained expressions of this distribution in this section, which has the form of a mixed Erlang (ME) distribution. We start by deriving the PDF of this new distribution in an exact closed form. The obtained simple form will help us to derive the other mathematical functions to characterize the ME distribution.
3.1. PDF of the ME distribution
It is known that the sum of independent distributions is the convolution random variable and its PDF can be determined by the n convolution of the PDF of the summands Xj, which is an approach used. Here we take the advantage of Laplace transform over convolution to obtain our expression.
Theorem 1. Let Sn — ME (a, ft, n), a > 0, ft > 0 and n E N*. Then the PDF of Sn is given by
where
fsn (t) = E f (t) i=0
(n)an-i в1
A = ¥—and Yi - Erl(n + i, a) (7)
i (a + ft)n y ' w
Proof. Let Xj — EE (a, ft), j = 1,2,..., n be n iid distributions and let Sn = £ Xj — ME (a, ft, n).
j=1
We have fSn (t) is the convolution of the PDF of Xj. Thus the Laplace transform of fSn (t) is the product of identical distribution of EE and we get
From Equation (6) L{fXi(t)} = We get
L{ fsn (t)} = [L{ fXi (t)}]'
(а+в+t) (a+e)(t+a)2'
Hfs,(»}- a2n(a + в +f)"
(a + ft)n (t + a)2n
and
fs (t) = c-i\ a2n (a + в + ОМ = a2n 1 f (a + ft + t)n
Ja + ft)n(t + a)2n ) (a + ft)n { (t + a)2n a2ne-af -1 ( (ft + t)n
(a + ft)n { t2n
n in I An n
However, (ft + t)n = E (n)ftltn-1, then ЩУ- = E (n)ftlt-n-1. Also L {t-n-1}
__t(i+n-1)
(i)в l , then t2n = E (i)в 1 . Also L ^ f = (i+n-1)r
i=0 i=0
Thus we conclude that
fsn (t) = fx+fn £ (I)ftl t'+n-1y.' Next, we rearrange the sum to pull out the closest PDF which
)n . i=0
„n+ii (n+
is a Erlang distribution of the form a—1 e— = fy.(t), where Y — Erl(n + i,a). So we can rewrite
EE (n)ftian-i an+4(n+i-1)e-a* fSn(t) = E0 (a + ft)n X (n + i - 1)!
E f (t) i=0
where A- — (nK 'в
(a+ß)n
■
In the following, we give another proof of the previous theorem by using the approach of convolution of PDF of independent random variables instead of the Laplace inverse approach.
Proof. [Alternate Proof]Let Xj ^ EE (a, ß), j = 1,2,..., n and Sn is the convolution random variable of EE. Then
fsn (t) = (fxi * fx2 * ... * fxn )(t) (8)
However, from Equation (5), the PDF of EE can be expressed as
a ß
fXj (t) = a+ß fE(a)(t) + a+ß -^(^w
Substitute fx• (t) in 8 to obtain
fSn(t) = ® (*/e(*)+ frlfr«))) (i)
/
where © (g) means that the expression is convoluted n times by itself. Furthermore, convolution is associative with scalar multiplication, thus
1 n / \
fSn (t) = (a + ft)n ® (afE(«) + ftfErl(2,a)) (t)
Now, using the generalized binomial expansion over the convolution operation, we obtain
1 " n—i i
fSn (i) = (I+ftF Ç (n) ® afE(x)(t) * ®ßfErl(2A)(t)
)n L-tW ^ "VE(a)W * ^K/trl(2,a) ) i=0
1 J^ ■ ■ (n — i i \
= l^-n E (n)an-ift' ( © fE(a) * ©fErl(2,a) J (t)
Also the convolution of n — i identical Exponential distribution is the Erlang distribution Erl(n —
ni
i, a) or © fE(a) = fErl(n-i,a) and the convolution of i Erlang distributions Erl(2, a) is the Erlang
distribution Erl(2i,a) or ®fErl(2A) = fErl(2i,x)- Thus, we get
1— П ^ n
i=0
On the other hand Erl(n — i,a) * Erl(2i,a) = Erl(n + i,a) and thus
1 n n ' ( \
fSn (t) = (a + ß)n E (ni)a" 'ß* yfErl(n—iA) * fErl(2i,a)) (t)-
n (n)an—i ßi fSn (t) = E a + ß)n /Erl(n+i,a)(i)
i=0 n
= E AifY (t)
j=0
(n)an-i ft*
where A, = (()+ft)« and Yj - Erl(n + i,a).
■
In the following corollary, we give the PDF of ME in one expression, related to regularized confluent hypergeometric function.
Corollary 1. Let Sn - ME (a, ft, n), a > 0, ft > 0 and n G N*. Then the PDF of Sn is given by
a2nin-lp — «t ^
fsn (t) = ^TftF"1 F (-n; n,
where 1Fi (a; b; x) is the regularized confluent hypergeometric function.
n (ll)Kn-i ai
Proof. From Theorem 1 we have fSn (t) = E AifYi(t) with Ai = (()+a)n and Yi ~ Erl(n + i, a).
i=0 ( a)
( i\n+'~1 _at
However, the PDF of Y. from Equation (1) is given by fy.(t) = (a — I(o,^)(t)• Thus
f (n)an_'P\ (at)n+ae_at fSn(t) .to (a + p)n X (n + i _ 1)!
_ a2ne_at ft F n+i_i
f 7ZTT1 rrrt
(a + a)ni=o(n + i - 1)!
iFi (-n; n, -ta)
a2ntn-1 e-a f
(a + a)n
n
_(tpy _
where E = 1F1 (-n; n,-tp) is the regularized confluent hypergeometric function
i=0 (n+l 1)!
which is defined as 1F1 (a; b, x) = 1 ^(i)^ having 1F1 (a; b, x) be the Kummer confluent hyper-geometric function.
3.2. CDF, MGF and other functions for ME distribution
In Theorem 1, we found a closed expression of the PDF for sum of identical EE random variables
n
and we gave the PDF expression as f Aify. (t). This expression shows that our distribution is
i=0 '
also a mixed distribution of the Erlang distribution. We take an advantage of this expression to find the other statistical characterization as CDF, MGF, moment of order k, reliability and hazard functions for ME distribution. Next, we derive exact closed expressions of these functions.
Theorem 2. Let Sn ~ ME(a, p, n), a > 0, p > 0 and n e N*. Then the CDF of Sn is given by
n
FSn (t) = f A>FYi (t) i=0
where Ai is defined in Equation (7) and Fy. is the CDF of Y' ~ Erl(n + i,a).
n
Proof. From Theorem 1, the PDF of Sn is fSn (t) = f Aify. (t). The CDF of Snis defined as
n i=0 i
t t t
//» n n /* n
fSn (x)dx = f Aify. (x)dx = f Ai fy. (x)dx = f AiFy. (t). 0 0 i=0 i=0 0 i=0
Lemma 1. f A. = 1.
i=0
Proof. Let Fy. (t) and FSn (t) be the CDF of Y. and Sn respectively. However, the limit at infinity of any CDF is 1. Starting from the expression of the CDF in Theorem 2, Fsk (t) = f n=0 AiFy. (t), we have lim FSn (t) = Jim fn=0 AiFy. (t), thus f"=0 Ai = 1. " i
■
Next, we give another expression for the CDF of Sn. Corollary 2. Let Sn ~ ME(a, p, n), a > 0, p > 0 and n e N*. Then the CDF of Sn is
F (t) = ^ an f (n)Plr(n + i, at) Sn() (a + p)n ¿0 a i (n + i _ 1)!
n
where r(-, ■) is the upper incomplete gamma function.
Proof. From Theorem 2, the CDF of Sn is FSn (t) = E=0 A-Fy (t). However, from Equation (2)
we have F^ (t) = 1 - r^-l^. Therefore,
r(n + i, at) \
(n +i - 1)0 ,
ъ (i) = E a — A
but from Lemma 1, E=0 Ai = 1 This leads to FSn (t) = 1 - E=0 Ai ^++-1)!. Moreover, from
(n)an-i fti
Equation (7) A- = v(>a+py and we get
F (t) = 1 £ A r(n + *,at)
FSn(i) = 1 — £ Ai (n + i — 1)!
= an £ (n)вГ(п + i, at)
= (a + e)nE0 ai (n + i — 1)!
Theorem 3. Let Sn - ME(a, ft, n), a > 0, ft > 0 and n G N*. Then the MGF of Sn is
n
<Sn (t) = E Ai <Y (t) i=0
where Ai is defined in Equation (7) and <Y. is the MGF of Yi — Erl(n + i, a).
n
Proof. Referring to Theorem 1 the PDF of Sn is fsn = E A-f^(t).Thus,
i=0
, ^TO + TO
ФSn (t) = / etxfSn (x)dx =1 etx (Aify (x)) dx = a. / ef (x)dx
+ TO
but / ef (x)dx = фу, (t), thus фsn (t) = £ Aiфу, (t).
i=0
Corollary 3. Let Sn - ME(a, в, n), a > 0, ß > 0 and n G N*. Then the MGF of Sn is
ф (t) = a2n £ (n)в
$Sn(t) = T^en £ i=0
(a + e)ni=0 (a — t)n+i'
Proof. From Theorem 3 <sn (t) = En=0 Ai<y (t) and the MGF of Erlang distribution Yi is given
^)n+- which leads to <Sn(t) = E"=0 A- (^)
in Equation (3) as фу(t) = {^)n+l which leads to фSn(t) = £n=0 Ai (^)п+-. Moreover, from
(n)an —i в
Equation (7) Ai = v(a+^n then
ф (t) £ (n)an—iв ( a X+l
фsn (t) = £ TOW" la—t J
a2n Д (n)e
E
(a + e)ni=0 (a — t)n+i
Theorem 4. Let Sn ~ ME(a, p, n), a > 0, p > 0 and n e N*. Then the reliability function of Sn is
n
RSn (t) = f ArRy, (t) i=0
and the hazard function of Sn is given as
() = su Aihi (t)Ryi (t)
hSn (t) £n=0 A'Ry' (t)
where hy (t) and Ry (t) are the hazard and reliability functions of Y' ~ Erl(n + i, a) respectively, and Ai is defined in Equation (7).
Proof. RSn (t) = 1 _ FSn (t), and from Theorem 2 we have Fgn (t) = En=0 AiFy. (t)
n
RSn (t) = 1 _ f AiFy, (t), i=0
nn
but Fy. (t) = 1 _ Ry. (t), and from Lemma 1, we have f A. = 1, thus RSn (t) = f A.Ry. (t).
i i i=0 n i=0 i
On the other hand, the expression of hazard function is given by
h,S (t)= Ml = fn=0 Aify{ (t) hSn ( ) RSn (t) fn=0 A'Ry, (t)
however, fy. (t) = h,(t)Ry (t), then
fn=0 Ah (t)Ry (t)
hsn (t) =
П=о AURYi (t) ■
Theorem 5. Let Sn ~ ME(a,p,n), a > 0, p > 0 and n e N*. Then the moment of order k of Sn is
E\Sk] = f A EYk] = k!an_k f p' (n\ (n +' + k E[Sn] = h = ia+Whn+,
where Y, ~ Erl(n + i, a), and A, is defined in Equation (7).
n
Proof. From Theorem 3, we have $Sn (t) = f A,(t). Now the moment of Sn of order k is
given by E[Sn] = = £ Аг ^ = £ A^]. Moreover, E[Y?] = ^
i=0
t=0 i=0
(n)a n-i ßi
from Equation (7) Ai = у(;я+ß)„ then
and
t=0 i=0
ESk.= £ (П)"n-ißl Г(п + i + k) E[Sn] £0 (a + ß)n акГ(п + i)
_ an-k £ ßi (n)Г(п + i + k)
as
(a + ß)n =0 a r(n + i) = k!an-k £ ß (n\(n + i + k
= (a + ß)n ¿0 n + i
(n)Г(п + i + k) _ щ-щk! (n +i + k - 1)! Jn\ fn + i + k
k!
(n_')!i
r(n + i) (n + i _ 1)!k! 'V'V'VV n + i
■
We end this part to point out the importance of writing the PDF of our ME distribution as linear combination PDF of the known Erlang distribution. This expression facilitates in determining the other statistical expressions as CDF, reliability and hazard functions, moment generating function (MGF), and kth moments. This procedure was adopted by Smaili et al. [10] and [11]. Later, these expressions are used to give an estimated model for a real-life data.
4. Real Life Data
To illustrate the new results presented in this paper, we fit the ME distribution to two examples of real data. The MLE and MME approaches are employed to estimate the parameters of the real-life data and MATHEMATICA software is used. We analyze real data sets to show that the ME distribution can be a better model than other existing distributions. We consider the distributions in recent papers that proposed their distribution to fit the model data. For each data set, we compare the fitted distributions using the three criteria: AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected) and BIC (Bayesian Information Criterion). Let us be precise that log(L) is the log-likelihood taking with the estimate values, AIC = 2k - 2 log(L), AICC = AIC + 2nk(kk+11) and BIC = -2 log(L) + k log(n), where k denotes the number of estimated parameters and n denotes the sample size. The best fitted distribution corresponds to lower AiC, AICC and BiC. Also the histogram and the estimated PDFs and CDFs for the best fitted models to the two data are displayed in Figures 1 and 2, respectively.
Data set 1: The data set contains n = 63 measures related to the strength of 1.5cm glass bers. It is reported in Smith and Naylor (1987): 0.55, 0.93,1.25,1.36,1.49,1.52,1.58,1.61,1.64,1.68,1.73, 1.81, 2, 0.74,1.04,1.27,1.39,1.49,1.53,1.59,1.61,1.66,1.68,1.76,1.82, 2.01, 0.77,1.11,1.28,1.42,1.5, 1.54, 1.6, 1.62, 1.66, 1.69, 1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.5, 1.55, 1.61, 1.62, 1.66, 1.7, 1.77, 1.84, 0.84,1.24, 1.3, 1.48,1.51,1.55,1.61,1.63,1.67, 1.7, 1.78,1.89.
We chose the analysis done by Chesneau in [3] for this data. Chesneau compared the Lindley, Exponential, Exponentiated Exponential (EExp), and Exponential Hypoexponential distribution (EHypo). The corresponding PDF of EExp and EHypo are given by
fEExp(x) = Aae-Ax (1 - e-Ax)°-1
fEHyp(x) = Aa (1 + 10a) e-Ax (1 - e-Ax)°-1 (1 - (1 - e-Ax^
respectively. We derive two estimated distributions of the ME distribution using MLE and MME and used as a competitive distribution of the previous ones. See Table 1. This table shows that the ME model gives a better fit to this data than the other distributions. The plots in Figures 1 also indicate the same thing. So, the ME model could be chosen as the best model.
Table 1: MLE and MME of ME Distribution with MLEs of competitor distributions and AIC, A1CC and BIC of data set 1
Model Estimated Parameters AIC AICC BIC
ME(MLE) w = 13, W = 13.771948428082293, ft = 20.34250599538599 50.547 50.747 63.1195
ME(MME) w = 15, W = 16.653713280301165, ft = 25.644927330965096 53.678 53.8784 66.251
Lindley d = 0.996116 164.56 164.62 166.70
Exponential A = 0.663647 179.66 179.73 181.80
EExp W = 31.3489, A = 2.61157 66.76 66.96 71.05
EHypo W = 24.0816, A = 1.83894 59.67 59.87 63.96
Data set 2: The data consist of 101 observations. The data was presented in Birnbaum & Saunders (1969) and correspond to the fatigue time of 101 6061-T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second (cps). The data are: 70, 90, 96, 97, 99,100, 103, 104, 104, 105,107, 108, 108, 108, 109,109, 112, 112, 113, 114,114, 114, 116, 119, 120, 120, 120, 121,121,123, 124, 124,124, 124, 124,128,128, 129,129,130, 130, 130,131,131, 131, 131, 131, 132, 132,132,133, 134, 134,134, 134, 134,136,136, 137,138,138, 138, 139,139,141, 141, 142, 142, 142, 142,142,142, 144, 144,145, 146, 148,148,149, 151,151,152, 155, 156,157,157, 157, 157, 158,159, 162,163,163, 164, 166,166,168, 170, 174,196, 212.
We chose the analysis done by Yousof et al. in [13] for this data. They compared the Weibull, Wei-Weibull (WW) and their Weibull-Weibull logarithmic (WWL) distribution to fit this data. The
(a) Fitted pdfsfor data 1 (b) Fitted cdfsfor data 1
Figure 1: The two figures show a bestfittingfor the EE distribution
CDF of WW and WWL distribution are given by
FWW (x; a, в, Л, y) = 1 — e
-a( eAx7- 1\в
Fwwl (x; a, в, Л, y, р)
1 ( y \в—m (Ax7—1—«(e^7—1)
paßYЛxY—1 (еЛ^ — 1j I A v '
respectively for a, ft, A, 7 > 0 and 0 < p < 1. We derive two estimated distributions of the ME distribution using MLE and MME and used as a competitive distribution of the previous ones, see Table 2. This table shows that the ME model gives a better fit to this data than the other distributions. The plots in Figure 2 also indicate the same thing. So, the ME model for the second time could be chosen as the best model.
Table 2: MLE and MME of ME Distribution with MLEs of competitor distributions and AIC, A1CC and BIC of data set 2
Model Estimated Parameters AIC AICC BIC
ME(MLE) и = 32, И = 0.31305352844373824, ß = 0.13952982315141454 916.557 916.679 931.018
ME(MME) и = 28, И = 0.3149943298488485, ß = 0.27793752048684467 918.586 933.047 918.709
Weibull Л = 0.0036, 7 = 1.1516 1167.38 1167.5 1172.61
WW Л = 0.0036, 7 = 1.1516 1167.38 1167.5 1172.61
WWL и = 0.0060, ß = 3.1600, Л = 0.0873, 7 = 0.6264, р = 0.8732 948.49 948.91 961.56
We see in Tables 1 and 2 that the ME distribution has the smallest AiC, AiCC and BiC for the two data sets, compared with lately proposed distributions, indicating that the ME distribution is a serious competitor to the other considered distributions.
5. Conclusion
A new distribution, Mixed Erlang (ME) distribution, has been proposed and its properties are studied. We derived exact closed expressions of the PDF, CDF, reliability function, hazard function, MGF, and kth moments. We have studied the maximum likelihood estimators and method of moments estimators and the parameters estimation is carried out in the presence of real data. We presented two real life data sets, and our ME distribution was compared with lately proposed distributions and showed that the ME distribution is a serious competitor to the others.
0.010
0.005
0.015
0.020
50
100
150
200
100
150
200
(a) Fitted pdfs for data 2
(b) Fitted cdfs for data 2
Figure 2: The two figures show a best fitting for the EE distribution
References
[1] Abdelkader, Y.H.: Erlang distributed activity times in stochastic activity networks. Kyber-netika. 39(3), 347-358 (2003)
[2] Bocharov, P.P., D'Apice, C., Pechinkin, A.V.: Queueing theory. Walter de Gruyter. (2011)
[3] Chesneau, C.: A new family of distributions based on the hypoexponential distribution with fitting reliability data. Statistica. 78(2), 127-147 (2018)
[4] Forbes, C., Evans, M., Hastings, N., Peacock, B.: Statistical distributions. John Wiley & Sons (2011)
[5] Gómez, Y.M., Bolfarine, H., Gómez, H.W.: A new extension of the exponential distribution. Revista Colombiana de Estadística. 37(1), 25-34 (2014)
[6] Gupta, R.D., Kundu, D.: Exponentiated exponential family: An alternative to Gamma and Weibull distribution'. Biometrical Journal. 43(1), 117-130 (2001)
[7] Jasiulewicz, H., Kordecki, W.: Convolutions of Erlang and of Pascal distributions with applications to reliability. Demonstr. Math. 36(1), 231-238 (2003)
[8] Kadri, T., Smaili, K., Kadry, S. In: Markov modeling for reliability analysis using Hypoexponential distribution. In Numerical Methods for Reliability and Safety Assessment, pp. 599-620. Springer, Cham (2015)
[9] Mudholkar, G.S., Srivastava, D.K.: Exponentiated Weibull family for analysing bathtub failure data. IEEE Trans. Reliability. 42, 299-302 (1993)
[10] Smaili, K., Kadri, T., Kadry, S.: A modified-form expressions for the hypoexponential distribution. British Journal of Mathematics & Computer Science. 4(3), 322-332 (2014)
[11] Smaili, K., Kadri, T., Kadry, S.: Finding the PDF of the hypoexponential random variable using the Kad matrix similar to the general Vandermonde matrix. Comm. Statist. Theory Methods. 45(5), 1542-1549 (2016)
[12] Trivedi, K.S.: Probability & statistics with reliability, queuing and computer science applications. John Wiley & Sons (2008)
[13] Yousof, H.M., Rasekhi, M., Afify, A.Z., Alizadeh, M., Ghosh, I. and Hamedani, G.G.: The Beta Weibull-G Family of Distributions: Theory, Characterizations and Applications. Pakistan Journal of Statistics. 33, 95-116 (2017)
