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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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