# 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.

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

© The Author(s) 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

Department of Business Administration, Faculty

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
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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)

distribution, and assume Θ = (𝜐, 𝜔, 𝜑)T is a vector of the distribution parameter with known 𝜑 (since t < 𝜑), then the loglikelihood function denoted by 𝓁(Θ) can be written as
(
)
n
(
) ∑
t
𝓁(Θ) = n loge (𝜐) + loge (𝜔) − loge (t(n) ) −
loge 1 − i
− (𝜔 − 1)
t(n)
i=1
(
{
(
)}
)}𝜔
n
n {
∑
∑
t
t
×
loge − loge 1 − i
−𝜐
− loge 1 − i
t(n)
t(n)
i=1
i=1
(39)
Since 𝜑 is(supposed known,)the score vector can be defined
𝜕
𝜕
𝓁(Θ), 𝜕𝜔
𝓁(Θ)
𝜕𝜐
T
U𝜔 (Θ) =
n
n ∑
−
𝜐 i=1
{
�
��𝜔̂
t
− loge 1 − i
t(n)
(43)
U𝜐𝜔 (Θ) = −
n
𝜐2
n
∑
i=1
(44)
{
)}𝜔
)}
{
(
(
t
t
loge − loge 1 − i
− loge 1 − i
t(n)
t(n)
(45)
U𝜔𝜔 (Θ) = −
n
∑
n
−𝜐
2
𝜔
i=1
)}𝜔 [
{
(
)}]2
(
t
t
loge − loge 1 − i
− loge 1 − i
t(n)
t(n)
{
(46)
For large n, the bivariate normal distribution with zero
mean vector 0, and variance-covariance matrix√I2 (𝚯)−1 can
̂ − 𝚯) ,
be used to approximate the distribution of n(𝚯
where I(𝚯) = limn→∞ n−1 In (𝚯) is the expected. The estimated asymptotic bivar iate nor mal distr ibution
̂ can be used to construct approximate
N2 (0, I(𝚯)−1 ) of 𝚯
confidence intervals for the distribution parameters. A
large sample
intervals for 𝜐 and 𝜔
√ 100(1 − 𝜸)% confidence
√
𝜐 ± z 𝛾 var(̂
̂ ± z 𝛾 var(𝜔)
̂ , respectively, where
are ̂
𝜐) and 𝜔
2
2
the var(⋅)’s are the diagonal elements of In (𝚯)−1 , and z 𝜸 is
2
𝜸
the upper 2 th percentile of the standard normal
distribution.
where the components cor-
responding to the GWU(𝜐, 𝜔, 𝜑) distribution by differentiating
Eq. (39) determined the parameters of new model. The elements of the score model are given by
U𝜐 (Θ) =
i=1
��
̂ of 𝜔 .
The solution of Eq. (43) represents the MLE 𝜔
𝜐 of 𝜐 can be found by substituting the estimate
The MLE ̂
𝜔
̂ in Eq. (42).
The elements of the observed information matrix
In (Θ) = {Irs } for r;s = 𝜐, 𝜔 are found by
f1;n (t) = f (t;𝜐n , 𝜔, 𝜑)
Maximum likelihood estimation
n
�
�
���
�
��
�
�
n
�
t
t
−
loge − loge 1 − i
× loge − loge 1 − i
t(n)
t(n)
i=1
U𝜐𝜐 (Θ) = −
where f (t;𝜐𝓁 , 𝜔, 𝜑) is the pdf of the GWU distribution with
parameters 𝜐𝓁 = (𝓁 + 1)𝜐, 𝜔 and 𝜑, and the pdf of the smallest order statistics T(1) is given by:
by U(Θ) =
(42)
��𝜔̂
(
)}𝜔
t
− loge 1 − i
t(n)
(40)
{
(
)}
(
)}𝜔
n
n {
∑
t
t
n ∑
+
−𝜐
− loge 1 − i
loge − loge 1 − i
𝜔 i=1
t(n)
t
(n)
i=1
)}
(
{
ti
× loge − loge 1 −
t(n)
(41)
Simulation study
A simulation study is conducted to examine the performance of the MLE method to estimate the GWU distribution parameters. Monte Carlo technique is employed to
simulate the GWU random number generation depending
on the inversion method, i.e., varieties of the GWU distribution are generated using (29). The simulation study is conducted for nine-parameter incorporation, 𝜐 = 0.5, 1.0, 1.5 and
𝜔 = 0.5, 1.0, 1.5 and consider sample sizes with n = 30, 50
and 100. The process is repeated 5000 times for each parameter incorporation and each sample size.
13
112
1
The average bias AB = 5000
Mathematical Sciences (2019) 13:105–114
and
i=1
1 ∑5000
mean square error MSE = 5000 i=1 (estimate − actual)2 of
∑5000
̂ + 2q,
AIC = − 2(𝚯)
(estimate − actual)
̂ + q ln(n),
BIC = − 2(𝚯)
the MLEs are used to measure the performance.
It can be observed in Table 3 that for all the different
sets when we increase n, the bias for 𝜐 and 𝜔 decreases. The
MSEs tend to toward zero when we increase the sample size
n. The simulation results validate the performance of the
maximum likelihood estimators.
̂ refers to the log-likelihood function that evaluwhere (𝚯)
ates at the MLE, while q is the number of parameters and n
is the sample size.
Voltage data
The data have been obtained from Meeker and Escobar [10].
They represent the lifetimes of 30 electronic components
taken from power-line voltage spikes during electric storms.
The data have been previously used by Nadarajah et al. [12]
and Tahir et al. [16] (Table 4).
The total time on test (TTT) transform plot illustrates
that data set has a bathtub-shaped FRF. So, the GWU distribution may be suitable for modeling this data set. Table 5
provides the MLEs of the unknown parameters with corresponding standard errors for different distributions. The
values of these statistics in data sets are smaller for the GWU
Application
Voltage data were employed to illustrate the flexibility of
the GWU distribution, in addition to compare the behavior of the new model with other generalization of Weibull
distribution. The model selection is achieved by using the
maximized log-likelihood, Akaike information criterion AIC
and Bayesian information criterion (BIC) given by:
Table 3 MLE, bias and MSE
for various parameter values
when 𝜑 = 1
Sample size
Actual values
Bias
n
𝜐
𝜔
B(̂𝜐)
B(𝜔)
̂
MSE(̂𝜐)
MSE(𝜔)
̂
30
0.5
1.0
1.5
0.5
1.0
1.5
0.5
1.0
1.5
0.5
1.0
1.5
0.5
1.0
1.5
0.5
1.0
1.5
0.5
1.0
1.5
0.5
1.0
1.5
0.5
1.0
1.5
0.5
0.5
0.5
1.0
1.0
1.0
1.5
1.5
1.5
0.5
0.5
0.5
1.0
1.0
1.0
1.5
1.5
1.5
0.5
0.5
0.5
1.0
1.0
1.0
1.5
1.5
1.5
0.01219
0.04255
0.07695
0.00979
0.03798
0.08354
0.01502
0.03740
0.06931
0.00828
0.03238
0.04621
0.00687
0.02107
0.05013
0.00641
0.02259
0.04742
0.01051
0.02847
0.02957
0.00561
0.01278
0.02433
0.00513
0.01473
0.02716
0.11761
0.03608
0.02466
0.04648
0.04534
0.04657
0.06757
0.07139
0.06745
0.10939
0.02617
0.01424
0.02832
0.02710
0.02772
0.03924
0.04102
0.04072
0.12317
0.02028
0.00824
0.01289
0.01104
0.01188
0.01681
0.01706
0.01876
0.02083
0.04569
0.10246
0.01468
0.04540
0.10968
0.01519
0.04537
0.10517
0.01239
0.02506
0.05540
0.00904
0.02529
0.05355
0.00943
0.02542
0.05776
0.00644
0.01220
0.02356
0.00472
0.01197
0.02496
0.00458
0.01194
0.02552
0.01009
0.00588
0.00624
0.02446
0.02448
0.02503
0.05393
0.05480
0.05508
0.00586
0.00318
0.00320
0.01350
0.01335
0.01328
0.03009
0.02978
0.03020
0.00326
0.00147
0.00152
0.00632
0.00638
0.00616
0.01469
0.01430
0.01429
50
100
13
MSE
Mathematical Sciences (2019) 13:105–114
Table 4 Lifetimes of 30
electronic components from
Meeker and Escobar [10]
Table 5 MLEs of the
parameters (the standard errors
are given in parentheses) to the
voltage data and the values of
the − log likelihood, AIC and
BIC statistics
275
106
88
Dist.
113
13
300
247
147
300
28
23
212
143
181
300
300
30
300
23
likelihood
Estimates
65
300
300
− Log
10
2
80
300
261
245
173
293
266
AIC
BIC
135.29
274.59
278.79
300.03
139.22
282.43
286.63
0.005
(0.001)
–
169.92
347.84
353.44
7.701
(0.219)
0.004
(0.0000)
–
171.33
352.71
358.32
0.337
(0.057)
2.874
–
300.00
152.58
311.15
315.34
0.006
(0.055)
–
–
–
184.31
368.63
371.43
𝜐
𝜔
𝜆
𝛾
𝜑
GWU
0.603
(0.037)
0.624
(0.025)
–
–
300.03
WU
0.377
(0.102)
0.214
(0.029)
–
–
BW
0.079
(0.030)
0.066
(0.036)
7.936
(0.765)
KuW
0.052
(0.024)
0.229
(0.091)
WP
0.772
(0.252)
W
1.265
(0.252)
Bold values show the best model which is suggested in this paper
Fig. 3 Voltage data set. a TTT- transform plot represents to the voltage data set. b The empirical and estimated cdf of the distribution. c Probability plots. d The expected and observed number of observations
13
114
distribution compared to those values of other fitted distributions. The approximate 95% two-sided asymptotic confidence interval of the parameters 𝜐 , 𝜔 and 𝜑, respectively, is:
(0.391,0.815) , (0.417, 0.831) and (2030.72, 369.34).
It is concluded that the proposed model provides a goodness of fit for the given data set, which leads to ability to
choose the GWU distribution as the best model (Fig. 3).
Conclusion
The GWU distribution was introduced with the new three
parameters. The new model can have bathtub and increasing
shapes for the FRF. The new model displays the lifetime data
on a bathtub shape as a versatility of FRF. Many structural
properties of the GWU distribution have been studied and
derived, encompassing the moment, moment-generating
function and order statistics. The MLE was used to estimate
the asymptotic interval and point estimates of the parameters. One set of real-life data was employed to demonstrate
the flexibility of GWU distribution. The encourage results
of new model were compared with the results of the others distributions in the literature. Discover revealed that the
GWU distribution gives us a better fit than the other lifetime
models. It is expected that the new model can be employed
in other several sciences.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativeco
mmons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate
credit to the original author(s) and the source, provide a link to the
Creative Commons license, and indicate if changes were made.
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Publisher’s Note Springer Nature remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
Likelihood theory, numerical methods, simulation methods
April 8, 2015
1
Numerical maximization of the likelihood
For many statistical models, closed form expressions for the maximum likelihood estimators can
not be derived. For example, suppose that t1 , t2 , . . . , tn is a sample from a Weibull distribution
with parameters a and b. The density function for the Weibull distribution is given by
a
f (t) =
b
a−1
t a
t
e−( b ) .
b
(1)
The Weibull distribution can be derived from the assumption that the rate of mortality λ(t) =
(1/b)(t/b)a−1 . When the parameter a = 1, the rate of mortality is constant and the models
becomes equivalent to the exponential model. For other values of a, the rate of mortality
either increases or deceases with age. Note that the parameterization used here (and by R) is
somewhat differerent from the parameterization used in notat 2 in ST0103.
Given this model, the likelihood function for the observed data is thus
a
n
Y
a ti a−1 − tbi
L(a, b) =
e
b b
i=1
Y ti a−1 P
a
n −an
e− (ti /b) ,
=a b
b
(2)
(3)
and the log likelihood
l(a, b) = n ln a − na ln b − (a − 1)
X
ln ti −
X
(ti /b)a .
(4)
Noting that
∂
∂ a ln(ti /b)
(ti /b)a =
e
= ea ln(ti /b) ln(ti /b) = (ti /b)a ln(ti /b),
(5)
∂a
∂a
and setting the partial derivatives of the log likelihood function l(a, b) equal to zero yields
X
X
n
− n ln b −
ln ti −
(ti /b)a ln(ti /b) = 0
(6)
a
and
na a X
+
(ti /b)a .
(7)
b
b
These equations are non-linear in the unknowns a and b and no closed form solution can be
found.
The maximum likelihood estimates must therefore instead be computed using numerical
methods. This can by first defining a function in R which computes the likelihood (or the log
likelihood) for given values of the unknown parameters a and b and for a given set of data. We
shall then use a special function in R which finds the parameter values which maximises the
likelihood. This function (the R function optim) works by making repeated calls to a given
−
1
function for different parameter values. Based on the evaluated value of likelihood at these
parameter values, the optimisation algorithm usually finds it’s way at least to a local maximum
in the parameter space.
Let us start by defining a function in R which computes the likelihood. This should be
a function of the unknown parameters a and b and since the likelihood also depends on the
observed sample t1 , t2 , . . . , tn we need an additional argument containing the observations. Our
R function may then compute either the likelihood of the log likelihood. In practice, optimisation
on the log likelihood works best. By default, optim minimises the given function. Thus if we let
our function compute the negative log likelihood and minimise this we maximise the likelihood.
Based on equation (4) we can define the likelihood function in R as follows
l

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