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A New Generalized Weighted Weibull Distribution

Salman Abbas

Department of Mathematics,

COMSATS University Islamabad, Wah Capmus, Pakistan

[email protected]

Gamze Ozal

Department of Statistics

Hacettepe University, Ankara, Turkey

[email protected]

Saman Hanif Shahbaz

Department of Statistics

King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

[email protected]

Muhammad Qaiser Shahbaz

Department of Statistics

King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

[email protected]

Abstract

In this article, we present a new generalization of weighted Weibull distribution using Topp Leone

family of distributions. We have studied some statistical properties of the proposed distribution

including quantile function, moment generating function, probability generating function, raw moments, incomplete moments, probability, weighted moments, Rayeni and q − th entropy. We have

obtained numerical values of the various measures to see the effect of model parameters. Distribution of order statistics for the proposed model has also been obtained. The estimation of the model

parameters has been done by using maximum likelihood method. The effectiveness of proposed

model is analyzed by means of a real data sets. Finally, some concluding remarks are given.

Keywords: Weighted Weibull, Topp Leone family, Order Statistics, Entropy

1.

Introduction

In some situations, it is found that the classical distributions are not suitable for the

data sets related to the field of engineering, financial, biomedical, and environmental

sciences. The extension of classical models is, therefore, continually needed to obtain

suitable model for applications in these areas. Researchers have obtained several extended models for use in these situations but still the room is available to obtain new

models with much wider applicability.

The Weibull distribution is not a suitable model to explain the non-monotone hazard rate function (hrf), such as unimodal, U-shaped or bathtub form. Hence, there

are many generalizations of the Weibull distribution in the literature. Some notable

models are, exponentiated Weibull distribution by Mudholkar and Srivastava (1993),

extended Weibull distribution by Ghitany et al. (2005), beta Weibull distribution

by Lee et al. (2007), the flexible Weibull distribution by Bebbington et al. (2007),

Kumaraswamy Weibull distribution by Cordeiro et al. (2010), truncated WeibulPak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

Salman Abbas, Gamze Ozal, Saman Hanif Shahbaz, Muhammad Qaiser Shahbaz

l distribution by Zhang and Xie (2011), The Topp-Leone Generated Weibull distribution by Aryal et al. (2016), Generalized weibull distributions by Lai, (2014),

The Kumaraswamy-transmuted exponentiated modified Weibull distribution by AlBabtain et al. (2017), Generalized Flexible Weibull Distribution by Ahmad and Iqbal

(2017), A reduced new modified Weibull distribution by Almalki (2018), and the

transmuted exponentiated additive Weibull distribution by Nofal et al. Nofal et al..

Nasiru (2015) proposed a new Weighted Weibull (WW) distribution and discussed

its statistical properties using Azzalani’s family of weighted distributions by Azzalini

(1985). The density (pdf ) and distribution function (cdf ) of the WW distribution are

given by

f (x) = (1 + λγ )αγxγ−1 e−αx

γ (1+λγ )

;

x, α, λ > 0,

(1)

and

γ (1+λγ )

F (x) = 1 − e−αx

x, α, λ > 0,

(2)

where, α is a scale parameter and λ and γ are shape parameters. The corresponding

survival function of the WW distribution is given by

F̄ (x) = e−αx

γ (1+λγ )

.

(3)

Note that the WW distribution reduces to Weibull distribution for λ = 0 .

Topp-Leone family of distributions is proposed by Al-Shomrani et al. (2016). The cdf

of the proposed family is given by

FT L−G (t) = [G(t)]b [2 − G(t)]b = [1 − (Ḡ(t))2 ]b :,

x <,
b>0

(4)

and the corresponding pdf is obtained as

fT L−R (t) = 2bg(t)Ḡ(t)[1 − (Ḡ(t))2 ]b−1 ,

b > 0.

(5)

where g(t) = G0 (t) and Ḡ(t) = 1 − G(t).

In this paper a new generalization of the WW distribution, the Topp Leone Weighted Weibull (TLWW) distribution, is obtained. The aim of this generalization is to

provide a flexible extension of the WW distribution which can be used in much wider

situations.

The paper is organized as follows: The pdf and cdf of the proposed model is introduced and several mathematical characteristics are studied in Section 2. Distribution

of the order statistics is obtained in Sec. 4. Estimation of the model parameters are

done in Sec. 5. The influence of the estimators are evaluated in Sec. 6. The validity

of prosed model is on real data is presented in Sec. 7. Some concluding remarks are

given in Sec. 8.

2.

Topp-Leone Weighted Weibull Distribution and its Properties

In this section, we have obtained the pdf and cdf of new model. For this, we consider

the survival function of the WW distribution given in (3) and have used it in (4).

164

Pak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

A New Generalized Weighted Weibull Distribution

The cdf of the TLWW distribution is obtained as

γ (1+λγ )

F (x) = [1 − (e−αx

)2 ]b = [1 − e−2αx

γ (1+λγ )

]b ,

b > 0.

(6)

The density function is obtained by differentiating (6) and is given as

f (x) = 2bαγ(1 + λγ )xγ−1 e−2αx

γ (1+λγ )

[1 − e−2αx

γ (1+λγ )

]b−1 ,

x, α, λ, γ, b > 0,

(7)

where, α, γ, b are shape parameters and λ is scale parameter. A random variable X

having pdf (7) is denoted as X ∼ T LW W (α, λ, γ, b). The proposed model reduces to

Topp Leone Weibull distribution for λ = 0. For λ = 0 and γ = 1, it reduces to Topp

Leon Exponential distribution.

The reliability function (rf ), which is also known as survival function, is the probability of an item not failing prior to some time t. The rf of the TLWW distribution

is obtained as R(x) = 1 − H(x) and is given as

S(x) = 1 − [1 − e−2αx

γ (1+λγ )

]b .

The hazard rate function which is also known as, force of mortality in actuarial statistics, Mill’s ratio in statistics and intensity function in extreme value theory are

important characteristics in reliability theory, It is roughly explained as the conditional probability of failure, given it has survived to the time t. The hrf of random

variable X is defined as h(x) = f (x)/R(x) and for TLWW distribution it is given as

γ

γ

γ (1+λγ )

2bαγ(1 + λγ )xγ−1 e−2αx (1+λ ) [1 − e−2αx

h(x) =

1 − [1 − e−2αxγ (1+λγ ) ]b

]b−1

.

The cumulative hrf of the TLWW distribution is given by

H(x) = − log |1 − [1 − e−2αx

2.1

γ (1+λγ )

]b |.

Limiting Behavior

The behaviors of the pdf , cdf and hrf of TLWW distribution are investigated when

x → 0 and x → ∞. Therefore, lim f (x) and lim f (x) are given in the following

x→0

x→∞

γ

γ

γ

γ

lim f (x) = lim 2bαγ(1 + λγ )xγ−1 e−2αx (1+λ ) [1 − e−2αx (1+λ ) ]b−1 = 0,

x→0

x→0

γ

γ

γ

γ

lim f (x) = lim 2bαγ(1 + λγ )xγ−1 e−2αx (1+λ ) [1 − e−2αx (1+λ ) ]b−1 = ∞

x→∞

x→∞

From above it is clear that the proposed model has a unique mode. The limiting

behavior of cdf and hrf is given below

γ

γ b

lim f (x) = lim 1 − e−2αx (1+λ ) = 0,

x→0

x→0

Pak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

165

Salman Abbas, Gamze Ozal, Saman Hanif Shahbaz, Muhammad Qaiser Shahbaz

γ

γ b

lim f (x) = lim 1 − e−2αx (1+λ ) = 1,

x→∞

x→∞

γ

γ

2bαγ(1 + λγ )xγ−1 e−2αx (1+λ ) [1 − e−2αx

lim h(x) = lim

x→0

x→0

1 − [1 − e−2αxγ (1+λγ ) ]

lim h(x) = lim

x→∞

2.2

x→∞

γ

γ

γ (1+λγ )

2bαγ(1 + λγ )xγ−1 e−2αx (1+λ ) [1 − e−2αx

1 − [1 − e−2αxγ (1+λγ ) ]

]b−1

γ (1+λγ )

]b−1

= 0,

= 0.

Shape

The distribution and density functions of the proposed model can be expressed in

the form of exponentiated G-distribution. Prudnikov et al. (1986) presented a series

representation and is given as

∞

X

(1)j Γ(α + 1) j

(1 + x) =

x,

j!Γ(α + 1 − j)

j=0

α

the distribution function of TLWW distribution is written as follow

F (x) =

∞

X

(−1)j

j=0

Γ(b + 1)

γ

γ

(eαx (1+λ ) )2j .

j!Γ(b + 1 − j)

The density of TLWW distribution can also be written in the form of exponentiated

distributions and is given as

f (x) =

∞

X

(−1)j 2Γ(b + 1)

j=0

j!Γ(b − j)

αγxγ−1 (1 + λγ ) e−2αx

γ (1+λγ )

(j+1)

(8)

We, now, present the plots for the density function of the TLWW distribution for

different values of parameters in Figures 1 to 4. Figure 1 shows that as the value of b

increase there is a smooth increase in the peak of the curve and starting point is also

shifted. One can clearly see from Figure 2 that λ plays opposite role as compared

with the b. As the values of b increase, the curve shrinks and a rapid decrease occurs

in the peak of curve. Figure 3 shows that the distribution is positively skewed for

the smaller values of γ. From Figure 4 it can be seen that α plays important role

in shape of the distribution. For smaller values of α, the curve is smooth but as the

values of α increase great change appears and curve’s peak increases abruptly.

166

Pak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

A New Generalized Weighted Weibull Distribution

3

2.5

b=2

b=2.5

b=3.0

b=3.5

2.5

λ=1.5

λ=2.5

λ=3.5

lλ=4.5

2

2

f(x)

f(x)

1.5

1.5

1

1

0.5

0.5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

0.4

x

0.5

0.6

0.7

0.8

0.9

1

x

Figure 1: Plots of pdf for α = 2, γ = Figure 2: Plots of pdf for b = 2, α =

2.5, λ = 1.5

2, γ = 2.5

2.5

3

γ=2.5

γ=3

γ=3.5

γ=4

α=0.5

α=1.5

α=2.5

α=3.5

2.5

2

2

f(x)

f(x)

1.5

1.5

1

1

0.5

0.5

0

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

x

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Figure 3: Plots of pdf for b = 2, α = Figure 4: Plots of pdf for b = 2, λ =

2, λ = 1.5

1.5, γ = 2.5

2.3 Quantile Function

The quantile function of the TLWW distribution is obtained as Q(u) = F −1 (u) and

given as

1

Q(u) = −

ln{1 − u b }

2α(1 + λγ )

! γ1

, f or α > 0, u ∈ (0, 1).

Median of the distribution can be obtained by replacing u = 0.5 in above equation.

The quantile function is used to observe the effect of shape parameters on skewness

and kurtosis. The Bowley’s measure of skewness (S) is given as

S=

Q( 14 ) + Q( 43 ) − 2Q( 12 )

Q( 43 ) − Q( 14 )

and the Moors’s coefficient of kurtosis (K) is given as

K=

Q( 87 ) − Q( 58 ) + Q( 83 ) − Q 18

.

Q( 86 ) − Q( 82 )

Pak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

167

Salman Abbas, Gamze Ozal, Saman Hanif Shahbaz, Muhammad Qaiser Shahbaz

2.4

Moments

The raw or non-central moment for any probability distribution is obtained by using

Z ∞

0

xq dF (x).

µq =

−∞

Using pdf of the TLWW distribution in (8), we obtain the raw moments of the TLWW

distribution as

Z ∞ X

∞

(−1)j 2Γ(b + 1)

γ

γ (j+1)

0

xq

.

µq =

αγxγ−1 (1 + λγ ) e−2αx (1+λ )

j!Γ(b

−

j)

0

j=0

From the transformation w = (j + 1)2αxγ (1 + λγ ) and after some calculations, the

q − th moment of TLWW distribution is obtained as

µ0q =

Γ(b+1)

(−1)j j!Γ(b−j)

1

j+1

γq +1

∞

X

q

bj Γ(1 + ),

γ

j=0

1

2α(1+λγ )

γq

where bj =

. The coefficient of variation (CV ), coefficient of skewness (CS), and coefficient of kurtosis (CK) of the TLWW distribution

are obtained as follows

r

µ2

− 1,

CV =

µ1

CS =

CK =

µ3 − 3µ2 µ1 + 2µ31

3

(µ2 − µ1 ) 2

,

µ4 − 4µ3 µ1 + 6µ2 µ21

.

(µ2 − µ21 )2

Now, the first incomplete moment is used to derive the mean deviation, Bonferroni, and Lorenz curves. These curves have great influences in economics, reliability,

demography, insurance, and medicine. The incomplete moment of the TLWW distribution is obtained by using (7) and is given below.

Z

ϕs (t) =

t

x

s

0

∞

X

(−1)j 2Γ(b + 1)

j!Γ(b − j)

j=0

αγxγ−1 (1 + λγ ) e−2αx

γ (1+λγ )

(j+1)

dx

Simplifying, the incomplete moments is given by

ϕs (t) =

∞

X

j=0

A∗j

s

s

γ

γ(1 + ) − γ(1 + ), 2α(1 + λ )(j + 1) .

γ

γ

γs +1

γs

Γ(b+1)

1

1

where A∗j = (−1)j j!Γ(b−j)

.

j+1

(2α(1+λγ ))

The mean deviation about mean [m1 = E(|X − µ01 |)] and [m2 = E(|X − M |)] mean

deviation about median of X are given as m1 = 2µ01 F (µ01 ) − 2ϕ1 (µ01 ) and m2 =

168

Pak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

A New Generalized Weighted Weibull Distribution

µ01 − 2ϕ1 (M ), respectively, where µ01 = E(X), M = M edian(X) = Q(0.5), and

F (µ01 ) is calculated from (7) and ϕ1 (t) is the first incomplete moment given by (19)

with s = 1.

These equations for ϕ1 (t) can be used to obtain Bonferroni and Lorenz curves given

1 (q)

0

probability π as B(π) = ϕπµ

and L(π) = ϕ1µ(q)

0

0 , respectively, where µ1 = E(X) and

1

1

q = Q(π) is quantile function of X at π.

The (q, r)th probability weighted moment (PWM) of X is defined as

Z ∞

xq [F (x)]r f (x)dx.

ρq,r =

−∞

Using (5) and (6), we can write after some algebra,

r

[F (x)] f (x) =

∞

X

a(j, m)h2j+1 (x),

j,m=0

where

a(j, m) =

∞

X

(−1)j+m

j,m=0

Γ(b + 1)(rb + 1)

j!m!Γ(b − j)(rb + 1 − j)

and

γ (1+λγ )(2j+1)

h2j+1 (x) = 2αγxγ−1 (1 + λγ )e−2γx

,

After making transformation, the (q, r)th PWM of X can be expressed as

ρq,r (x) = a(j, m)

1

2j + 1

γq +1

1

2α(1 + λγ )

γq

q

Γ(1 + ).

γ

Now, we provide numerical values for the mean, variance, coefficient of skewness, and

coefficient of kurtosis in Tables 1 to 4, respectively.

Pak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

169

Salman Abbas, Gamze Ozal, Saman Hanif Shahbaz, Muhammad Qaiser Shahbaz

Table 1: Mean of TLWW distribution

parameters

γ

b λ

4

5

6

1 0.6409 0.6958 0.7363

2 0.3754 0.3972 0.4122

1 3 0.2533 0.2662 0.2754

4 0.1904 0.1998 0.2066

5 0.1524 0.1599 0.1653

6 0.1270 0.1332 0.1378

1 0.7429 0.7859 0.8167

2 0.4351 0.4486 0.4572

2 3 0.2936 0.3007 0.3055

4 0.2206 0.2257 0.2292

5 0.1766 0.1805 0.1833

6 0.1472 0.1505 0.1528

1 0.7929 0.8288 0.8541

2 0.4644 0.4731 0.4781

3 3 0.3134 0.3171 0.3195

4 0.2355 0.2380 0.2397

5 0.1885 0.1904 0.1917

6 0.1571 0.1587 0.1598

1 0.8248 0.8557 0.8774

2 0.4830 0.4885 0.4912

4 3 0.3259 0.3274 0.3282

4 0.2450 0.2457 0.2462

5 0.1961 0.1966 0.1970

6 0.1634 0.1638 0.1641

1 0.8477 0.8750 0.8940

2 0.4965 0.4994 0.5004

5 3 0.3350 0.3347 0.3344

4 0.2518 0.2512 0.2508

5 0.2015 0.2010 0.2007

6 0.1680 0.1675 0.1672

1 0.8654 0.8897 0.9066

2 0.5068 0.5078 0.5075

6 3 0.3420 0.3404 0.3391

4 0.2570 0.2554 0.2544

5 0.2057 0.2044 0.2035

6 0.1715 0.1703 0.1696

170

for α = 1 and various values of

7

0.7674

0.4232

0.2824

0.2118

0.1694

0.1412

0.8397

0.4630

0.3090

0.2318

0.1854

0.1545

0.8729

0.4814

0.3212

0.2410

0.1928

0.1606

0.8935

0.4927

0.3288

0.2466

0.1973

0.1644

0.9080

0.5007

0.3341

0.2506

0.2005

0.1671

0.9190

0.5068

0.3382

0.2537

0.2029

0.1691

8

0.7919

0.4316

0.2879

0.2159

0.1727

0.1439

0.8576

0.4674

0.3117

0.2338

0.1871

0.1559

0.8875

0.4837

0.3226

0.2419

0.1936

0.1613

0.9058

0.4937

0.3293

0.2469

0.1976

0.1646

0.9187

0.5007

0.3339

0.2505

0.2004

0.1670

0.9285

0.5060

0.3375

0.2531

0.2025

0.1688

9

0.8118

0.4383

0.2923

0.2192

0.1754

0.1461

0.8720

0.4708

0.3139

0.2354

0.1884

0.1570

0.8990

0.4854

0.3237

0.2427

0.1942

0.1618

0.9156

0.4943

0.3296

0.2472

0.1978

0.1648

0.9272

0.5006

0.3338

0.2504

0.2003

0.1669

0.9360

0.5054

0.3370

0.2527

0.2022

0.1685

Pak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

A New Generalized Weighted Weibull Distribution

Table 2: Variance of TLWW distribution for α

parameters

γ

b λ

4

5

6

7

1 0.0196 0.0173 0.0151 0.0131

2 0.0249 0.0267 0.0278 0.0286

1 3 0.0169 0.0184 0.0194 0.0201

4 0.0114 0.0124 0.0132 0.0137

5 0.0081 0.0088 0.0094 0.0098

6 0.0060 0.0066 0.0070 0.0073

1 0.0144 0.0117 0.0095 0.0079

2 0.0285 0.0294 0.0299 0.0302

2 3 0.0208 0.0216 0.0222 0.0225

4 0.0145 0.0150 0.0154 0.0157

5 0.0104 0.0108 0.0111 0.0113

6 0.0078 0.0081 0.0083 0.0085

1 0.0109 0.0085 0.0067 0.0054

2 0.0295 0.0301 0.0305 0.0307

3 3 0.0226 0.0231 0.0234 0.0236

4 0.0159 0.0163 0.0165 0.0166

5 0.0116 0.0118 0.0119 0.0121

6 0.0087 0.0089 0.0090 0.0091

1 0.0085 0.0065 0.0050 0.0040

2 0.0300 0.0304 0.0307 0.0308

4 3 0.0237 0.0239 0.0241 0.0242

4 0.0169 0.0170 0.0171 0.0172

5 0.0123 0.0124 0.0125 0.0125

6 0.0093 0.0093 0.0094 0.0094

1 0.0068 0.0051 0.0039 0.0031

2 0.0302 0.0306 0.0308 0.0309

5 3 0.0244 0.0245 0.0246 0.0246

4 0.0175 0.0176 0.0176 0.0176

5 0.0128 0.0128 0.0128 0.0128

6 0.0097 0.0097 0.0097 0.0097

1 0.0055 0.0041 0.0031 0.0025

2 0.0303 0.0306 0.0308 0.0309

6 3 0.0249 0.0250 0.0249 0.0249

4 0.0180 0.0180 0.0179 0.0179

5 0.0132 0.0132 0.0131 0.0130

6 0.0100 0.0099 0.0099 0.0099

Pak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

= 1 and various values of

8

0.0114

0.0291

0.0206

0.0142

0.0101

0.0075

0.0066

0.0304

0.0228

0.0159

0.0115

0.0086

0.0045

0.0308

0.0237

0.0167

0.0121

0.0091

0.0033

0.0309

0.0243

0.0172

0.0125

0.0094

0.0025

0.0310

0.0246

0.0176

0.0128

0.0097

0.0020

0.0310

0.0249

0.0178

0.0130

0.0098

9

0.0100

0.0295

0.0211

0.0145

0.0104

0.0077

0.0056

0.0306

0.0230

0.0161

0.0116

0.0087

0.0037

0.0309

0.0238

0.0168

0.0122

0.0092

0.0027

0.0310

0.0243

0.0173

0.0126

0.0095

0.0021

0.0310

0.0246

0.0176

0.0128

0.0097

0.0016

0.0311

0.0249

0.0178

0.0130

0.0098

171

Salman Abbas, Gamze Ozal, Saman Hanif Shahbaz, Muhammad Qaiser Shahbaz

Table 3: Coefficient of skewness of the

various values of parameters

γ

b λ

4

5

6

1 0.2970 0.2545 0.2224

2 0.4807 0.4656 0.4547

1 3 0.5821 0.5754 0.5701

4 0.6295 0.6255 0.6223

5 0.6545 0.6519 0.6498

6 0.6692 0.6674 0.6658

1 0.2252 0.1878 0.1609

2 0.4358 0.4252 0.4184

2 3 0.5555 0.5516 0.5487

4 0.6123 0.6102 0.6085

5 0.6426 0.6413 0.6402

6 0.6605 0.6596 0.6588

1 0.1904 0.1569 0.1334

2 0.4119 0.4048 0.4007

3 3 0.5410 0.5392 0.5379

4 0.6028 0.6021 0.6014

5 0.6360 0.6356 0.6352

6 0.6556 0.6554 0.6552

1 0.1695 0.1386 0.1173

2 0.3963 0.3917 0.3894

4 3 0.5313 0.5311 0.5310

4 0.5964 0.5967 0.5969

5 0.6315 0.6319 0.6320

6 0.6523 0.6527 0.6528

1 0.1557 0.1266 0.1066

2 0.3849 0.3823 0.3814

5 3 0.5242 0.5252 0.5260

4 0.5916 0.5928 0.5935

5 0.6281 0.6291 0.6297

6 0.6498 0.6506 0.6511

1 0.1460 0.1181 0.0992

2 0.3761 0.3750 0.3753

6 3 0.5186 0.5207 0.5222

4 0.5879 0.5898 0.5910

5 0.6255 0.6270 0.6278

6 0.6479 0.6490 0.6497

172

TLWW distribution for α = 1 and

7

0.1973

0.4464

0.5659

0.6197

0.6480

0.6645

0.1407

0.4136

0.5464

0.6071

0.6392

0.6581

0.1159

0.3980

0.5369

0.6009

0.6348

0.6549

0.1016

0.3882

0.5309

0.5969

0.6320

0.6528

0.0921

0.3812

0.5265

0.5940

0.6300

0.6513

0.0855

0.3759

0.5232

0.5918

0.6284

0.6502

8

0.1772

0.4399

0.5624

0.6175

0.6465

0.6634

0.1250

0.4100

0.5446

0.6060

0.6385

0.6576

0.1025

0.3960

0.5361

0.6004

0.6345

0.6547

0.0896

0.3873

0.5307

0.5968

0.6320

0.6528

0.0811

0.3812

0.5269

0.5943

0.6302

0.6515

0.0752

0.3765

0.5239

0.5923

0.6288

0.6505

9

0.1608

0.4345

0.5594

0.6157

0.6452

0.6625

0.1124

0.4072

0.5431

0.6050

0.6378

0.6571

0.0919

0.3946

0.5354

0.5999

0.6342

0.6545

0.0801

0.3868

0.5305

0.5967

0.6320

0.6528

0.0724

0.3813

0.5271

0.5945

0.6304

0.6516

0.0671

0.3771

0.5245

0.5927

0.6291

0.6507

Pak.j.stat.oper.res. Vol.15 No.1 2019 pp161-178

A New Generalized Weighted Weibull Distribution

Table 4: Coefficient of kurtosis of TLWW distribution for α = 1 and various

values of parameters

γ

b λ

4

5

6

7

8

9

1 1.8001 1.7602 1.7333 1.7144 1.7006 1.6903

2 2.0269 2.0074 1.9933 1.9827 1.9743 1.9675

1 3 2.1708 2.1619 2.1549 2.1492 2.1445 2.1406

4 2.2366 2.2316 2.2275 2.2240 2.2212 2.2187

5 2.2700 2.2668 2.2641 2.2619 2.2599 2.2583

6 2.2889 2.2867 2.2848 2.2832 2.2818 2.2807

1 1.7357 1.7077 1.6903 1.6788 1.6708 1.6651

2 1.9677 1.9547 1.9463 1.9404 1.9360 1.9325

2 3 2.1343 2.1292 2.1254 2.1224 2.1200 2.1180

4 2.2137 2.2111 2.2089 2.2071 2.2056 2.2044

5 2.2547 2.2531 2.2517 2.2505 2.2495 2.2487

6 2.2780 2.2769 2.2759 2.2751 2.2744 2.2738

1 1.7095 1.6879 1.6749 1.6666 1.6609 1.6568

2 1.9372 1.9287 1.9238 1.9206 1.918 …

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