# Generalization of the Weibull distribution: the odd Weibull family

Use load(“LungCancerData.RData”) to load the posted Lung Cancer Data. Use “time” for survival time. A failure density function (model) is assigned to each group (find in the table below).( Odd Weibull (OW))

1. Study the parameters of the assigned time to failure density function. Each parameter must be investigated separately while you fix the other parameters. Visualize the model while you change one parameter at a time.

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2. Estimate the parameters of the model using MLE (Maximum Likelihood Estimation) for the Lung Cancer dataset as follows.

a) For the entire data.

b) For Female Patients.

c) For Male Patients.

3. Estimate a set of two parameters at a time by fixing the other parameters and repeat this until you study all possible combinations of two parameters.

4. Using the estimated parameters in part 2, find and visualize the reliability and hazard for groups a, b, and c in question 2.

Presentation must include brief theory, simulations, observations, and conclusions. Please submit a report (pdf), your presentation (ppt), and the code.

Mathematical Sciences (2019) 13:105–114
https://doi.org/10.1007/s40096-019-0283-7
ORIGINAL RESEARCH
A new uniform distribution with bathtub‑shaped failure rate
with simulation and application
Jamal N. Al abbasi1 · Mundher A. Khaleel2 · Moudher Kh. Abdal‑hammed3 · Yue Fang Loh4 · Gamze Ozel5
Received: 11 October 2018 / Accepted: 8 April 2019 / Published online: 20 April 2019
Abstract
Bathtub failure rate shape is widely used in industrial and medical applications. In this paper, a three-parameter lifetime
distribution, so-called the generalized Weibull uniform distribution that extends the Weibull distribution, is proposed and
studied. This distribution has bathtub-shaped or decreasing failure rate function which enables it to fit real lifetime data sets.
Various structural properties of the new distribution are derived, including explicit expressions for the quantile function,
moments, moment-generating function and order statistics. Parameter estimations are provided by a maximum likelihood
estimation, and the performance of the maximum likelihood estimation is evaluated using a simulation study. An application
to real-life data demonstrates that the proposed distribution can be very useful in fitting real data.
Keywords Weibull distribution · Reliability · Failure rate · Bathtub shape · Quantile function
Mathematics Subject Classification 62E15 · 62F10 · 62P30
Introduction
In recent years, many classical distributions have been generalized by adding more shape parameters since numerous
application in the field of engineering, financial, biomedical
and environmental sciences indicated that classical distributions are not suitable to explain the data sets. Hence, there
is continual need for the extension of these distributions to
make effective progress in the application of these areas.
The Weibull distribution with two parameters is a flexible
* Gamze Ozel
gamzeozl@hacettepe.edu.tr
1
Department of Statistics, Al Nahrain University, Baghdad,
Iraq
2
Department of Mathematics, Faculty of Computer Science
and Mathematics, University of Tikrit, Tikrit, Iraq
3
of Administration and Economics, University of Tikrit,
Tikrit, Iraq
4
Department of Actuarial Science and Applied Statistics,
Faculty of Business and Information Science, UCSI
University, Kuala Lumpur, Malaysia
5
Department of Statistics, Hacettepe University, Ankara,
Turkey
distribution to model different types for lifetime data that
exhibit monotone-shaped failure rates. The Weibull distribution can be specified through its cumulative distribution
function (cdf) given by
𝜔
G(Z) = 1 − e−𝜐Z , Z, 𝜐, 𝜔 > 0.
(1)
where 𝜐, 𝜔 refer to the scale and shape parameters, respectively. According to Eq. (1), the probability density function
(pdf) of the Weibull distribution is given as follows:
𝜔
(2)
From (1) and (2), the failure rate function (FRF) is obtained
as
g(Z) = 𝜐𝜔Z 𝜔−1 e−𝜐Z .
h(Z) = 𝜐𝜔Z 𝜔−1 .
(3)
Indeed, when 𝜔 > 1, the FRF is increasing, whereas 𝜔 < 1, it is decreasing, and when 𝜔 = 1, it is constant. The Weibull distribution, on the other hand, is incapable to model data that exhibit bathtub-shaped or unimodal-shaped FRFs. The later can be considered as a common situation in industrial and medical applications, for example, electric machine life cycles, human mortality, life cycle of cancer patients, etc. Furthermore, the Weibull distribution may not be a suitable distribution for many of the engineering applications requiring load-bearing capability in the area of material science like strength of brittle materials which include most 13 Vol.:(0123456789) 106 Mathematical Sciences (2019) 13:105–114 ceramics, glass, optical fiber and some polymers, and in other numerous applications. However, the strength values are limited where the Weibull distribution requires an unlimited value. Therefore, it is necessary to find generalizations of the Weibull distribution. Several trials have been conducted to define new generalizations of the Weibull distribution by adding several additional shape parameter(s), for instance, see [7, 13, 15, 19]. Broadly speaking, for bringing more flexibility to baseline distribution can be added an extra parameter, so it can be more benefit in exploring tail properties and exhibiting bathtub-shaped failure rate function. Some models for bathtub and unimodal-shaped FRFs are summarized in Table 1. Note that the generalization is also complex and causes estimation problems, especially when the number of parameters is four or more. In this study, the main purpose is to obtain the more flexible distribution with a few parameters. Recently, a family of distributions has been proposed by Bourguignon [3] called as the Weibull-G “W-G” family by using the Weibull G(t;𝜼) distribution as in (1) and replacing the argument t with G(t;𝜼) , ̄ ̄ where G(t;𝜼)) = 1 − G(t;𝜼)). Its cdf is defined as −𝜐 F(t;𝜐, 𝜔, 𝜼) = 1 − e [ G(t;𝜼) ̄ G(t;𝜼) ]𝜔 , t ∈  ⊆ ; 𝜐, 𝜔 > 0,
(4)
where G(t;𝜼) refers to a parent cdf, which relies on a parameter vector 𝜼 . The corresponding pdf to (4) is
] [ ]𝜔
G(t;𝜼)𝜔−1 −𝜐 G(t;𝜼)
̄
f (t;𝜐, 𝜔, 𝜼) = 𝜐𝜔g(t;𝜼)
.
e G(t;𝜼)
𝜔+1
̄
G(t;𝜼)
[
Assuming that the parent distribution is uniform in the interval (0, 𝜑), 𝜑 > 0, then, the cdf is given by
t
,
𝜑
G(t;𝜑) =
(6)
0 < t < 𝜑 < ∞. Then, the corresponding pdf becomes 1 , 𝜑 g(t;𝜑) = (7) 0 < t < 𝜑 < ∞. Inserting (6) in (4) yields a three-parameter Weibull uniform (WU) distribution with the cdf given by [ ( F(t) = 1 − exp − 𝜐 t 𝜑−t )𝜔 ] , 0 < t < 𝜑 < ∞, 𝜐, 𝜔 > 0.
(8)
This distribution is also known as the Kies distribution [8].
The pdf of the WU distribution is given by
f (t) =
𝜐𝜔𝜑
(𝜑 − t)2
(
t
𝜑−t
)𝜔−1
[ (
exp − 𝜐
t
𝜑−t
)𝜔 ]
.
(9)
More recently, Cordeiro et al. [6] proposed a generalized
Weibull-G “GW-G” family of distributions by substituting the
argument t with − loge {1 − G(t;𝜼)} in Eq. (1) and defined the
cdf of their family of distributions by
𝜔
F(t;𝜐, 𝜔, 𝜼) = 1 − e−𝜐[− loge {1−G(t;𝜼)}] ,
t ∈  ⊆ ;
𝜐, 𝜔 > 0.
(10)
Then, the pdf of the GW-G family is given by
(5)
Table 1  Some models for bathtub-shaped or unimodal-shaped FRFs
Distribution
Distribution function, F(t)
Failure rate function, h(t)
Weibull uniform (WU), Kies [8]
)𝜔 ]
t
,
1 − exp − 𝜐 𝜑−t
𝜐𝜔𝜑
(𝜑−t)2
Exponentiated Weibull (EW), Mudholkar and Srivastava [11]
(
𝜔 )𝜆
1 − e−𝜐t ,
Additive Weibull (AW), Xie and Lai [18]
Modified Weibull (MW), Lai et al. [9]
[
(
𝜔
00
−𝜐t𝜔 e𝜆t
Odd Weibull (OW), Cooray [4]
1−e
,
t>0
[
( 𝜐t𝜔
)𝜆 ]−1
1− 1+ e −1
,t > 0
Flexible Weibull (FW), Bebbington et al. [2]
1 − e−e
SZModified Weibull (SZMW), Sarhan and Zaindin [14]
Kumaraswamy Weibull (KwW), Cordeiro et al. [5]
𝜐t− 𝜔t
𝜔
1 − e−𝜐t −𝛾t ,
t>0
[
]𝛾
(
𝜔 )𝜆
−𝜐t
1− 1− 1−e
, t>0
Weibull Pareto (WP), Alzaatreh et al. [1]
1−e
New Weibull Pareto (NWP), Tahir et al. [17]
1−e
Weibull power (WPo), Tahir et al. [16]
1−e
13
t>0
,
[
( )]𝜔
−𝜐 loge 𝜑t

[( )𝜐
−𝜐
t
𝜑
[
]𝜔
−1
t𝜆
𝜑𝜆 −t𝜆
]𝜔
,
,
,
t>𝜑>0
{
t
𝜑−t
}𝜔−1
𝜔
𝜔
𝜐𝜔𝜆t𝜔−1 e−𝜐t (1−e−𝜐t )𝜆−1
1−(1−e−𝜐t𝜔 )𝜆
𝜔−1
𝜆−1
𝜐𝜔t
+ 𝛾𝜆t
𝜐(𝜔 + 𝜆t)t𝜔−1 e𝜆t
𝜔
𝜔
𝜔−1 𝜐t
𝜐𝜔𝜆t
e (e𝜐t − ]1)𝜆−1
[
−1
𝜔
× 1 + (e𝜐t − 1)𝜆
(
)
𝜔
𝜐 + t𝜔2 e𝜐t− t
𝜐𝜔t𝜔−1 + 𝛾
𝜔
𝜔
𝜐𝜔𝛾𝜆t𝜔−1 e−𝜐t (1−e−𝜐t )𝜆−1
1−(1−e𝜐t𝜔 )𝜆
𝜐𝜔
t
[
𝜐 loge
( )]𝜔−1
t
𝜑
[( )𝜐
]𝜔−1
−1
[
]𝜔
t>𝜑>0
𝜐𝜔t𝜐−1
𝜑𝜐
0 0.
(14)
Here, the shapes of the pdf and FRF of the GWU distribution are discussed. The derivative loge f (t) with respect to t
is given by
(15)
So, the roots of Eq. (15) represent the critical points of the
pdf as follows:
{
)}𝜔
(
)
(
t
t
+ loge 1 −
+ (𝜔 − 1) = 0.
𝜐𝜔 − loge 1 −
𝜑
𝜑
(16)
Herein, there may be more than one root of Eq. (16). If t = t0
that is a point of inflexion, a local maximum or a local mini2
mum relying on whet her d 2 loge f (t;𝜐, 𝜔, 𝜑) = 0 ,
dt
d2 log f (t;𝜐, 𝜔, 𝜑) < 0 , d2 log f (t;𝜐, 𝜔, 𝜑) > 0 , where
e
e
dt 2
dt 2
d2 log f (t;𝜐, 𝜔, 𝜑) is given by
e
dt 2
{
(
)}{
(
)
}
{
(
)}𝜔
t
t
t
loge 1 −
loge 1 −
− (𝜔 − 1) − 𝜐𝜔 − loge 1 −
𝜑
𝜑
𝜑
)
}
(
{
t
+ (𝜔 − 1) − (𝜔 − 1)
× loge 1 −
𝜑
)2 {
( )}2
(
𝜑2 1 − 𝜑t
loge 𝜑−t
𝜑
It is difficult to determine the behavior of the density function from Eq. (16); despite this, in Fig. 1, some of the possible
shapes are illustrated of the pdf when 𝜑 = 2.5 and for different values of 𝜐 and 𝜔. For fixed 𝜔 < 1, the density is always U-shaped, and as 𝜐 tends to 0 (∞), the density function f(t) tends to J (reversed-J) shape, for fixed 𝜔 > 1, the right (left)skewed increases as 𝜐 decreases (increases), and when 𝜔 = 1,
the density function is monotonically decreasing if 𝜐 > 1,
constant if 𝜐 = 1 and monotonically increasing if 𝜐 < 1. Some calculation indicates the limiting behaviors of the pdf, and the limit of the f (t;𝜐, 𝜔, 𝜑) as T → 𝜑− is 0. Further, the limit of f(t) as T → 0+ is given by ⎧∞ ⎪ lim f (t;𝜐, 𝜔, 𝜑) = ⎨ 0 t→0+ ⎪ 𝜑𝜐 ⎩ if 0 < 𝜔 < 1 if 𝜔 > 1
if 𝜔 = 1
Shape of failure rate function
Based on Eq. (14), the cumulative hazard function H(t) is simply given by
13
108
Mathematical Sciences (2019) 13:105–114
)}𝜔
{
(
t
.
H(t;𝜐, 𝜔, 𝜑) = 𝜐 − loge 1 −
𝜑
The FRF has other important system’s quantity characterizing life phenomena and characteristic of interest of a random
variable defined by
h(t;𝜐, 𝜔, 𝜑) =
f (t;𝜐, 𝜔, 𝜑)
,
R(t;𝜐, 𝜔, 𝜑)
h(t) of the GWU distribution takes the following form:
h(t;𝜐, 𝜔, 𝜑) =
𝜐𝜔
d
H(t;𝜐, 𝜔, 𝜑) = (
)
dt
𝜑 1− t
{
)}𝜔−1
(
t
.
− loge 1 −
𝜑
𝜑
{ O b v i(o u s l y,)} b e c a u s e
− loge 1 − 𝜑t
𝜔−1
through
the
(17)
term
, the shape of FRF relies only on the
parameter 𝜔, also there is no direct effect on the shapes by the
remaining two parameters. In order to discuss the shapes of
h(t), with respect to t, the derivative of loge h(t) is given by

𝜔−1
d
1
loge h(t;𝜐, 𝜔, 𝜑) = �
� ⎢1 −
� ⎥. (18)

dt
𝜑 1 − 𝜑t ⎢
loge 1 − 𝜑t ⎥

So, the roots of Eq. (18) represent the critical points of the
FRF as follows:
)
(
t
− (𝜔 − 1) = 0
loge 1 −
(19)
𝜑
One can obtain the turning point of the FRF t0 by solving
Eq. (19) as follows:
]
[
t0 = 𝜑 1 − e𝜔−1
(20)
It can be shown from Eq. (17) that when 𝜔 ≥ 1, h(t) tends
to increase in t, and when 0 < 𝜔 < 1, h(t) tends to decrease for t < t0 and increase for t > t0. This implies a bathtub shape for
the FRF. However, at a minimum value t = t0 for the h(t), the
value of t0 decreases as 𝜔 increases, and h(t0 ) can be obtained
after simplification using Eqs. (17) and (20) as
h(t0 ;𝜐, 𝜔, 𝜑) =
Fig. 1  Plots of f(t) for selected parameter values of GWU distribution
13
𝜐𝜔(1 − 𝜔)𝜔−1
.
𝜑e𝜔−1
Although bathtub-shaped FRF are common in reliability
and survival analysis, these properties are related to several
lifetime distributions, as the GWU distribution.
The limiting behaviors of the FRF can be easily shown from
Eq. (17), while the limit of the as h(t) as T → 𝜃 − is ∞. Further,
the limit of h(t) as T → 0+ is given by
Mathematical Sciences (2019) 13:105–114
⎧∞

lim h(t;𝜐, 𝜔, 𝜑) = ⎨ 0
t→0+
⎪ 𝜑𝜐

109
if 0 < 𝜔 < 1 if 𝜔 > 1
if 𝜔 = 1
Figure 2 displays several of the possible shapes of
h(t;𝜐, 𝜔, 𝜑) for selected values of (𝜐, 𝜔, 𝜑).
Figure 2 reveals that the proposed distribution can produce
flexible FRF shapes such as increasing, decreasing, bathtub,
J, snf reversed-J. This fact implies that the GWU distribution
can be very useful to fit different data sets with various shapes.
Now, suppose Z is a random variable having the Weibull
distribution in Eq. (1) with parameter 𝜐 and 𝜔, the mgf of Z
is given by
(
)
v
∞ sv Γ 1 +

𝜔
(22)
MZ (s) =
v
v!𝜐 𝜔
v=0
Transformation
Since the relationship between the random variable Z has
the Weibull distribution in Eq. (1) and the random[ variable
]
T that follows GWU distribution given by T = 𝜑 1 − e−Z ,
the first non-central moment of T is given by
[
]
[
]
𝜇1� = 𝜑 1 − E(e−Z ) = 𝜑E 1 − MZ (− 1)
[
]
The random variable T = 𝜑 1 − e−Z , when a random variable Z is Weibull-distributed with scale parameter 𝜐 > 0 and
shape parameter 𝜔 > 0, follows the GWU(v, w, 𝜑) distribution
that helps us to simulate the random variable T in easy way
by first[ simulating
] the random variable Z and then calculate
T = 𝜑 1 − e−Z which has the GWU distribution.

⎡∞
v ⎤
v+1 Γ 1 + v ⎤
∞ (− 1)v Γ 1 +
(−
1)

𝜔 ⎥
𝜔 ⎥
𝜇1� = 𝜑⎢1 −
n
n
⎥ = 𝜑⎢

𝜔
𝜔
v!𝜐
v!𝜐
v=0

⎢ v=1

Moment
Assume Z is a random variable possessing the moment-generating function (mgf) MZ (s) = E(esZ ). One can express MZ (s)
in a Maclaurin series expansion as
MZ (s) =
∞ v �

s 𝜇v
v=0
(21)
v!
v
so that 𝜇v′ is the coefficient of sv! in expansion (21).
Setting s = − 1 in (22), we have MZ (− 1), and hence, the first
non-central moment of T becomes
(23)
So, the rth non-central moment of the GWU distribution
can be obtained directly from Eq. (22) for s = − r as
[
]
[
]
𝜇r� = 𝜑r 1 − E(e−rZ ) = 𝜑r 1 − MZ (−r)
)
(
v
∞ (− 1)v+1 r v Γ 1 +

𝜔
.
= 𝜑r
n
v!𝜐 𝜔
v=1
(24)
Further, the central moments 𝜇r of T are obtained from
Eq. (24) as follows:
r

( )
r
𝜇r =
(− 1)r 𝜇i� 𝜇r−i ,
i
i=0
(25)
For r = 2, 3, 4 , Eq. (25) gives 𝜇2 , 𝜇3 and 𝜇4 , which are of
the most interest because of the relations between variance,
kurtosis and skewness of the random variables Z. Note that
𝜇0� = 1 and 𝜇1� = 𝜇. Table 2 displays the numerical values for
mean, variance, skewness and kurtosis for several values of
parameters for the GWU distribution.
Moment‑generating function
The result in (25) can be used in (21) to rewrite the mgf as
(
)
w

∞ (− 1)w+1 vw Γ 1 +
v
v

𝜔
𝜑s
(26)
MT (s) =
w
v!
w!𝜐 𝜔
v=0
w=1
Fig. 2  Plots of h(t;𝜐, 𝜔, 𝜑) for selected parameter values of GWU distribution
For this section, Eq. (26) represents
[ ] the main result. The
characteristic function Φ(s) = E eisT of T can be √
obtained
from this equation with is in place of s, where i = − 1.
13
110
Mathematical Sciences (2019) 13:105–114
Table 2  Mean, variance, skewness and kurtosis for some values of 𝜐
and 𝜔 with 𝜑 = 1 (*:undefined)
𝜐
𝜔
Mean
Variance
Skewness
Kurtosis
1.0
1.2
1.5
2.0
3.0
4.0
5.0
8.0
10.0
1.0
1.2
1.5
2.0
3.0
5.0
10.0
1.0
1.2
1.5
2.0
5.0
10.0
1.0
1.2
1.5
2.0
5.0
10.0
1.0
1.5
2.0
5.0
10.0
1.0
1.5
2.0
5.0
10.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.5
1.5
1.5
1.5
1.5
1.5
2.0
2.0
2.0
2.0
2.0
2.0
5.0
5.0
5.0
5.0
5.0
10.0
10.0
10.0
10.0
10.0
0.500
0.454
0.400
0.333
0.250
0.200
0.167
0.111
0.091
0.512
0.471
0.422
0.361
0.284
0.203
0.123
0.527
0.492
0.449
0.395
0.250
0.170
0.546
0.517
0.482
0.438
0.313
0.237
0.592
0.563
0.543
0.480
0.435
0.611
0.596
0.586
0.553
0.528

0.389
0.338
0.293
0.258
0.188
0.158
0.430
0.434
0.430
0.413
0.372
0.303
0.208
0.446
0.450
0.449
0.439
0.355
0.271
0.460
0.465
0.467
0.464
0.412
0.350
0.475
0.484
0.489
0.493
0.486
0.473
0.479
0.483
0.492
0.496

0.231
0.245
0.241
0.209
0.187

0.881
0.156
0.223
0.254
0.226
− 0.035
0.009
0.062
0.127
0.252
0.259
− 0.061
− 0.024
0.022
0.080
0.221
0.268
− 0.129
− 0.090
− 0.062
0.029
0.094
− 0.157
− 0.138
− 0.124
− 0.078
− 0.042

0.232
0.240
0.226
0.210

0.199
0.242
0.242
0.121
0.120
0.126
0.143
0.229
0.260
0.127
0.123
0.123
0.131
0.197
0.247
0.148
0.136
0.130
0.126
0.137
0.159
0.151
0.146
0.133
0.127
Setting 𝜆 = 12 in (27) gives the median of the GWU distribution as follows:
} ]]
[
[ {
− loge (2) 𝜔
t 1 = 𝜑 1 − exp −
2
𝜐
(28)
Further, setting 𝜆 = 14 and 𝜆 = 34 gives the lower quartile
and upper quartile, respectively. Simulating the GWU random variable is straightforward. If U is a random number
following a uniform distribution  (0, 1), then the random
variable
} ]]
[
[ {
− loge (1 − u) 𝜔
T = Q(U) = 𝜑 1 − exp −
𝜐
(29)
will follow a GWU distribution with parameters 𝜐 , 𝜔 and 𝜑.
Order statistics
The pdf fj;n (t) of the jth orders T(j) is given by
(
)
n−1
fj;n (t) = n
f (t)F(t)j−1 {1 − F(t)}n−j , j = 1, 2, … , n.
j−1
(30)
Using
j−1
F(t)
)}𝜔 ]j−1
{
(
−𝜐 − loge 1− 𝜑t
[
= 1−e
=
j−1

(
)
(
{
)}𝜔
t
j−1
𝓁 −𝓁𝜐 − loge 1− 𝜑
(− 1) e
𝓁
(31)
𝓁=0
and
n−j
{1 − F(t)}
)}𝜔
{
(
−(n−j)𝜐 − loge 1− 𝜑t
=e
(32)
and substituting (31) and (32) into (30), we get
)}𝜔−1
(
{
(
) j−1
n−1 ∑
𝜐𝜔
t
fj;n (t) = n
(−1)𝓁 (
) 𝜐 − loge 1 −
j − 1 𝓁=0
𝜑
𝜑 1 − 𝜑t
{
(
)}𝜔
−𝜐(n−j+𝓁+1)𝜐 − loge 1− 𝜑t
× e
Quantile function and simulation
The quantile function of the GWU distribution is obtained
by inverting (12) as
} ]]
[ {
− loge (1 − 𝜆) 𝜔
Q(𝜆) = 𝜑 1 − exp −
𝜐
[
13
(27)
(33)
)
(
) j−1 (
(− 1)𝓁
n−1 ∑ j−1
f (t;𝜐n,𝓁 , 𝜔, 𝜑)
fj;n (t) = n
𝓁
j − 1 𝓁=0
(n − j + 𝓁 + 1)
(34)
where f (t;𝜐n,𝓁 , 𝜔, 𝜑) is the GWU PDF with parameters
𝜐n,𝓁 = (n − j + 𝓁 + 1)𝜐 , 𝜔 and 𝜑.
Therefore, the rth non-central moment of the jth OS of
GWU distribution is given by
Mathematical Sciences (2019) 13:105–114
𝜇r� = n
j;n
(
×
n−1
j−1
)
) j−1 (
∑ j−1
𝓁
(
v
∞ (−1)v+1 r v Γ 1 +

𝜔
𝓁=0
(−1)𝓁
𝜑r
(n − j + 𝓁 + 1)
)
111
Setting (40) and (41) to zero and simplifying, we obtain
n

̂
𝜐=
(35)
∑n �
i=1
t
− loge 1 − t i
(n)
v
v=1 v!{(n − j + 𝓁 + 1)𝜐} 𝜔
So, the pdf of the largest order statistics T(n) is given by
fn;n (t) = n
n−1

(
𝓁=0
)}w−1
(
{
)
n−1
𝜐𝜔
t
𝓁
(−1) (
) − loge 1 −
𝓁
𝜑
𝜑 1 − 𝜑t
{
(
)}𝜔
−𝜐(𝓁+1) − loge 1− 𝜑t
× e
(36)
fn;n (t) = n
n−1

(
n−1
𝓁
𝓁=0
)
(− 1)𝓁
f (t;𝜐𝓁 , 𝜔, 𝜑)
(𝓁 + 1)
(37)
n
=
∑n �
𝜔
̂
i=1
n

(38)
where f (t;𝜐n , 𝜔, 𝜑) is the pdf of the GWU distribution with
parameters 𝜐n = n𝜐 , 𝜔 and 𝜑.
t
− loge 1 − t i
(n)
��𝜔̂
Consider a random sample t1 , t2 , … , tn from GWU(𝜐, 𝜔, 𝜑)