Kumarswamy Truncated Lomax Distribution with Applications
MANAL H. ALABDULHADI
Department of Mathematics
College of Science and Arts, Qassim University
Bukairiayh 51452
SAUDI ARABIA
Abstract: - This paper introduces a new flexible generalized family of distributions, named Kumarswamy Trun-
cated Lomax Distribution. We study its statistical properties including quantile function, skewness, kurtosis,
moments, generating functions, incomplete moments and order statistics. Maximum likelihood estimation of the
model parameters is investigated. An application is carried out on real data set to illustrate the performance and
flexibility of the proposed model.
Key-Words: - Truncated Lomax distribution; Kumaraswamy generated family of distributions; Lorenz curve;
Moments; Estimation; Simulation.
Received: October 22, 2022. Revised: April 25, 2023. Accepted: May 17, 2023. Published: May 25 2023.
1 Introduction
Lifetime models are extremely valuable for ex-
plaining and forecasting real-world occurrences in
medicine, research, economics, and a variety of other
fields. These models are often useful in product life-
time evaluations such as reliability and survival anal-
yses. Many researchers have been undertaken in or-
der to develop appropriate lifespan models that best
represent real-world occurrences. Throughout the last
decade, research has mostly concentrated on estab-
lishing new families of distributions by adding more
parameters to the standard and common families of
continuous distributions. These additional param-
eters have produced far more flexible models that
match real-world datasets better than standard mod-
els, [1].
Some of these approaches for generated family of
distributions are as follows: exponentiated version of
the M family in [2], Kumaraswamy Kumaraswamy
−Gin [3], generalized Kumaraswamy −Gin [4], al-
pha power transformation family in [5], Weibull odd
Burr III −Gin [6], odd Frechet −Gin [7], exponen-
tiated Kumaraswamy −Gin [8], type-I half logistic
Burr X −Gin [9], transmuted Burr X−Gin [10],
exponentiated generalized −Gin [11], gamma Ku-
maraswamy −Gin [12], odd-generalized N-H −G
in [13], extended-gamma −Gin [14], Kumaraswamy
type I half logistic −Gin [15], T−Xfamily
in [16], gamma −Gin [17], Kumaraswamy Pois-
son −Gin [18], exponentiated power-generalized
Weibull power series −Gin [19], beta generalized
Marshall-Olkin Kumaraswamy −Gin [20], odd Burr
X−Gin [21], the Weibull −Gin [22], type-II half
logistic−Gin [23], sec−Gin [24], truncated Cauchy
power Weibull −Gin [25], exponentiated truncated
inverse Weibull −Gin [26], odd Perks −Gin [27],
sine Topp-Leone −Gin [28], Kumaraswamy gener-
alized Marshall-Olkin −Gin [29], a new power Topp-
Leone−Gin [30], truncated inverted Kumaraswamy
−Gin [31], transmuted odd Frechet −Gin [32],
Kumaraswamy Marshal-Olkin −Gin [33], Kavya-
Manoharan transformation family, [34], among oth-
ers. Recently, [35], studied the Kumaraswamy (K)
generated family of distributions and it has the follow-
ing cumulative distribution function (CDF) and prob-
ability density function (PDF) as below:
F(y;µ, η, ω) = 1 −[1 −G(y;ω)µ]η,
y∈R, µ, η > 0,(1)
and
f(y;µ, η, ω) = µηg(y;ω)G(y;ω)µ−1
[1 −G(y;ω)µ]η−1,
y∈R, µ, η > 0.
(2)
where ωand H(.)are the vector of parameters and
the CDF for the baseline distribution.
Many studies on the K-family of distributions have
made significant contributions by providing new dis-
tributions. For instance, the K Weibull distribution
was presented by [36]. In [37], the authors proposed
K generalized gamma distribution. In [38], [39], the
authors presented the K Weibull exponential and K
quasi Lindley distributions, respectively. The K gen-
eralized power Lomax distribution was developed by
[40].
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A truncated distribution is a conditional distribution
with a narrower scope than the parent distribution.
The truncated distributions have had widespread ap-
plication, primarily in reliability and life-testing stud-
ies. The recently truncated L (TL) distribution pub-
lished by [41], with one shape parameter has attracted
our interest. The CDF and PDF for the TL distribu-
tion are
G(y;λ) = 1−(1 + y)−λ
1−2−λ,0< y < 1, λ > 0,
(3)
and
g(y;λ) = λ(1 + y)−λ−1
1−2−λ,0< y < 1, λ > 0.
(4)
The goal of creating this new generalization of the
TL distribution is to improve the performance and
the flexibility of TL distribution. The PDF of the
KTL distribution is bathtub, uni-modal, decreasing,
and right skewed, while the HRF is U-shaped, bath-
tub, increasing and J-shaped.
The remainder of the manuscript is arranged as
follows: the construction of the KTL distribution is
proposed in Section 2. Section 3 deduces some basic
statistical features, like; quantile function, skewness,
kurtosis, moments, generating functions, incomplete
moments and order statistics. Maximum likelihood
estimate is covered in Section 4. In section 5, we con-
sider three real-world applications to assess the adapt-
ability of the KTL model, followed by some final con-
cluding remarks in Section 6.
2 The KTL Distribution
The KTL distribution is constructed by inserting the
Equations (3) and (4) into the Equations (1) and (2).
The random variable (RV) Yis said to have KTL
model denoted by KTL ∼ψ= (µ, η, λ)if the CDF,
PDF and survival function (SF) of Yare provided as
below
F(y;ψ) = 1 −1−1−(1 + y)−λ
1−2−λµη
,(5)
f(y;ψ) = λµη(1 + y)−λ−1
(1 −2−λ)µ1−(1 + y)−λµ−1
1−1−(1 + y)−λ
1−2−λµη−1
, y > 0, ψ > 0,
(6)
and
R(y;ψ) = 1−1−(1 + y)−λ
1−2−λµη
.
The HRF, the reversed HRF and cumulative HRF
for the KTL distribution are provided as below
h(y;ψ) =
λµη(1 + y)−λ−11−(1 + y)−λµ−1
(1 −2−λ)µ1−1−(1+ y)−λ
1−2−λµ,
τ(y;ψ) =
λµη(1 + y)−λ−11−(1 + y)−λµ−1
(1 −2−λ)µ1−1−1−(1+ y)−λ
1−2−λµη
×1−1−(1 + y)−λ
1−2−λµη−1
,
and
H(y;ψ) = −ηln 1−1−(1 + y)−λ
1−2−λµ.
The KTL distribution is very flexible and includes
three sub-models such as; When η= 1, we get the
exponentiated TL distribution; when µ= 2, we get
the Topp-Leone TL distribution and when η=µ= 1,
we get the TL distribution, [41].
Figure 1 and Figure 2 mention the PDF and HRF
curves for the KTL distribution with different values
of the parameter. The PDF is bathtub, uni-modal,
decreasing, and right skewed, while the HRF is U-
shaped, bathtub, increasing and J-shaped.
3 Important Mathematical
Characterizations
This section investigates some important mathemati-
cal characterizations of the KTL distribution.
3.1 Quntile Function
The quantile function of KTL distribution is provided
via
yu=1−1−2−λ1−(1 −u)
1
η
1
µ
−1
λ
−1,
(7)
where u∈(0,1). To get the first quantile (Q1), the
second quantile (Q2) (median) and the third quantile
(Q3) we insert u= 0.25, 0.5 and 0.75 respectively.
The Q1, Q2 and Q3 are provided via
Q1 = 1−1−2−λ1−(0.75)
1
η
1
µ
−1
λ
−1,
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Figure 1: Plots of PDF for the KTL distribution.
Figure 2: Plots of hrf for the KTL distribution.
Q2 = 1−1−2−λ1−(0.5)
1
η
1
µ
−1
λ
−1,
and
Q3 = 1−1−2−λ1−(0.25)
1
η
1
µ
−1
λ
−1.
The Bowley skewness (α1) and the Moors kurtosis
(α2) are provided via
α1=y0.75 +y0.25 −2y0.5
y0.75 −y0.25
.
and
α2=y0.875 −y0.625 +y0.375 −y0.125
y0.75 −y0.25
.
Table 1: Results of Some numerical values of Q1,Q2,
Q3,α1and α2for the KTL model at λ= 5
η µ Q1Q2Q3α1α2
2.5 3 0.132 0.203 0.298 0.141 1.258
3.1 0.137 0.209 0.305 0.139 1.257
3.2 0.142 0.215 0.311 0.136 1.256
3.3 0.146 0.22 0.317 0.134 1.255
3.4 0.151 0.226 0.323 0.132 1.254
3.5 0.155 0.231 0.329 0.13 1.253
3.6 0.16 0.236 0.335 0.129 1.252
3.7 0.164 0.241 0.341 0.127 1.251
3.8 0.168 0.246 0.346 0.125 1.251
3.9 0.172 0.251 0.351 0.124 1.25
4 3 0.107 0.161 0.229 0.122 1.249
3.1 0.111 0.166 0.235 0.119 1.249
3.2 0.115 0.171 0.241 0.117 1.248
3.3 0.12 0.176 0.247 0.115 1.247
3.4 0.124 0.181 0.252 0.113 1.247
3.5 0.128 0.186 0.258 0.111 1.246
3.6 0.132 0.19 0.263 0.109 1.246
3.7 0.136 0.195 0.268 0.108 1.245
3.8 0.14 0.199 0.273 0.106 1.245
3.9 0.144 0.204 0.278 0.104 1.244
5 3 0.097 0.144 0.203 0.113 1.245
3.1 0.101 0.149 0.209 0.111 1.244
3.2 0.105 0.154 0.215 0.108 1.244
3.3 0.109 0.159 0.22 0.106 1.243
3.4 0.113 0.163 0.226 0.104 1.243
3.5 0.117 0.168 0.231 0.102 1.242
3.6 0.121 0.173 0.236 0.101 1.242
3.7 0.125 0.177 0.241 0.099 1.242
3.8 0.128 0.181 0.246 0.097 1.241
3.9 0.132 0.186 0.251 0.096 1.241
From Table 1 we note that when η=2.5,4, 5, and µ
increases, Q1, Q2 and Q3 increase, whereas α1and
α2decrease.
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3.2 Ordinary and Incomplete Moments
The nth ordinary moments of the KTL distribution
can be calculated from the next equation
µ′
n=∞
0
ynf(y;ψ)dy, (8)
by employing (6) in (8) we have
µ′
n=λµη
(1 −2−λ)µ∞
0
yn(1 + y)−λ−1
×1−(1 + y)−λµ−1
×1−1−(1 + y)−λ
1−2−λµη−1
dy,
By utilizing the binomial theorem to the last term in
the above equation
1−1−(1 + y)−λ
1−2−λµη−1
=
∞
i=0
(−1)iη−1
i
×1−(1 + y)−λ
1−2−λµi
.
(9)
then,
µ′
n=
∞
i=0
(−1)iη−1
iλµη
(1 −2−λ)µ(i+1)
∞
0
yn(1 + y)−λ−11−(1 + y)−λµ(i+1)−1dy,
Again utilizing the binomial theorem (9) to the last
term in the above equation, then we have
µ′
n=
∞
i,j=0
Ψi,j ∞
0
yn(1 + y)−λ(j+1)−1dy, (10)
where,
Ψi,j =(−1)i+jλµη
(1 −2−λ)µ(i+1) .
×η−1
i µ(i+ 1) −1
j.
By using the prime beta function the integration
(10) leads to
µ′
n=
∞
i,j=0
Ψi,j B(n+ 1, λ (j+ 1) −n), λ (j+ 1) > n.
(11)
To get the first four moments, we put n= 1, 2, 3 and
4 as below
µ′
1=
∞
i,j=0
Ψi,j B(2, λ (j+ 1) −1) , λ (j+ 1) >1,
µ′
2=
∞
i,j=0
Ψi,j B(3, λ (j+ 1) −2) , λ (j+ 1) >2,
µ′
3=
∞
i,j=0
Ψi,j B(4, λ (j+ 1) −3) , λ (j+ 1) >3,
and
µ′
4=
∞
i,j=0
Ψi,j B(5, λ (j+ 1) −4) , λ (j+ 1) >4.
The moment generating function of the KTL dis-
tribution is provided via
MY(t) = EetY =
∞
n=0
tn
n!µ′
n
=
∞
i,j,n=0
tn
n!Ψi,j B(n+ 1, λ (j+ 1) −n), λ (j+ 1) > n.
Table 2 shows the numerical values of the first four
moments µ′
1, µ′
2, µ′
3, µ′
4,also the numeri-
cal values of variance (σ2), coefficient of skewness
(CS), coefficient of kurtosis (CK) and coefficient of
variation (CV) for the TKL model.
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Table 2: Results of some moments, CS, CK, and CV for the KTL model at λ= 5
η µ µ′
1µ′
2µ′
3µ′
4σ2CS CK CV
2.5 3 0.233 0.284 3.308 109.787 0.23 28.354 2013 2.063
3.1 0.214 0.184 1.741 53.521 0.138 32.129 2745 1.732
3.2 0.202 0.126 0.933 26.384 0.085 35.223 3550 1.447
3.3 0.192 0.092 0.51 13.148 0.055 36.995 4296 1.214
3.4 0.185 0.071 0.285 6.625 0.037 36.829 4796 1.033
3.5 0.179 0.058 0.163 3.375 0.026 34.425 4876 0.897
3.6 0.175 0.05 0.096 1.739 0.019 30.067 4483 0.797
3.7 0.17 0.044 0.059 0.906 0.015 24.586 3735 0.726
3.8 0.166 0.04 0.038 0.478 0.013 18.993 2850 0.677
3.9 0.163 0.037 0.026 0.256 0.011 14.062 2024 0.643
4 3 0.997 4.604 64.441 2159 3.609 7.679 147.862 1.905
3.1 0.707 2.586 32.25 1004 2.086 9.116 211.41 2.043
3.2 0.535 1.494 16.459 474.179 1.208 10.825 302.571 2.052
3.3 0.432 0.892 8.563 227.106 0.705 12.781 428.679 1.945
3.4 0.368 0.553 4.541 110.298 0.418 14.939 596.518 1.758
3.5 0.327 0.358 2.455 54.306 0.252 17.213 809.594 1.536
3.6 0.3 0.245 1.354 27.102 0.155 19.443 1063 1.315
3.7 0.281 0.177 0.763 13.709 0.098 21.37 1337 1.116
3.8 0.268 0.136 0.441 7.029 0.064 22.649 1592 0.948
3.9 0.257 0.11 0.262 3.653 0.044 22.912 1771 0.814
5 3 3.511 20.843 300.086 10090 8.516 6.725 96.039 0.831
3.1 2.166 11.215 145.352 4547 6.522 5.572 83.156 1.179
3.2 1.405 6.182 71.966 2084 4.209 5.959 98.287 1.461
3.3 0.965 3.494 36.393 971.414 2.563 6.843 129.083 1.659
3.4 0.706 2.028 18.784 460.134 1.53 8.03 176.217 1.753
3.5 0.55 1.213 9.891 221.389 0.911 9.461 243.054 1.735
3.6 0.455 0.752 5.312 108.158 0.545 11.105 333.745 1.625
3.7 0.394 0.486 2.909 53.639 0.331 12.914 451.78 1.458
3.8 0.355 0.33 1.627 26.999 0.204 14.798 597.73 1.271
3.9 0.329 0.237 0.93 13.791 0.129 16.601 765.724 1.089
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Note that when η=2.5 and µincreases, σ2, CS, CK
and CV decrease. When η=4 and µincreases, σ2and
CV decrease, whereas CS and CK increase. When
η=5 and µincreases, σ2decreases and CV increases
then decreases, whereas CS and CK increase.
3.3 Order Statistics
Assume that Y1, Y2, ..., Ynbe independent and iden-
tically distributed (iid) RVs and their corresponding
continuous F(y). Suppose that Y1:n< Y2:n< ... <
Yn:nthe relevant ordered random sample from a pop-
ulation of size n. The pdf of the mth ORS, is provided
via
fm:n(y) = Q(n, m)f(y)F(y)m−1(1 −F(y))n−m,(12)
where Q(n, m) = n!
(m−1)!(n−m)! . For the KTL distri-
bution, the pdf of the mth ORS is computed via
fm:n(y) = Q(m, n)λµη(1 + y)−λ−1
(1 −2−λ)µ1−(1 + y)−λµ−1
×1−1−(1 + y)−λ
1−2−λµη(n−m+1)−1
×1−1−1−(1 + y)−λ
1−2−λµηm−1
.
(13)
The first ORS of the KTL distribution can be com-
puted by inserting For m=1 in Equation (13), as
f1:n(y) = nλµη(1 + y)−λ−1
(1 −2−λ)µ1−(1 + y)−λµ−1
×1−1−(1 + y)−λ
1−2−λµη(n−2)−1
.
The first ORS of the KTL distribution can be com-
puted by inserting For m=n in Equation (13), as
fn:n(y) = nλµη(1 + y)−λ−1
(1 −2−λ)µ1−(1 + y)−λµ−1
×1−1−(1 + y)−λ
1−2−λµη−1
×1−1−1−(1 + y)−λ
1−2−λµηn−1
.
4 Approach of ML Estimation
We used the ML estimates (MLEs) approach in this
part to estimate the unknown parameters of the KTL
distribution. Assume that y1, ..., ynis the nth random
sample (RS) from the KTL distribution (6). The KTL
distribution’s log-likelihood function is provided via
log L=nlog (λ) + nlog (µ) + nlog (η)
−nµ log 1−2−λ−(λ+ 1)
n
i=1
log (1 + yi)
+ (µ−1)
n
i=1
log 1−(1 + yi)−λ
+ (η−1)
n
i=1
log 1−1−(1 + yi)−λ
1−2−λµ.
(14)
By differentiating Equation (14) with respect to µ, λ
and ηrespectively as
∂log L
∂µ =n
µ−nlog 1−2−λ+
n
i=1
log 1−(1 + yi)−λ
−(η−1)
n
i=1
ln 1−(1+ yi)−λ
1−2−λ
1−(1+ yi)−λ
1−2−λ−µ
−1
,
(15)
∂log L
∂λ =n
λ+nµ2−λlog (2)
1−2−λ−
n
i=1
log (1 + yi)
−(µ−1)
n
i=1
(1 + yi)−λlog (1 + yi)
1−(1 + yi)−λ
−µ(η−1)
n
i=1 1−(1+ yi)−λ
1−2−λµ−1
1−1−(1+ yi)−λ
1−2−λµ
×(1 + yi)−λ1−2−λlog (1 + yi)
(1 −2−λ)2
+
2−λ1−(1 + yi)−λlog (2)
(1 −2−λ)2
,
(16)
and
∂log L
∂η =n
η+
n
i=1
log 1−1−(1 + yi)−λ
1−2−λµ.
(17)
The MLEs of the parameters µ, λ and ηsymbolize
by ˆµ, ˆ
λand ˆηare calculated by equating the equations
(15), (16) and (17) to 0 and simultaneously solving
these nonlinear systems of equations.
5 Simulation
A brief simulation is run to evaluate the performance
of the ML approach for estimating parameters. The
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KTL is investigated for such reasons using Monte
Carlo simulations. The computations in this section
are performed utilizing Mathematica 9. The proce-
dure is structured as follows:
1. We generate random samples from the KTL dis-
tribution by utilizing
yu=1−1−2−λ1−(1 −u)
1
η
1
µ
−1
λ
−1,
2. Monte Carlo simulations were performed 10000
times with n= 100, 200 and 300.
3. The chosen values for the parameters are listed in
Table 3, Table 4, Table 5, Table 6 and Table 7.
4. Mean square error (MSE), lower limit (LL), up-
per limit (UL), and average length (AL) formulas
of 90% and 95% are determined.
The simulation results for the KTL distribution are re-
ported in Table 3, Table 4, Table 5, Table 6 and Table
7 with different values of µ, λ, η and n. According
to the tables, the estimated MSE and AL decrease in
most of the situations when nincreases. We note that
the estimation method works sufficiently well for es-
timating the parameters µ, λ and η
6 Applications
In this section, we exhibit applications to two real-
world datasets to demonstrate the applicability and
importance of the KTL distribution. The goodness-
of-fit statistics for the KTL distribution and other
competing distributions are examined, and the MLEs
of their parameters are shown.
We will also compare the fits of the KTL dis-
tribution to other models: TL, unit exponential
Pareto (UExP), [42], exponential Pareto (ExP), [43],
the unit-Weibull (UW), [44], Kumaraswamy (K),
[45], Marshall-Olkin Kumaraswamy (MOK), [46],
Marshall-Olkin extended Topp Leone (MOETL),
[47], unit-Gompertz (UGo), [48], unit generalized log
Burr XII (UGLB), [49], Topp Leone (ToLe), [50], and
unit gamma Gompertz (UGGo), [51], distribution.
To examine the fit of all competitive models,
Cramer-von Mises (CV), Anderson-Darling (A-D),
and Kolmogorov Smirnov (KS) statistics with their
p-values (PV) were used.
6.1 The rate of COVID-19 recovery in
Turkey
We’re curious in studying on the pace of COVID-19
in Turkey recovery during the epidemic. In Turkey,
the daily ratio of total recoveries to total confirmed
cases was calculated. From March 27 to April 20,
a total of 25 observations were made, [42]. The
information is as follows: 0.0074, 0.0095, 0.0113,
0.015, 0.018, 0.0212, 0.0229, 0.0231, 0.0328, 0.0385,
0.0439, 0.0464, 0.0483, 0.0507, 0.0515, 0.0568,
0.0605, 0.0648, 0.0737, 0.0818, 0.0955, 0.1099,
0.127, 0.1388, 0.1476.
Table 8 proposes ten different distributions for
comparison with the KTL. It is observed that the KTL
achieves the minimal value in all metrics of goodness
of fit, with a big value of PV. This demonstrates its ap-
plicability and efficiency for modelling the dataset, as
opposed to the other competing distributions.Figure 3
shows the fitted KTL PDF, CDF, and P-P plots of the
data set The rate of COVID-19 recovery in Turkey.
We consider the KTL model to analyze this dataset
compared with the other competitive models. The
MLEs of the parameters of each fitted distribution
with their stranded errors (SE) and the goodness of
fit statistics are reported in Table 8. According to the
table, it is clear that all the models work quite well for
analyzing this dataset since the PV of those models
are greater than 0.05. However, the KTL model pro-
vides the best fit among all competitive models since
it has the smallest values of CV, A-D and KS statis-
tics, as well as having the largest PV.
6.2 The rate of COVID-19 recovery in
France
The World Health Organization verified that the first
death from COVID-19 occurred in France. This data
set comprises 38 observations estimated as the daily
ratio of total recoveries to the cumulative number of
confirmed cases and different cumulative numbers of
confirmed death cases in France from 1 January to 7
February 2022. a total of 38 observations were made,
[42]. The information is as follows: 0.195, 0.2338,
0.2368, 0.1073, 0.1592, 0.2784, 0.0689, 0.1791,
0.1121, 0.1865, 0.2631, 0.0716, 0.1411, 0.1477,
0.1874, 0.0853, 0.0922, 0.1711, 0.1962, 0.2146,
0.1041, 0.1524, 0.1811, 0.0643, 0.2698, 0.1245,
0.176, 0.2363, 0.0712, 0.1361, 0.1386, 0.3316, 0.077,
0.1367, 0.1549, 0.2178, 0.0951, 0.1346
Table 9 shows the MLEs estimation of the pa-
rameters and the goodness-of-fit metrics. Table 9
shows that the KTL achieves the lowest values of the
goodness-of-fit metrics when compared to the other
nine recommended models. Figure 4 displays the his-
togram with fitted pdf, fitted CDF, and P-P plot of the
KTL model.
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Table 3: Simulation results for the KTL distribution at µ=λ=η=0.3
nparameter MLE MSE 90%95%
LL UL AL LL UL AL
100
µ0.3055 0.0034 0.1690 0.4420 0.2730 0.1428 0.4682 0.3253
λ0.2758 0.0098 -0.0509 0.6025 0.6534 -0.1134 0.6651 0.7785
η0.3063 0.0007 0.2337 0.3788 0.1450 0.2198 0.3927 0.1728
200
µ0.3284 0.0072 0.2206 0.4362 0.2156 0.2000 0.4569 0.2569
λ0.3912 0.0536 0.0766 0.7059 0.6293 0.0163 0.7662 0.7499
η0.3026 0.0012 0.2522 0.3530 0.1008 0.2425 0.3626 0.1201
300
µ0.3194 0.0015 0.2382 0.4007 0.1625 0.2226 0.4162 0.1936
λ0.3173 0.0070 0.1148 0.5198 0.4051 0.0760 0.5586 0.4826
η0.3050 0.0003 0.2643 0.3458 0.0815 0.2565 0.3536 0.0971
Table 4: Simulation results for the KTL distribution at µ=λ=η=0.5
nparameter MLE MSE 90%95%
LL UL AL LL UL AL
100
µ0.5552 0.0870 0.2923 0.8182 0.5260 0.2419 0.8686 0.6267
λ0.5444 0.3956 -0.0483 1.1372 1.1855 -0.1618 1.2507 1.4125
η0.5435 0.0190 0.3795 0.7076 0.3280 0.3481 0.7390 0.3908
200
µ0.5016 0.0074 0.3613 0.6420 0.2807 0.3344 0.6689 0.3345
λ0.4869 0.0261 0.1460 0.8278 0.6818 0.0807 0.8931 0.8124
η0.5212 0.0053 0.4203 0.6222 0.2020 0.4009 0.6416 0.2406
300
µ0.5165 0.0107 0.3910 0.6420 0.2510 0.3670 0.6661 0.2991
λ0.5870 0.0460 0.2328 0.9411 0.7083 0.1650 1.0089 0.8439
η0.4949 0.0018 0.4177 0.5721 0.1544 0.4029 0.5869 0.1840
Table 5: Simulation results for the KTL distribution at µ=0.8 and λ=η=0.3
nparameter MLE MSE 90%95%
LL UL AL LL UL AL
100
µ0.8310 0.1532 0.2126 1.4493 1.2367 0.0942 1.5678 1.4735
λ0.4102 0.3108 -0.1986 1.0190 1.2177 -0.3152 1.1356 1.4508
η0.3166 0.0018 0.2490 0.3842 0.1352 0.2360 0.3971 0.1611
200
µ0.8261 0.0494 0.4259 1.2864 0.8605 0.3435 1.3688 1.0253
λ0.2882 0.0151 0.0278 0.5286 0.5008 -0.0202 0.5766 0.5967
η0.3154 0.0009 0.2680 0.3628 0.0948 0.2589 0.3719 0.1130
300
µ0.8171 0.0696 0.5496 1.3047 0.7551 0.4773 1.3770 0.8997
λ0.3010 0.0347 0.1055 0.6165 0.5110 0.0566 0.6655 0.6089
η0.3069 0.0005 0.2701 0.3437 0.0736 0.2630 0.3507 0.0877
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Table 6: Simulation results for the KTL distribution at µ=0.8 and λ=η=0.5
nparameter MLE MSE 90%95%
LL UL AL LL UL AL
100
µ0.8975 0.0665 0.3814 1.4136 1.0323 0.2825 1.5125 1.22995
λ0.6950 0.1692 -0.0599 1.4499 1.5098 -0.2045 1.5944 1.79892
η0.4708 0.0031 0.3563 0.5854 0.2291 0.3343 0.6073 0.272959
200
µ0.7585 0.0159 0.4925 1.0246 0.5321 0.4415 1.0755 0.634012
λ0.5209 0.0347 0.1353 0.9065 0.7712 0.0614 0.9803 0.918915
η0.5060 0.0032 0.4158 0.5962 0.1805 0.3985 0.6135 0.215007
300
µ0.8595 0.0124 0.6053 1.1136 0.5083 0.5567 1.1623 0.605648
λ0.5041 0.0190 0.2093 0.7988 0.5895 0.1529 0.8552 0.702393
η0.5208 0.0012 0.4456 0.5961 0.1506 0.4311 0.6105 0.179386
Table 7: Simulation results for the KTL distribution at µ=0.9, λ=0.6 and η=0.4
nparameter MLE MSE 90%95%
LL UL AL LL UL AL
100
µ1.0393 0.1903 0.2677 1.8110 1.5433 0.1199 1.9587 1.8388
λ0.8659 0.3727 -0.2526 1.9844 2.2370 -0.4667 2.1986 2.6654
η0.4007 0.0018 0.3099 0.4915 0.1816 0.2925 0.5089 0.2164
200
µ0.9218 0.0417 0.5210 1.3226 0.8016 0.4443 1.3993 0.9550
λ0.7150 0.1307 0.1462 1.2839 1.1378 0.0372 1.3929 1.3556
η0.4005 0.0017 0.3367 0.4642 0.1275 0.3245 0.4764 0.1519
300
µ0.8874 0.0173 0.5927 1.1820 0.5893 0.5363 1.2384 0.7021
λ0.6478 0.0227 0.2536 1.0420 0.7884 0.1782 1.1175 0.9393
η0.4010 0.0011 0.3491 0.4528 0.1036 0.3392 0.4627 0.1235
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Table 8: The CV, A-D, KS, PV, MLEs, and SEs for the
data set of The rate of COVID-19 recovery in Turkey
Models CV A-D KS PV MLEs (SEs)
KTL 0.025 0.187 0.083 0.954 µ13.234 (3583)
η1.757 (65.008)
λ2.473 (821.933)
TL 0.126 1.426 0.151 0.620 λ18.633 (3.727)
UExP 0.030 0.229 0.101 0.941 α1.342 (0.209)
β0.115 (0.054)
λ2.063 (1.069)
UW 0.065 0.386 0.136 0.692 α0.005 (0.003)
β4.160 (0.418)
K 0.031 0.231 0.102 0.933 α1.416 (0.230)
β50.941 (31.323)
MOK 0.036 0.237 0.113 0.872 α0.138 (0.159)
β1.876 (0.346)
λ47.547 (52.990)
UGo 0.124 0.726 0.160 0.493 α0.017 (0.013)
β1.146 (0.180)
MOETL 0.056 0.348 0.106 0.915 α0.006 (0.005)
β2.066 (0.298)
UGLB 0.038 0.251 0.099 0.947 α1.839 (3.079)
β2.724 (1.134)
λ3.605 (1.664)
UGGo 0.039 0.255 0.219 0.156 α1.288 (0.283)
β29.959 (14.070)
λ0.805 (0.400)
ExP 0.030 0.231 0.103 0.930 α1.432 (0.224)
β0.117 (0.909)
λ2.501 (27.807)
Figure 3: Estimated pdf, CDF and P-P plots of com-
petitive model for the data set of The rate of COVID-
19 recovery in Turkey
The MLEs of the parameters of each fitted distri-
butions with their stranded errors (SE), and the good-
ness of fit statistics are reported in Table 9. From the
table, it is found that the UEPD, MOETL, UGIBXI
and K models provide sufficient results for analyzing
this data set besides KTL model. However, the KTL
distribution is the best among all tested models.
Table 9: The CV, A-D, KS, PV, MLEs, and SEs for the
data set of The rate of COVID-19 recovery in France
Models CV A-D KS PV MLEs (SEs)
KTL 0.023 0.200 0.0613 0.999 µ5.707 (9.199)
η4.395 (2.897)
λ7.846 (19.41)
TL 0.742 14.901 0.333 0.001 λ6.398 (1.415)
UExP 0.037 0.297 0.072 0.981 α2.172 (0.265)
β0.354 (0.112)
λ2.640 (0.503)
UW 0.084 0.546 0.120 0.599 α0.028 (0.015)
β4.869 (0.597)
K 0.567 0.231 0.071 0.949 α2.658 (0.339)
β91.92 (50.57)
MOK 0.170 1.041 0.154 0.295 α0.149 (0.114)
β3.814 (0.900)
λ191.13 (88.89)
UGo 0.17 1.041 0.154 0.295 α0.007 (0.004)
β2.319 (0.229)
MOETL 0.064 0.453 0.077 0.965 α0.004 (0.002)
β4.270 (0.410)
UGLB 0.043 0.321 0.079 0.957 α2.093 (1.282)
β3.081 (1.080)
λ2.204 (0.828)
UGGo 0.063 0.437 0.153 0.301 α2.523 (0.426)
β157.24 (77.15)
λ1.385 (0.709)
Figure 4: Estimated pdf, CDF and P-P plots of com-
petitive model for the data set of The rate of COVID-
19 recovery in France
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7 Conclusion
In this paper, we proposed a new generalization of
the Truncated Lomax Distribution using the tractable
features of the Kumaraswamy generated family. Sev-
eral of its statistical properties are studied. The maxi-
mum likelihood method is used to estimate the value
of the parameters. It is found that the estimation
of the model parameters performs quite well. Fi-
nally, the flexibility of the introduced family was il-
lustrated by a real dataset, which showed that the Ku-
maraswamy truncated Lomax Distribution is better
than every other lifetime models used in this study.
Acknowledgment
Researches would like to thank the Deanship of Sci-
entific Research, Qassim University for funding pub-
lication of this project.
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