Alpha–beta-power family of distributions with applications to exponential distribution

Abstract

This article proposes a new way to increase the flexibility of a family of statistical distributions by adding
two additional parameters. The newly proposed family is called the alpha–beta power transformation family
of distributions. A specific model, the alpha–beta power exponential distribution, was thoroughly investigated.
The hazard rates of the proposed distribution can be decreasing, bathtub-shaped, and unimodal. The derived
structural features of the proposed model include explicit formulations for the quantiles, the moments, the
moment-generating function, and the incomplete and conditional moments. In addition, estimates of maximum
likelihood, least squares, weighted least squares, and minimum distance of Cramér von Mises, von Anderson–
Darling, and right-tail Anderson–Darling are obtained for the unknown parameters. Two real data sets were
examined to demonstrate the usefulness of the proposed approach.

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Alexandria Engineering Journal 100 (2024) 15–31
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Original Article
Alpha–beta-power family of distributions with applications to exponential
distribution
H.E. Semary a,d, Zawar Hussain b, Walaa A. Hamdi c, Maha A. Aldahlan c, Ibrahim Elbatal a,
Vasili B.V. Nagarjuna e,∗
aDepartment of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
bDepartment of Statistics, Quaid-e-Azam University, Islamabad, Pakistan
cDepartment of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia
dStatistics and Insurance Department, Faculty of Commerce, Zagazig University, Zagazig 44519, Egypt
eDepartment of Mathematics Vellore Institute of Technology, Andhra Pradesh, Amaravati, India
ARTICLE INFO
Keywords:
Exponential distribution
Bathtub shape
Moments
Maximum likelihood estimation
ABSTRACT
This article proposes a new way to increase the flexibility of a family of statistical distributions by adding
two additional parameters. The newly proposed family is called the alpha–beta power transformation family
of distributions. A specific model, the alpha–beta power exponential distribution, was thoroughly investigated.
The hazard rates of the proposed distribution can be decreasing, bathtub-shaped, and unimodal. The derived
structural features of the proposed model include explicit formulations for the quantiles, the moments, the
moment-generating function, and the incomplete and conditional moments. In addition, estimates of maximum
likelihood, least squares, weighted least squares, and minimum distance of Cramér von Mises, von Anderson–
Darling, and right-tail Anderson–Darling are obtained for the unknown parameters. Two real data sets were
examined to demonstrate the usefulness of the proposed approach.
1. Introduction and motivation
In statistics, the creation of probability distributions is a popular
subject. In distribution theory, adding one or more additional pa-
rameters to an existing model increases its flexibility and is a usual
procedure. Since 1980, the methods for putting up new statistical distri-
butions have been reduced to combining existing distributions to create
new families of distributions or adding one or more parameters to an
existing statistical distribution. By including an additional parameter
in addition to the two in the standard exponentiated distribution, [1]
proposed the exponentiated Weibull (EW) family. The EW distribution
has the following cumulative distribution function (CDF)
𝐺(𝑧) = 1 − 𝑒−𝜃𝑧𝜆𝜎
, 𝑧, 𝜃, 𝜆, 𝜎 > 0.(1)
Due to presence of additional shape parameter, the proposed EW
model is much flexible than the traditional Weibull model. Using
the expression in (1), several exponentiated distributions have pro-
posed in the literature. For example, exponentiated exponential (Ex-E)
by [2,3] studied the exponentiated Gumbel (Ex-Gu), [4] defined the
exponentiated gamma (Ex-Ga), exponentiated generalized gamma (Ex-
GGa) of [5], exponentiated inverted Weibull (EIW) proposed by [6],
∗Corresponding author.
E-mail addresses: hesemary@imamu.edu.sa (H.E. Semary), zhlangah@yahoo.com (Z. Hussain), whamdi@uj.edu.sa (W.A. Hamdi), maal-dahlan@uj.edu.sa
(M.A. Aldahlan), iielbatal@imamu.edu.sa (I. Elbatal), nagarjuna.vasili@vitap.ac.in (V.B.V. Nagarjuna).
exponentiated modified Weibull (EMW) by [7,8] introduced the ex-
ponentiated Kumaraswamy (Ex-K), [9] introduced the unit exponen-
tiated half logistic power series class of distributions, [10] studied
the exponentiated truncated inverse Weibull-generated family of distri-
butions, [11] discussed the exponentiated power generalized Weibull
power series family of distributions, [12] proposed the exponentiated
version of the M family of distributions, [13] introduced the expo-
nentiated general family of distributions, [14] studied the weighted
exponentiated family of distributions, [15] proposed the exponenti-
ated Weibull Rayleigh, [16] investigated the exponentiated Weibull
Weibull, [17] discussed the exponentiated extended family of distribu-
tions and [18] introduced the exponentiated generalized Kumaraswamy
distribution, among others.
Various academics have proposed various methodologies for adding
a parameter into probability distributions. These novel families have
been employed to analyze modeling data in numerous practical ar-
eas, such as engineering, economics, biological research, environmen-
tal sciences, and numerous additional areas. Some well-popular gen-
erated families of distributions are the Dinesh-Umesh-Sanjay (DUS)
transformation- G by [19], generalized DUS- G by [20], Cos- G by [21],
https://doi.org/10.1016/j.aej.2024.05.024
Received 15 March 2024; Received in revised form 25 April 2024; Accepted 7 May 2024

Alexandria Engineering Journal 100 (2024) 15–31
16
H.E. Semary et al.
the Kavya-Manoharan (KM) transformation- G by [22], the new ex-
tended cosine- G by [23], the Compounded bell-G by [24], the sine
Kumaraswamy- G by [25], T-X family by [26], sec- G by [27], the
truncated Cauchy power Weibull- G by [28], Weibull-G by [29], a
new power Topp-Leone- G by [30], the sine exponentiated Weibull-
G by [31], the Gompertz- G by [32], type I and type II half-logistic-
G by [33,34], the odd Perks- G by [35], the Type II half-logistic
odd Fréchet- G by [36], beta-G by [37], generalized odd Burr III- G
by [38], odd generalized N-H- G by [39], logarithmically-exponential-
G by [40], generalized inverted Kumaraswamy-G by [41], Type II
exponentiated half logistic-G by [42], generalized truncated Fréchet-
G by [43], Marshall–Olkin odd Burr III-G by [44], anew truncated
muth-G by [45], odd inverse power generalized Weibull-G by [46],
unit exponentiated half logistic power series-G by [47], and for more
information see [48–52].
Recently, [53] proposed method for introducing new statistical fam-
ily of distributions which is called alpha power transformation (APT)
family distributions. The idea of APT is not new; it is essentially a
Poisson-G family [54] with 𝛼=𝑒𝜃. The CDF of Poisson-G is given by
𝐺(𝑧) = 𝑒𝜃 𝐹 (𝑧)−1
𝑒𝜃−1 , where 𝜃 > 0. Then, the CDF the APT are defined as
below:
𝐺(𝑧) = 𝛼𝐹(𝑧)− 1
𝛼− 1 ,𝑧∈𝑅, 𝛼 > 0, 𝛼 ≠1.(2)
Several researchers have employed the transformation to obtain
novel distributions produced by the APT-G of distributions. Ref. [55–
65] proposed the APT Pareto, APT Weibull, APT Lindley, APT extended
exponential, APT inverse Lindley, APT Topp-Leone Weibull, exponen-
tial APT-G, transmuted APT-G, APT extended power Lindley, APT
Kumaraswamy-Burr III and APT Weibull-G distributions respectively.
In this article, we propose a new method for generating lifetime
distributions. We call this new method is alpha beta power e trans-
formation (ABPT) method. The new suggested family defined by the
following CDF
𝐺(𝑧) = 𝛼𝐹(𝑧)−𝛽𝐹(𝑧)
𝛼−𝛽,𝑧∈𝑅, 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1.(3)
Clearly, when 𝛽= 1, then the CDF of ABPT in (3) reduces to the CDF
of APT in (2). The probability density function (PDF) corresponding to
(3) is provided via
𝑔(𝑧) = 𝑓(𝑧)
𝛼−𝛽log(𝛼)𝛼𝐹(𝑧)− log(𝛽)𝛽𝐹(𝑧),𝑧∈𝑅, 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1.(4)
The survival function (SF), hazard rate function (HRF), reversed HRF
and cumulative HRF of ABPT family are given respectively by:
𝑆(𝑧) = 𝛼−𝛽−𝛼𝐹(𝑧)+𝛽𝐹(𝑧)
𝛼−𝛽,𝑧∈𝑅, 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1,
ℎ(𝑧) = 𝑓(𝑧)log(𝛼)𝛼𝐹(𝑧)− log(𝛽)𝛽𝐹(𝑧)
𝛼−𝛽−𝛼𝐹(𝑧)+𝛽𝐹(𝑧),𝑧∈𝑅, 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1,
𝜏(𝑧) = 𝑓(𝑧)log(𝛼)𝛼𝐹(𝑧)− log(𝛽)𝛽𝐹(𝑧)
𝛼𝐹(𝑧)−𝛽𝐹(𝑧),𝑧∈𝑅, 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1,
and
𝐻(𝑧) = − log𝛼−𝛽−𝛼𝐹(𝑧)+𝛽𝐹(𝑧)
𝛼−𝛽,𝑧∈𝑅, 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1.
The main idea of this article, is using the proposed new trans-
formation method to the exponential distribution to introduce a new
lifetime distribution named as alpha–beta power exponential (ABPE)
distribution. The proposed model is much flexible and is able to model
real phenomena with decreasing, unimodal or bathtub HRF. Several
statistical features of the suggested model are computed. Six different
approaches of estimation are discussed to asses the behavior of pa-
rameters for the newly model. Two real data sets were examined to
demonstrate the usefulness of the proposed approach
The subsequent section of this article is organized in the following
manner: Section 2provides the formulation of the ABPE model. Sec-
tion 3study several statistical properties of the ABPE model. Section 4
focus on estimating the unknown parameters for the ABPE model using
six different approaches of estimation. In Section 5, a simulation study
is done to show the performance of the six methods of estimation to
estimate the parameters for the ABPE model. Section 6provides an
analysis of real data sets. Section 7contains concluding remarks.
2. Formulation of the alpha–beta power exponential model
In this section, we are motivated to introduce a new extension of
the exponential distribution which has the next CDF and PDF 𝐹(𝑧;𝜃)=
1 − 𝑒−𝜃𝑧, 𝑧, 𝜃 > 0and 𝑓(𝑧;𝜃)=𝜃𝑒−𝜃𝑧 . A random variable 𝑍is said to
have the ABPE distribution, if its CDF is given by
𝐺(𝑧) = 𝛼1−𝑒−𝜃 𝑧−𝛽1−𝑒−𝜃𝑧 
𝛼−𝛽,𝑧, 𝜃, 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1.(5)
The PDF of the ABPE model is given by
𝑔(𝑧) = 𝜃 𝑒−𝜃𝑧
𝛼−𝛽log(𝛼)𝛼1−𝑒−𝜃𝑧− log(𝛽)𝛽1−𝑒−𝜃 𝑧,𝑧, 𝜃 , 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1.
(6)
The SF, HRF, reversed HRF and cumulative HRF of ABPE distribu-
tion are provided respectively via:
𝑆(𝑧) = 𝛼−𝛽−𝛼1−𝑒−𝜃 𝑧+𝛽1−𝑒−𝜃𝑧 
𝛼−𝛽,𝑧, 𝜃, 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1,
ℎ(𝑧) =
𝜃𝑒−𝜃𝑧 log(𝛼)𝛼1−𝑒−𝜃 𝑧− log(𝛽)𝛽1−𝑒−𝜃𝑧 
𝛼−𝛼(1−𝑒−𝜃𝑧)−𝛽+𝛽(1−𝑒−𝜃 𝑧),𝑧, 𝜃 , 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1,
𝜏(𝑧) =
𝜃𝑒−𝜃𝑧 log(𝛼)𝛼1−𝑒−𝜃 𝑧− log(𝛽)𝛽1−𝑒−𝜃𝑧 
𝛼(1−𝑒−𝜃𝑧)−𝛽(1−𝑒−𝜃 𝑧),𝑧, 𝜃 , 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1,
and
𝐻(𝑧) = − log 𝛼−𝛽−𝛼1−𝑒−𝜃𝑧+𝛽1−𝑒−𝜃𝑧 
𝛼−𝛽,𝑧, 𝜃, 𝛼, 𝛽 > 0, 𝛼 ≠𝛽≠1.
Fig. 1 shows the density curves of the ABPE distribution, while
Figs. 2 and 3show the plots of the curves for HRF of the ABPE
distribution with various combinations of parameters. The HRF of the
ABPE distribution is decreasing, unimodal or bathtub for different
parametric values. But the PDF of the ABPE distribution is unimodal,
right skewed, and decreasing-increasing-decreasing (D-I-D).
3. Various statistical and mathematical features
In this section, the fundamental mathematical aspects of the ABPE
distribution are discussed. These properties include the quantile func-
tion, the median, moments, and the formula from which random num-
bers are obtained.
3.1. Quantile function
The 𝑞𝑡ℎ quantile denoted by 𝑧𝑞of the ABPE model is derived from
the next formula
𝑃𝑍⩽𝑧𝑞=𝑞, 0⩽𝑞⩽1.
Using the CDF in (5) of the ABPE model in the previous equation,
we get
𝛼1−𝑒−𝜃𝑧−𝛽1−𝑒−𝜃 𝑧=(𝛼−𝛽)𝑞. (7)
The expression obtained in Eq. (7) does not have a closed form
solution. Hence, the numerical solution of Eq. (7) can be derived by
utilizing computer software to obtain numerical value of the quantile
or to generate random numbers from the ABPE model.

Alexandria Engineering Journal 100 (2024) 15–31
17
H.E. Semary et al.
Fig. 1. Different shapes of PDF of the ABPE distribution for different values of parameters.
Fig. 2. Decreasing and decreasing-increasing HRF of the ABPE distribution.
Fig. 3. Increasing and bathtub HRF of the ABPE distribution.

Alexandria Engineering Journal 100 (2024) 15–31
18
H.E. Semary et al.
3.2. Various types of moments
Assume that the random variable 𝑍has the ABPE model. Then the
𝑟𝑡ℎ moments of 𝑍can be computed as
𝜇∕
𝑟=∫∞
0
𝑧𝑟𝑔(𝑧)𝑑𝑧 =𝜃
𝛼−𝛽log(𝛼)∫∞
0
𝑧𝑟𝑒−𝜃𝑧𝛼1−𝑒−𝜃 𝑧𝑑 𝑧
− log(𝛽)∫∞
0
𝑧𝑟𝑒−𝜃𝑧𝛽1−𝑒−𝜃 𝑧𝑑 𝑧.(8)
Using the series representation 𝛼𝑣=∞
𝑖=0
(log 𝛼)𝑖
𝑖!𝑣𝑖, in (8), we get
𝜇∕
𝑟=𝜃
𝛼−𝛽∞

𝑖=0
(log(𝛼))𝑖+1
𝑖!∫∞
0
𝑧𝑟𝑒−𝜃𝑧1 − 𝑒−𝜃 𝑧𝑖𝑑 𝑧
−
∞

𝑗=0
(log(𝛽))𝑗+1
𝑗!∫∞
0
𝑧𝑟𝑒−𝜃𝑧1 − 𝑒−𝜃 𝑧𝑗𝑑 𝑧.(9)
The expression in (9), can be written as
𝜇∕
𝑟=𝜃
𝛼−𝛽∞

𝑖,𝑘=0
(log(𝛼))𝑖+1
𝑖!𝑘!∫∞
0
𝑧𝑟𝑒−𝜃(𝑘+1)𝑧𝑑𝑧
−
∞

𝑗,𝑙=0
(log(𝛽))𝑗+1
𝑗!𝑙!∫∞
0
𝑧𝑟𝑒−𝜃(𝑙+1)𝑧𝑑𝑧.(10)
Finally, we get
𝜇∕
𝑟=𝛤(𝑟+ 1)
𝜃𝑟(𝛼−𝛽)∞

𝑖,𝑘=0
(log(𝛼))𝑖+1
𝑖!𝑘!(𝑘+ 1)𝑟+1 −
∞

𝑗,𝑙=0
(log(𝛽))𝑗+1
𝑗!𝑙!(𝑙+ 1)𝑟+1 .(11)
The 𝑛th incomplete moment of 𝑍is provided via
𝐾𝑛(𝑦)=∫𝑦
0
𝑧𝑛𝑔(𝑧)𝑑𝑧.
From (10) we can write the previous equation as the following
𝐾𝑛(𝑦)=𝜃
𝛼−𝛽∞

𝑖,𝑘=0
(log(𝛼))𝑖+1
𝑖!𝑘!∫𝑦
0
𝑧𝑛𝑒−𝜃(𝑘+1)𝑧𝑑𝑧
−
∞

𝑗,𝑙=0
(log(𝛽))𝑗+1
𝑗!𝑙!∫𝑦
0
𝑧𝑛𝑒−𝜃(𝑙+1)𝑧𝑑𝑧.
Then,
𝐾𝑛(𝑦)=1
𝜃𝑛(𝛼−𝛽)∞

𝑖,𝑘=0
(log(𝛼))𝑖+1𝛾(𝑛+ 1, 𝜃 (𝑘+ 1)𝑦)
𝑖!𝑘!(𝑘+ 1)𝑛+1
−
∞

𝑗,𝑙=0
(log(𝛽))𝑗+1𝛾(𝑛+ 1, 𝜃 (𝑙+ 1)𝑦)
𝑗!𝑙!(𝑙+ 1)𝑛+1 .
The 𝑛th conditional moment of 𝑍is provided via
𝜁𝑛(𝑦)=∫∞
𝑦
𝑧𝑛𝑔(𝑧)𝑑𝑧.
From (10) we can write the previous equation as the following
𝜁𝑛(𝑦)=𝜃
𝛼−𝛽∞

𝑖,𝑘=0
(log(𝛼))𝑖+1
𝑖!𝑘!∫∞
𝑦
𝑧𝑛𝑒−𝜃(𝑘+1)𝑧𝑑𝑧
−
∞

𝑗,𝑙=0
(log(𝛽))𝑗+1
𝑗!𝑙!∫∞
𝑦
𝑧𝑛𝑒−𝜃(𝑙+1)𝑧𝑑𝑧.
Then,
𝜁𝑛(𝑦)=1
𝜃𝑛(𝛼−𝛽)∞

𝑖,𝑘=0
(log(𝛼))𝑖+1𝛤(𝑛+ 1, 𝜃 (𝑘+ 1)𝑦)
𝑖!𝑘!(𝑘+ 1)𝑛+1
−
∞

𝑗,𝑙=0
(log(𝛽))𝑗+1𝛤(𝑛+ 1, 𝜃 (𝑙+ 1)𝑦)
𝑗!𝑙!(𝑙+ 1)𝑛+1 ,
where 𝛾(., .)and 𝛤(., .)are the lower and upper gamma functions.
The moment generating function of 𝑍is calculated as
𝑀𝑧(𝑡)=∫∞
0
𝑒𝑡𝑧𝑔(𝑧)𝑑 𝑧 =
∞

𝑟=0
𝑡𝑟
𝑟!∫∞
0
𝑧𝑟𝑔(𝑧)𝑑𝑧
∞

𝑟=0
𝑡𝑟
𝑟!𝜇∕
𝑟.(12)
Then,
𝑀𝑧(𝑡)=𝛤(𝑟+ 1)
𝜃𝑟(𝛼−𝛽)∞

𝑖,𝑘,𝑟=0
𝑡𝑟(log(𝛼))𝑖+1
𝑖!𝑘!𝑟!(𝑘+ 1)𝑟+1 −
∞

𝑗,𝑙,𝑟=0
𝑡𝑟(log 𝛼)𝑗+1
𝑗!𝑙!𝑟!(𝑙+ 1)𝑟+1 .
Table 1 show the numerical values of the moments 𝜇′
1,𝜇′
2,𝜇′
3and
𝜇′
4also the numerical values of variance (𝜎2), standard deviation (𝜎),
coefficient of skewness (CS), coefficient of kurtosis (CK) and coefficient
of variation (CV) associated with the ABPE distribution. From Table 1
we can note that when the parameters 𝛽and 𝛼increase and 𝜃is fixed
then the values of 𝜇′
1,𝜇′
2,𝜇′
3,𝜇′
4,𝜎2,𝜎and CV are increased but the
values of CS and CK are decreased. Fig. 4 shows 3D plots of mean,
variance, skewness, kurtosis, CV and index of dispersion (ID) for the
ABPE distribution at 𝜃=0.5.
3.3. Order statistics
Let 𝑍1, 𝑍2,…, 𝑍𝑣are 𝑣random samples from the ABPE distribution
with CDF and PDF in Eqs. (7) and (8). Suppose that 𝑍(1), 𝑍(2),…, 𝑋(𝑣)
are the corresponding order statistics. The PDF of the 𝑘th order statis-
tics is given by:
𝑔𝑍(𝑘)(𝑧) = 𝑣!
(𝑘− 1)!(𝑣−𝑘)! 𝑔(𝑧)[𝐺(𝑧)]𝑘−1[1 − 𝐺(𝑧)]𝑣−𝑘.(13)
By Substituting (7) and (8) in (13), we get the PDF of 𝑍(𝑘)of order
statistics for the ABPE distribution as follows:
𝑔𝑍(𝑘)(𝑧) = 𝑣!𝜃𝑒−𝜃𝑧
(𝑘− 1)!(𝑣−𝑘)!(𝛼−𝛽)𝑘log(𝛼)𝛼1−𝑒−𝜃𝑧− log(𝛽)𝛽1−𝑒−𝜃 𝑧
×𝛼1−𝑒−𝜃𝑧−𝛽1−𝑒−𝜃 𝑧𝑘−1 1 − 𝛼1−𝑒−𝜃𝑧 −𝛽1−𝑒−𝜃𝑧 
𝛼−𝛽𝑣−𝑘
.
(14)
4. Different estimation methods
In this part, we are going to explore various methods for esti-
mating the parameters 𝜃,𝛼, and 𝛽. These methods include maximum
likelihood estimation (MLE), least squares estimation (LSE), weighted
least squares estimation (WLSE), Cramer von Mises Minimum Distance
Estimation (CME), Anderson–Darling estimation (ADE), and right-tail
Anderson–Darling estimation (RTADE). We will solve the equations
generated from each estimators numerically, utilizing the capabilities
of the R software.
4.1. Maximum likelihood estimation method
The maximum likelihood estimation (MLE) is a commonly used
method to figure out unknown parameters. It works by maximizing
the log of the likelihood of the distribution. For instance, if we have
a random sample, denoted as 𝑧1,…, 𝑧𝑛, drawn from our ABPE distri-
bution with a sample size of 𝑛, then the log-likelihood function can be
expressed as:
𝑙𝑛=
𝑛

𝑖=1
log 𝑓(𝑧𝑖;𝜃, 𝛼, 𝛽 )=𝑛log(𝜃) − 𝜃
𝑛

𝑖=1
𝑧𝑖−𝑛log(𝛼−𝛽)
+
𝑛

𝑖=1
log log(𝛼)𝛼1−𝑒−𝜃𝑧𝑖− log(𝛽)𝛽1−𝑒−𝜃𝑧𝑖.
The approach determining partial derivatives then equating each par-
tial derivative by zero and solve them simultaneously obtain the values
of 𝜃,𝛼, and 𝛽that maximize them. The partial derivatives of 𝑙𝑛are as
follows:
𝜕𝑙𝑛
𝜕𝜃 =𝑛
𝜃−
𝑛

𝑖=1
𝑧𝑖+
𝑛

𝑖=1
𝑧𝑖𝑒−𝜃𝑧𝑖(log(𝛼))2𝛼1−𝑒−𝜃𝑧𝑖−(log(𝛽))2𝛽1−𝑒−𝜃𝑧𝑖
log(𝛼)𝛼1−𝑒−𝜃𝑧𝑖− log(𝛽)𝛽1−𝑒−𝜃𝑧𝑖,
𝜕𝑙𝑛
𝜕𝛼 =−𝑛
𝛼−𝛽+
𝑛

𝑖=1
𝛼−𝑒−𝜃𝑧𝑖log(𝛼) − 𝑒−𝜃𝑧𝑖log(𝛼)+1
log(𝛼)𝛼1−𝑒−𝜃𝑧𝑖− log(𝛽)𝛽1−𝑒−𝜃𝑧𝑖,

Alexandria Engineering Journal 100 (2024) 15–31
19
H.E. Semary et al.
Table 1
Results of 𝜇′
1,𝜇′
2,𝜇′
3,𝜇′
4,𝜎2,𝜎, CS, CK, and CV associated with the ABPE distribution.
𝛽 𝛼 𝜃 𝜇′
1𝜇′
2𝜇′
3𝜇′
4𝜎2𝜎 𝐶𝑆 𝐶 𝐾 𝐶𝑉
0.3
1.3 0.5 0.150 0.121 0.174 0.393 0.099 0.314 4.077 31.127 2.093
1.5 0.8 0.184 0.188 0.326 0.808 0.154 0.393 3.876 25.382 2.131
1.7 1.1 0.195 0.199 0.329 0.750 0.162 0.402 3.502 20.482 2.065
1.9 1.4 0.196 0.192 0.295 0.612 0.154 0.393 3.255 17.751 2.007
2.1 1.7 0.192 0.179 0.254 0.484 0.142 0.377 3.081 16.007 1.960
2.3 2 0.187 0.165 0.217 0.380 0.130 0.360 2.950 14.786 1.922
2.5 2.3 0.181 0.150 0.185 0.299 0.118 0.343 2.847 13.876 1.891
2.7 2.6 0.175 0.137 0.158 0.238 0.107 0.327 2.764 13.167 1.865
2.9 2.9 0.169 0.126 0.135 0.191 0.097 0.311 2.694 12.596 1.843
3.1 3.2 0.163 0.115 0.117 0.155 0.088 0.297 2.635 12.124 1.823
0.6
1.3 0.5 0.223 0.269 0.544 1.565 0.219 0.468 3.761 23.936 2.103
1.5 0.8 0.234 0.278 0.524 1.357 0.223 0.472 3.362 19.034 2.020
1.7 1.1 0.231 0.259 0.447 1.039 0.206 0.454 3.124 16.542 1.960
1.9 1.4 0.224 0.235 0.370 0.778 0.184 0.429 2.963 14.998 1.914
2.1 1.7 0.215 0.210 0.305 0.584 0.164 0.405 2.844 13.929 1.879
2.3 2 0.206 0.188 0.252 0.444 0.146 0.382 2.751 13.134 1.851
2.5 2.3 0.197 0.169 0.210 0.342 0.130 0.360 2.676 12.514 1.827
2.7 2.6 0.189 0.152 0.176 0.267 0.116 0.341 2.613 12.016 1.807
2.9 2.9 0.181 0.137 0.150 0.212 0.105 0.324 2.560 11.602 1.789
3.1 3.2 0.173 0.125 0.128 0.170 0.095 0.308 2.514 11.252 1.774
0.9
1.3 0.5 0.267 0.365 0.792 2.366 0.293 0.542 3.385 19.277 2.028
1.5 0.8 0.265 0.338 0.661 1.743 0.268 0.517 3.099 16.315 1.951
1.7 1.1 0.255 0.300 0.530 1.246 0.235 0.485 2.924 14.672 1.900
1.9 1.4 0.243 0.264 0.423 0.898 0.205 0.453 2.800 13.583 1.862
2.1 1.7 0.231 0.232 0.341 0.658 0.179 0.423 2.707 12.800 1.833
2.3 2 0.219 0.205 0.278 0.492 0.157 0.396 2.633 12.199 1.809
2.5 2.3 0.208 0.182 0.229 0.374 0.139 0.372 2.572 11.719 1.789
2.7 2.6 0.198 0.163 0.190 0.290 0.123 0.351 2.520 11.324 1.772
2.9 2.9 0.189 0.146 0.160 0.227 0.110 0.332 2.476 10.991 1.757
3.1 3.2 0.181 0.132 0.136 0.181 0.099 0.315 2.437 10.706 1.744
1.2
1.3 0.5 0.299 0.436 0.981 2.982 0.347 0.589 3.152 16.794 1.970
1.5 0.8 0.288 0.384 0.767 2.045 0.301 0.548 2.931 14.729 1.902
1.7 1.1 0.273 0.332 0.595 1.410 0.257 0.507 2.791 13.507 1.858
1.9 1.4 0.257 0.287 0.466 0.994 0.221 0.470 2.690 12.670 1.826
2.1 1.7 0.242 0.249 0.370 0.718 0.190 0.436 2.612 12.048 1.801
2.3 2 0.229 0.218 0.298 0.531 0.166 0.407 2.549 11.561 1.780
2.5 2.3 0.217 0.193 0.244 0.400 0.146 0.382 2.497 11.165 1.763
2.7 2.6 0.205 0.171 0.202 0.308 0.129 0.359 2.453 10.835 1.748
2.9 2.9 0.195 0.153 0.169 0.240 0.115 0.339 2.414 10.554 1.735
3.1 3.2 0.186 0.138 0.143 0.191 0.103 0.321 2.380 10.311 1.723
and
𝜕𝑙𝑛
𝜕𝛽 =𝑛
𝛼−𝛽−
𝑛

𝑖=1
𝛽−𝑒−𝜃𝑧𝑖log(𝛽) − 𝑒−𝜃𝑧𝑖log(𝛽)+1
log(𝛼)𝛼1−𝑒−𝜃𝑧𝑖− log(𝛽)𝛽1−𝑒−𝜃𝑧𝑖.
The above three equations are not closed forms. Therefore, we use
the iterating method such as newton Raphson method to estimate
the model parameters numerically. In the present paper, the (SANN)
algorithm in R language is used to have numerical estimates of the
model parameters.
4.2. Least squares method
The LSE method, as introduced by [66], involves minimizing its
functions to obtain parameter estimates. The least squares function is
expressed as follows:
𝐿𝑆𝐸 (𝜃, 𝛼 , 𝛽) =
𝑛

𝑖=1 𝐺(𝑧𝑖, 𝜃, 𝛼, 𝛽 ) − 𝑖
𝑛+ 1 2
=
𝑛

𝑖=1 𝛼1−𝑒−𝜃𝑧𝑖−𝛽1−𝑒−𝜃𝑧𝑖
𝛼−𝛽−𝑖
𝑛+ 1 2
.
Moving forward, we will adopt the following abbreviations:
𝐺𝑖=𝐺(𝑧𝑖, 𝜃, 𝛼, 𝛽 ),(15)
𝐺𝜃𝑖=𝑧𝑖𝑒−𝜃𝑧𝑖
𝛼−𝛽log(𝛼)𝛼1−𝑒−𝜃𝑧𝑖− log(𝛽)𝛽1−𝑒−𝜃𝑧𝑖,(16)
𝐺𝛼𝑖=𝛽1−𝑒−𝜃𝑧𝑖+𝛽𝑒−𝜃 𝑧𝑖𝛼−𝑒−𝜃𝑧𝑖−𝛽𝛼−𝑒−𝜃𝑧𝑖−𝑒−𝜃𝑧𝑖𝛼1−𝑒−𝜃 𝑧𝑖
(𝛼−𝛽)2,(17)
and
𝐺𝛽𝑖=𝛼1−𝑒−𝜃𝑧𝑖+𝛼𝑒−𝜃 𝑧𝑖𝛽−𝑒−𝜃𝑧𝑖−𝛼𝛽 −𝑒−𝜃𝑧𝑖−𝑒−𝜃𝑧𝑖𝛽1−𝑒−𝜃𝑧𝑖
(𝛼−𝛽)2.(18)
The required procedure involves identifying partial derivatives, set-
ting them to zero, and solving them together. Since this system cannot
be expressed in a closed form, we resort to numerical techniques and
rely on R software. Utilizing the results from Eqs. (15) to (18), we
encounter the following partial derivative.
𝜕𝐿𝑆 𝐸(𝜃 , 𝛼, 𝛽)
𝜕𝜃 = 2
𝑛

𝑖=1
𝐺𝜃𝑖𝐺𝑖−𝑖
𝑛+ 1 ,
𝜕𝐿𝑆 𝐸(𝜃 , 𝛼, 𝛽)
𝜕𝛼 = 2
𝑛

𝑖=1
𝐺𝛼𝑖𝐺𝑖−𝑖
𝑛+ 1 ,
and
𝜕𝐿𝑆 𝐸(𝜃 , 𝛼, 𝛽)
𝜕𝛽 = 2
𝑛

𝑖=1
𝐺𝛽𝑖𝐺𝑖−𝑖
𝑛+ 1 .
4.3. Weighted least squares method
The WLSE method, introduced by [66], aims to minimize the WLSE
function to obtain parameter estimates. The weighted least squares

Alexandria Engineering Journal 100 (2024) 15–31
20
H.E. Semary et al.
Fig. 4. 3D plots of mean, variance, skewness, kurtosis, CV and ID for the ABPE distribution at 𝜃=0.5.
function is described as follows:
𝑊 𝐿𝑆𝐸 (𝜃, 𝛼 , 𝛽) =
𝑛

𝑖=1
(𝑛+ 1)2(𝑛+ 2)
𝑖(𝑛−𝑖+ 1) 𝐺(𝑧𝑖, 𝜃, 𝛼, 𝛽 ) − 𝑖
𝑛+ 1 2
=
𝑛

𝑖=1
(𝑛+ 1)2(𝑛+ 2)
𝑖(𝑛−𝑖+ 1) 𝛼1−𝑒−𝜃𝑧𝑖−𝛽1−𝑒−𝜃𝑧𝑖
𝛼−𝛽−𝑖
𝑛+ 1 2
.
Following the same procedure as conducted in the LSE, we arrive at the
subsequent partial derivatives.
𝜕𝑊 𝐿𝑆 𝐸(𝜃 , 𝛼, 𝛽)
𝜕𝜃 = 2
𝑛

𝑖=1
𝐺𝜃𝑖
(𝑛+ 1)2(𝑛+ 2)
𝑖(𝑛−𝑖+ 1) 𝐺𝑖−𝑖
𝑛+ 1 ,
𝜕𝑊 𝐿𝑆 𝐸(𝜃 , 𝛼, 𝛽)
𝜕𝛼 = 2
𝑛

𝑖=1
𝐺𝛼𝑖
(𝑛+ 1)2(𝑛+ 2)
𝑖(𝑛−𝑖+ 1) 𝐺𝑖−𝑖
𝑛+ 1 ,
and
𝜕𝑊 𝐿𝑆 𝐸(𝜃 , 𝛼, 𝛽)
𝜕𝛽 = 2
𝑛

𝑖=1
𝐺𝛽𝑖
(𝑛+ 1)2(𝑛+ 2)
𝑖(𝑛−𝑖+ 1) 𝐺𝑖−𝑖
𝑛+ 1 .
4.4. Cramer von Mises minimum distance estimation method
CME determines the estimates for 𝜃,𝛼, and 𝛽by minimizing the
Cramer von Mises minimum distance estimation function. This function
is defined as follows:
𝐶𝑀 𝐸 (𝜃, 𝛼, 𝛽 ) = 1
12𝑛+
𝑛

𝑖=1 𝐺(𝑧𝑖, 𝜃, 𝛼, 𝛽 ) − 2𝑖− 1
2𝑛2
=1
12𝑛+
𝑛

𝑖=1 𝛼1−𝑒−𝜃𝑧𝑖−𝛽1−𝑒−𝜃𝑧𝑖
𝛼−𝛽−2𝑖− 1
2𝑛2
.(19)

Alexandria Engineering Journal 100 (2024) 15–31
21
H.E. Semary et al.
Employing the abbreviations specified in Eqs. (15) through (18), the
subsequent partial derivatives emerge.
𝜕𝐶 𝑀𝐸(𝜃, 𝛼 , 𝛽)
𝜕𝜃 = 2
𝑛

𝑖=1
𝐺𝜃𝑖𝐺𝑖−2𝑖− 1
2𝑛,
𝜕𝐶 𝑀𝐸(𝜃, 𝛼 , 𝛽)
𝜕𝛼 = 2
𝑛

𝑖=1
𝐺𝛼𝑖𝐺𝑖−2𝑖− 1
2𝑛,
and
𝜕𝐶 𝑀𝐸(𝜃, 𝛼 , 𝛽)
𝜕𝛽 = 2
𝑛

𝑖=1
𝐺𝛽𝑖𝐺𝑖−2𝑖− 1
2𝑛.
These equations necessitate numerical solutions, prompting our utiliza-
tion of R software to concurrently resolve them and obtain the estimates
for 𝜃,𝛼, and 𝛽.
4.5. Anderson- Darling method
The Anderson–Darling Estimation Method, introduced by [67], in-
volves minimizing the Anderson–Darling function to obtain estimates
for 𝜃,𝛼, and 𝛽. This function can be expressed as:
𝐴(𝜃, 𝛼, 𝛽 ) = −𝑛−1
𝑛
𝑛

𝑖=1
(2𝑖− 1) log(𝐺(𝑧𝑖, 𝜃, 𝛼 , 𝛽)) + log(1 − 𝐺(𝑧𝑛+𝑖−1, 𝜃 , 𝛼, 𝛽))
= −𝑛−1
𝑛
𝑛

𝑖=1
(2𝑖− 1)log 𝛼(1−𝑒−𝜃𝑧𝑖)−𝛽(1−𝑒−𝜃𝑧𝑖)
𝛼−𝛽
+ log 
1 − 𝛼1−𝑒−𝜃𝑧𝑛+𝑖−1 −𝛽1−𝑒−𝜃𝑧𝑛+𝑖−1 
𝛼−𝛽.
In order to obtain estimates for 𝜃,𝛼, and 𝛽, we begin by calcu-
lating the partial derivatives of 𝐴(𝜃, 𝛼, 𝛽 ), utilizing the abbreviations
introduced in Eqs. (15) through (18) to get the following:
𝜕𝐴(𝜃, 𝛼 , 𝛽)
𝜕𝜃 = − 1
𝑛
𝑛

𝑖=1
(2𝑖− 1) 𝐺𝜃𝑖
𝐺(𝑧𝑖, 𝜃, 𝛼, 𝛽 )−
𝐺𝜃𝑛+𝑖−1
1 − 𝐺(𝑧𝑛+𝑖−1, 𝜃 , 𝛼, 𝛽),
𝜕𝐴(𝜃, 𝛼 , 𝛽)
𝜕𝛼 = − 1
𝑛
𝑛

𝑖=1
(2𝑖− 1) 𝐺𝛼𝑖
𝐺(𝑧𝑖, 𝜃, 𝛼, 𝛽 )−
𝐺𝛼𝑛+𝑖−1
1 − 𝐺(𝑧𝑛+𝑖−1, 𝜃 , 𝛼, 𝛽),
and
𝜕𝐴(𝜃, 𝛼 , 𝛽)
𝜕𝛽 = − 1
𝑛
𝑛

𝑖=1
(2𝑖− 1) 𝐺𝛽𝑖
𝐺(𝑧𝑖, 𝜃, 𝛼, 𝛽 )−
𝐺𝛽𝑛+𝑖−1
1 − 𝐺(𝑧𝑛+𝑖−1, 𝜃 , 𝛼, 𝛽).
4.6. The Right - tail Anderson–Darling estimation method
The Right-tail Anderson–Darling Method (RADE) derives estimates
for 𝜃,𝛼, and 𝛽by minimizing the subsequent function:
𝑅𝐴𝐷𝐸(𝜃 , 𝛼, 𝛽) = 𝑛
2− 2
𝑛

𝑖=1
𝐺(𝑧𝑖, 𝜃, 𝛼, 𝛽 )
−1
𝑛
𝑛

𝑖=1
(2𝑖− 1) log(1 − 𝐺(𝑧𝑛+𝑖−1 , 𝜃, 𝛼, 𝛽 ))
=𝑛
2− 2
𝑛

𝑖=1
𝛼1−𝑒−𝜃𝑧𝑖−𝛽1−𝑒−𝜃𝑧𝑖
𝛼−𝛽
−1
𝑛
𝑛

𝑖=1
(2𝑖− 1) 
log 
1 − 𝛼1−𝑒−𝜃𝑧𝑛+𝑖−1 −𝛽1−𝑒−𝜃𝑧𝑛+𝑖−1 
𝛼−𝛽
.
Following the typical procedure of computing the partial derivatives of
𝑅𝐴𝐷𝐸 using the abbreviations outlined in Eqs. (15) to (18).
𝜕𝑅𝐴𝐷𝐸 (𝜃, 𝛼, 𝛽 )
𝜕𝜃 = −2
𝑛

𝑖=1
𝐺𝜃𝑖+1
𝑛
𝑛

𝑖=1
(2𝑖− 1) 𝐺𝜃𝑛+𝑖−1
1 − 𝐺(𝑧𝑛+𝑖−1, 𝜃 , 𝛼, 𝛽),
𝜕𝑅𝐴𝐷𝐸 (𝜃, 𝛼, 𝛽 )
𝜕𝛼 = −2
𝑛

𝑖=1
𝐺𝛼𝑖+1
𝑛
𝑛

𝑖=1
(2𝑖− 1) 𝐺𝛼𝑛+𝑖−1
1 − 𝐺(𝑧𝑛+𝑖−1, 𝜃 , 𝛼, 𝛽),
and
𝜕𝑅𝐴𝐷𝐸 (𝜃, 𝛼, 𝛽 )
𝜕𝛽 = −2
𝑛

𝑖=1
𝐺𝛽𝑖+1
𝑛
𝑛

𝑖=1
(2𝑖− 1) 𝐺𝛽𝑛+𝑖−1
1 − 𝐺(𝑧𝑛+𝑖−1, 𝜃 , 𝛼, 𝛽).
5. Simulation results
In this section, we conduct a hands-on evaluation to determine the
effectiveness of various estimation approaches outlined in Section 4.
To measure their efficiency, we generate synthetic datasets using our
proposed model. Subsequently, we utilize these datasets to apply the
estimation techniques and derive suggested model estimators. The eval-
uation of these methods is based on two primary criteria, the relative
bias (Rbias) and the mean square error (MSE). The generated random
samples of sizes (𝑛=250, 300, 350, 400, 450, 500, 550, 600, 650, 700)
from the APBE distribution. We replicate this process 1000 times, each
time with different initial values for (𝜃,𝛼,𝛽). The estimations for the
mean, Rbias and mean square error (MSE) are available in Tables 2 to 6,
these tables also include ranks for the Rbias and MSE, and followed by
summary of ranks of all tables in Table 7. These tables display:
•As the sample size 𝑛increases for all estimation methods, the MSE
of all estimates also decreases. This suggests that larger sample
sizes lead to more precise estimates, resulting in a reduction of
both systematic and random errors.
•Upon analyzing the rankings and results of the simulations in
Table 7, it becomes apparent that the maximum likelihood esti-
mation technique consistently surpasses other methods in terms
of accurately estimating the parameters of interest.
6. Modeling to real data
In this section, we illustrate the importance and potentiality of the
ABPE distribution using two real data sets.
The first data set consists of 61 observations of the COVID-19 data
set from Italy, recorded between 13 June and 12 August 2021 [68] data
set is: 14, 5, 22, 52, 31, 18, 10, 21, 13, 17, 63, 37, 14, 24, 24, 13, 13,
15, 22, 36, 7, 16, 22, 35, 52, 19, 28, 20, 20, 27, 10, 28, 5, 11, 21, 31,
56, 7, 11, 3, 31, 31, 12, 25, 24, 7, 30, 27, 21, 12, 24, 9, 15, 17, 26, 22,
42, 40, 23, 28, 21.
The second set of data was studied by [69], and it represents the
test on the cycle at which the yarn failed of 100 cycles until failure of
the yarn. The data set is: 597, 286, 497, 176, 321, 251, 203, 249, 193,
76, 71, 93, 284, 135, 220, 236, 364, 229, 40, 396, 203, 341, 61, 90,
568, 157, 137, 185, 239, 353, 42, 182, 571, 246, 264, 229, 290, 188,
250, 315, 38, 146, 149, 400, 38, 86, 124, 151, 180, 121, 169, 131, 88,
198, 224, 65, 277, 180, 279, 292, 188, 338, 186, 15, 61, 55, 20, 350,
185, 337, 829, 198, 124, 55, 40, 264, 105, 20, 166, 135, 282, 81, 211,
98, 244, 653, 196, 398, 423, 195, 143, 400, 194, 180, 325, 264, 393,
262, 246, 175.
The descriptive analysis of all data sets is reported in Table 8.
Further, we shall compare the fits of the ABPE distribution with
other models: the half logistic modified Kies exponential (HLMK_E)
[70], modified Kies exponential (MK_E) [71], new alpha power expo-
nential (NAP_E) [72], alpha power inverted exponential (API_E) [73],
exponentiated exponential (E_E) [74], odd log–logistic Weibull (OLLW)
[75], odd log–logistic Exponentiated Weibull (OLLEW) [76], and Wei-
bull Poisson (WPo) [77] models.
The maximum likelihood estimators (MLEs) and standard errors
(SEs) of the model parameters are computed. In order to assess the
competitive models, various criteria are taken into account, includ-
ing the Akaike information criterion (𝜌1), correct Akaike information
criterion (𝜌2), Kolmogorov–Smirnov (𝜌3) test, 𝑝-value (𝜌4) test, the
Cramer-Von-Mises test (𝜌5), and the Anderson–Darling test (𝜌6).
Tables 9 and 11 show the MLEs with SEs for the datasets. Tables 10
and 12 also shows the numerical values for the Lnl, 𝜌1,𝜌2,𝜌3,𝜌4,

Alexandria Engineering Journal 100 (2024) 15–31
22
H.E. Semary et al.
Table 2
Simulation results at 𝜃= 0.6,𝛼= 0.5and 𝛽= 0.4.
n Estimate 𝜃 𝛼 𝛽
MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE
250
Mean 0.5736 0.6012 0.5701 0.5901 0.5732 0.5445 0.7233 0.7870 0.7147 0.7404 0.7505 0.6926 0.3797 0.4019 0.4003 0.3996 0.3555 0.3487
Rbias 0.044{3} 0.0021{1} 0.0498{5} 0.0165{2} 0.0446{4} 0.0925{6} 0.4466{3} 0.5741{6} 0.4294{2} 0.4808{4} 0.501{5} 0.3851{1} 0.0506{4} 0.0047{3} 0.0007{1} 0.001{2} 0.1113{5} 0.1282{6}
MSE 0.0652{6} 0.062{5} 0.048{3} 0.0578{4} 0.0456{2} 0.038{1} 0.2769{2} 0.6477{6} 0.4294{3} 0.5565{5} 0.4806{4} 0.2652{1} 0.2184{3} 0.2937{6} 0.245{4} 0.2598{5} 0.1437{1} 0.1502{2}
300
Mean 0.5637 0.6211 0.5925 0.6161 0.5923 0.5544 0.7050 0.8235 0.7502 0.7808 0.7457 0.7041 0.3933 0.3841 0.3838 0.4039 0.3774 0.3395
Rbias 0.0605{5} 0.0351{4} 0.0125{1} 0.0268{3} 0.0128{2} 0.076{6} 0.4099{2} 0.6469{6} 0.5004{4} 0.5615{5} 0.4914{3} 0.4082{1} 0.0167{2} 0.0397{3} 0.0406{4} 0.0097{1} 0.0564{5} 0.1514{6}
MSE 0.068{6} 0.0607{4} 0.0459{3} 0.0613{5} 0.0438{2} 0.0337{1} 0.2938{2} 0.6823{6} 0.4634{4} 0.6429{5} 0.4428{3} 0.2879{1} 0.1849{3} 0.2144{5} 0.1932{4} 0.217{6} 0.167{2} 0.1227{1}
350
Mean 0.5691 0.6215 0.5898 0.6151 0.5899 0.5632 0.7221 0.8413 0.7721 0.7895 0.7593 0.7240 0.3933 0.3665 0.3556 0.3904 0.3554 0.3391
Rbias 0.0516{5} 0.0358{4} 0.0169{2} 0.0252{3} 0.0168{1} 0.0614{6} 0.4442{1} 0.6826{6} 0.5442{4} 0.5789{5} 0.5186{3} 0.4481{2} 0.0168{1} 0.0838{3} 0.1111{4} 0.024{2} 0.1116{5} 0.1524{6}
MSE 0.0657{6} 0.0604{5} 0.0447{3} 0.059{4} 0.042{2} 0.035{1} 0.3163{1} 0.6949{6} 0.4771{4} 0.6379{5} 0.4407{3} 0.318{2} 0.1936{4} 0.1996{5} 0.1574{3} 0.2083{6} 0.1446{2} 0.1372{1}
400
Mean 0.5708 0.6155 0.5800 0.6075 0.5801 0.5568 0.6892 0.7832 0.7186 0.7456 0.7063 0.7030 0.3714 0.3810 0.3581 0.3844 0.3596 0.3225
Rbias 0.0486{5} 0.0259{2} 0.0334{4} 0.0125{1} 0.0331{3} 0.0721{6} 0.3784{1} 0.5664{6} 0.4371{4} 0.4912{5} 0.4126{3} 0.406{2} 0.0714{3} 0.0476{2} 0.1048{5} 0.0391{1} 0.101{4} 0.1937{6}
MSE 0.0527{5} 0.054{6} 0.0384{3} 0.0512{4} 0.0361{2} 0.0314{1} 0.2208{1} 0.5458{6} 0.3374{4} 0.4903{5} 0.3053{3} 0.2455{2} 0.1307{2} 0.1936{6} 0.1418{4} 0.1654{5} 0.1359{3} 0.0936{1}
450
Mean 0.5758 0.6109 0.5911 0.6046 0.5918 0.5649 0.6770 0.7730 0.7455 0.7401 0.7266 0.7114 0.3984 0.3689 0.3508 0.3809 0.3616 0.3244
Rbias 0.0403{5} 0.0181{4} 0.0149{3} 0.0077{1} 0.0137{2} 0.0584{6} 0.354{1} 0.546{6} 0.4909{5} 0.4803{4} 0.4533{3} 0.4227{2} 0.0041{1} 0.0777{3} 0.1229{5} 0.0477{2} 0.0961{4} 0.189{6}
MSE 0.0526{6} 0.0476{5} 0.0384{3} 0.0468{4} 0.0366{2} 0.0305{1} 0.2158{1} 0.5206{6} 0.4056{4} 0.4786{5} 0.3595{3} 0.2498{2} 0.1549{6} 0.1423{4} 0.1198{2} 0.1515{5} 0.1296{3} 0.1058{1}
500
Mean 0.57489 0.61448 0.58186 0.60864 0.58205 0.55411 0.68825 0.77906 0.72981 0.74171 0.70094 0.68822 0.36136 0.36247 0.33451 0.37556 0.35529 0.32230
Rbias 0.0418{5} 0.0241{2} 0.0302{4} 0.0144{1} 0.0299{3} 0.0765{6} 0.3765{2} 0.5581{6} 0.4596{4} 0.4834{5} 0.4019{3} 0.3764{1} 0.0966{3} 0.0938{2} 0.1637{5} 0.0611{1} 0.1118{4} 0.1942{6}
MSE 0.0481{4} 0.0516{6} 0.0369{3} 0.0494{5} 0.0348{2} 0.0288{1} 0.1926{1} 0.5131{6} 0.3167{4} 0.4619{5} 0.2797{3} 0.2087{2} 0.1146{3} 0.1401{5} 0.1048{2} 0.1432{6} 0.1172{4} 0.0955{1}
550
Mean 0.5849 0.6252 0.5861 0.6158 0.5850 0.5609 0.6814 0.7742 0.7324 0.7350 0.7115 0.6965 0.3747 0.3726 0.3369 0.3805 0.3496 0.3248
Rbias 0.0252{3} 0.0421{5} 0.0232{1} 0.0264{4} 0.025{2} 0.0651{6} 0.3628{1} 0.5483{6} 0.4648{4} 0.47{5} 0.4231{3} 0.3931{2} 0.0632{2} 0.0684{3} 0.1578{5} 0.0486{1} 0.126{4} 0.188{6}
MSE 0.0407{4} 0.0487{6} 0.0354{3} 0.0453{5} 0.0342{2} 0.0296{1} 0.1892{1} 0.4889{6} 0.3165{4} 0.4395{5} 0.2844{3} 0.2173{2} 0.1188{4} 0.1383{6} 0.1085{2} 0.1243{5} 0.1157{3} 0.0973{1}
600
Mean 0.5659 0.6106 0.5774 0.6048 0.5746 0.5491 0.6339 0.7580 0.6950 0.7174 0.6791 0.6519 0.3916 0.3473 0.3417 0.3685 0.3448 0.3271
Rbias 0.0569{5} 0.0176{2} 0.0377{3} 0.0081{1} 0.0423{4} 0.0848{6} 0.2678{1} 0.5159{6} 0.39{4} 0.4348{5} 0.3582{3} 0.3038{2} 0.0209{1} 0.1318{3} 0.1456{5} 0.0787{2} 0.1379{4} 0.1822{6}
MSE 0.0434{4} 0.0474{6} 0.0335{3} 0.0458{5} 0.0311{2} 0.0255{1} 0.1493{1} 0.4315{6} 0.2798{4} 0.3877{5} 0.2473{3} 0.1643{2} 0.1162{5} 0.107{4} 0.0802{2} 0.1201{6} 0.0808{3} 0.0743{1}
650
Mean 0.5912 0.6221 0.5937 0.6195 0.5936 0.5654 0.6687 0.7680 0.7239 0.7378 0.7020 0.6919 0.3712 0.3471 0.3339 0.3643 0.3495 0.3136
Rbias 0.0147{3} 0.0368{5} 0.0105{1} 0.0326{4} 0.0107{2} 0.0577{6} 0.3375{1} 0.5361{6} 0.4479{4} 0.4756{5} 0.404{3} 0.3837{2} 0.0721{1} 0.1323{4} 0.1652{5} 0.0893{2} 0.1261{3} 0.2159{6}
MSE 0.0345{4} 0.0437{6} 0.0327{3} 0.0433{5} 0.0314{2} 0.0262{1} 0.1605{1} 0.4271{6} 0.2899{4} 0.4015{5} 0.2609{3} 0.1928{2} 0.1003{4} 0.1019{5} 0.0844{2} 0.1084{6} 0.0965{3} 0.0749{1}
700
Mean 0.5805 0.6107 0.5845 0.6051 0.5813 0.5588 0.6401 0.7436 0.6746 0.7218 0.6622 0.6722 0.3829 0.3448 0.3510 0.3506 0.3522 0.3106
Rbias 0.0325{5} 0.0179{2} 0.0258{3} 0.0085{1} 0.0311{4} 0.0687{6} 0.2802{1} 0.4871{6} 0.3491{4} 0.4436{5} 0.3244{2} 0.3444{3} 0.0427{1} 0.138{5} 0.1225{3} 0.1235{4} 0.1195{2} 0.2236{6}
MSE 0.0353{4} 0.0415{6} 0.0285{3} 0.0406{5} 0.0267{2} 0.0229{1} 0.1404{1} 0.3665{6} 0.223{4} 0.3504{5} 0.1955{3} 0.1619{2} 0.0908{6} 0.0892{5} 0.0822{2} 0.0862{4} 0.0837{3} 0.0625{1}

Alexandria Engineering Journal 100 (2024) 15–31
23
H.E. Semary et al.
Table 3
Simulation results at 𝜃= 0.6,𝛼= 0.9and 𝛽= 0.4.
n Estimate 𝜃 𝛼 𝛽
MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE
250
Mean 0.6417 0.6796 0.6607 0.6938 0.6572 0.6339 1.3055 1.4859 1.3883 1.4925 1.3513 1.2735 0.4268 0.5099 0.4597 0.5195 0.4608 0.4523
Rbias 0.0695{2} 0.1327{5} 0.1012{4} 0.1563{6} 0.0954{3} 0.0565{1} 0.4505{2} 0.651{5} 0.5426{4} 0.6584{6} 0.5015{3} 0.415{1} 0.067{1} 0.2748{5} 0.1491{3} 0.2986{6} 0.152{4} 0.1306{2}
MSE 0.0247{1} 0.0504{5} 0.0363{4} 0.0516{6} 0.0331{3} 0.0298{2} 0.8082{2} 1.3139{6} 1.0369{4} 1.3065{5} 0.9046{3} 0.7596{1} 0.3032{1} 0.6926{6} 0.3967{2} 0.6723{5} 0.4051{4} 0.3987{3}
300
Mean 0.6361 0.6641 0.6479 0.6783 0.6470 0.6277 1.2738 1.4486 1.3599 1.4508 1.3431 1.2408 0.4172 0.4524 0.4265 0.4623 0.4110 0.4425
Rbias 0.0602{2} 0.1068{5} 0.0798{4} 0.1305{6} 0.0783{3} 0.0461{1} 0.4153{2} 0.6096{5} 0.5109{4} 0.612{6} 0.4923{3} 0.3787{1} 0.043{2} 0.1311{5} 0.0664{3} 0.1558{6} 0.0276{1} 0.1063{4}
MSE 0.0211{1} 0.0405{5} 0.0305{4} 0.0408{6} 0.0281{3} 0.0257{2} 0.7169{2} 1.1624{5} 0.9044{4} 1.1887{6} 0.8196{3} 0.6414{1} 0.271{2} 0.497{6} 0.3262{3} 0.4257{5} 0.2564{1} 0.3629{4}
350
Mean 0.6202 0.6616 0.6432 0.6726 0.6398 0.6208 1.1779 1.4201 1.3061 1.4130 1.2585 1.1761 0.4102 0.4228 0.4056 0.4420 0.4205 0.4293
Rbias 0.0337{1} 0.1026{5} 0.072{4} 0.121{6} 0.0663{3} 0.0347{2} 0.3088{2} 0.5779{6} 0.4512{4} 0.57{5} 0.3983{3} 0.3068{1} 0.0255{2} 0.057{4} 0.0139{1} 0.105{6} 0.0511{3} 0.0733{5}
MSE 0.0169{1} 0.0358{5} 0.0246{4} 0.0362{6} 0.0224{3} 0.0201{2} 0.5035{1} 1.0334{6} 0.7061{4} 1.019{5} 0.6349{3} 0.5148{2} 0.1924{1} 0.2952{5} 0.2035{3} 0.3019{6} 0.1962{2} 0.2319{4}
400
Mean 0.6245 0.6538 0.6382 0.6641 0.6348 0.6202 1.1988 1.3544 1.2680 1.3389 1.2293 1.1834 0.3884 0.4190 0.4079 0.4463 0.4193 0.4028
Rbias 0.0408{2} 0.0896{5} 0.0637{4} 0.1068{6} 0.0579{3} 0.0337{1} 0.332{2} 0.5049{6} 0.4089{4} 0.4877{5} 0.3659{3} 0.3149{1} 0.0289{3} 0.0475{4} 0.0197{2} 0.1156{6} 0.0481{5} 0.007{1}
MSE 0.0151{1} 0.0334{5} 0.0238{4} 0.0338{6} 0.0224{3} 0.019{2} 0.521{2} 0.8567{6} 0.68{4} 0.8348{5} 0.593{3} 0.4911{1} 0.1241{1} 0.2635{5} 0.1693{3} 0.2897{6} 0.1804{4} 0.1661{2}
450
Mean 0.6263 0.6605 0.6415 0.6693 0.6397 0.6270 1.2005 1.3786 1.2599 1.3726 1.2285 1.1878 0.3857 0.4182 0.4130 0.4331 0.4243 0.4120
Rbias 0.0438{1} 0.1008{5} 0.0691{4} 0.1155{6} 0.0662{3} 0.0449{2} 0.3339{2} 0.5318{6} 0.3999{4} 0.5251{5} 0.365{3} 0.3198{1} 0.0358{3} 0.0455{4} 0.0324{2} 0.0827{6} 0.0608{5} 0.03{1}
MSE 0.0155{1} 0.0317{5} 0.0224{4} 0.0319{6} 0.0211{3} 0.0185{2} 0.5069{2} 0.8717{5} 0.6084{4} 0.874{6} 0.5648{3} 0.4986{1} 0.1269{1} 0.2795{6} 0.1873{4} 0.2717{5} 0.186{3} 0.1693{2}
500
Mean 0.6155 0.6425 0.6291 0.6515 0.6268 0.6118 1.1592 1.3371 1.2226 1.3216 1.2000 1.1292 0.3801 0.3889 0.4008 0.4145 0.4013 0.4158
Rbias 0.0259{2} 0.0708{5} 0.0485{4} 0.0858{6} 0.0447{3} 0.0196{1} 0.288{2} 0.4857{6} 0.3585{4} 0.4685{5} 0.3333{3} 0.2547{1} 0.0498{6} 0.0277{3} 0.002{1} 0.0363{4} 0.0032{2} 0.0394{5}
MSE 0.0125{1} 0.0273{6} 0.0178{4} 0.0273{5} 0.0167{3} 0.0155{2} 0.4011{2} 0.8014{6} 0.5388{4} 0.7692{5} 0.4984{3} 0.38{1} 0.1056{1} 0.1949{5} 0.1723{3} 0.2319{6} 0.1492{2} 0.1938{4}
550
Mean 0.6206 0.6497 0.6377 0.6575 0.6349 0.6193 1.1285 1.2805 1.1823 1.2545 1.1704 1.1093 0.3966 0.4199 0.4264 0.4550 0.4212 0.4255
Rbias 0.0344{2} 0.0828{5} 0.0629{4} 0.0958{6} 0.0581{3} 0.0322{1} 0.2539{2} 0.4228{6} 0.3136{4} 0.3939{5} 0.3005{3} 0.2325{1} 0.0086{1} 0.0497{2} 0.0659{5} 0.1375{6} 0.053{3} 0.0637{4}
MSE 0.0125{1} 0.0267{5} 0.0172{4} 0.027{6} 0.0167{3} 0.0156{2} 0.3697{2} 0.6505{6} 0.4633{4} 0.6185{5} 0.4372{3} 0.3531{1} 0.1018{1} 0.2018{5} 0.1722{4} 0.2491{6} 0.1556{2} 0.166{3}
600
Mean 0.6221 0.6430 0.6310 0.6501 0.6294 0.6172 1.1376 1.2534 1.1801 1.2364 1.1668 1.1180 0.3875 0.4188 0.4030 0.4431 0.4036 0.4081
Rbias 0.0369{2} 0.0717{5} 0.0516{4} 0.0834{6} 0.0489{3} 0.0286{1} 0.264{2} 0.3926{6} 0.3112{4} 0.3738{5} 0.2964{3} 0.2423{1} 0.0311{4} 0.0471{5} 0.0076{1} 0.1076{6} 0.009{2} 0.0202{3}
MSE 0.0107{1} 0.0244{5} 0.0159{4} 0.0246{6} 0.0153{3} 0.0138{2} 0.3518{1} 0.6041{6} 0.4512{4} 0.5797{5} 0.4198{3} 0.3525{2} 0.0928{1} 0.2008{5} 0.1188{2} 0.2291{6} 0.1193{3} 0.1418{4}
650
Mean 0.6169 0.6305 0.6233 0.6373 0.6225 0.6112 1.1259 1.2140 1.1443 1.2130 1.1324 1.0986 0.3813 0.4117 0.4038 0.4191 0.4052 0.4022
Rbias 0.0282{2} 0.0509{5} 0.0388{4} 0.0622{6} 0.0375{3} 0.0187{1} 0.251{2} 0.3488{6} 0.2715{4} 0.3478{5} 0.2582{3} 0.2206{1} 0.0468{5} 0.0291{4} 0.0096{2} 0.0477{6} 0.0129{3} 0.0055{1}
MSE 0.0095{1} 0.0233{6} 0.0144{4} 0.0232{5} 0.0137{3} 0.0121{2} 0.3337{2} 0.5399{6} 0.3985{4} 0.5393{5} 0.3764{3} 0.3107{1} 0.0767{1} 0.1746{6} 0.1017{3} 0.1631{5} 0.0969{2} 0.1275{4}
700
Mean 0.6147 0.6367 0.6278 0.6433 0.6255 0.6129 1.0919 1.2246 1.1528 1.2198 1.1275 1.0769 0.3898 0.4063 0.3977 0.4182 0.4074 0.4073
Rbias 0.0245{2} 0.0612{5} 0.0464{4} 0.0721{6} 0.0425{3} 0.0215{1} 0.2132{2} 0.3607{6} 0.2809{4} 0.3553{5} 0.2528{3} 0.1965{1} 0.0254{5} 0.0159{2} 0.0058{1} 0.0455{6} 0.0186{4} 0.0181{3}
MSE 0.0088{1} 0.0215{6} 0.013{4} 0.0214{5} 0.0125{3} 0.011{2} 0.2743{2} 0.522{6} 0.3675{4} 0.5186{5} 0.3331{3} 0.2686{1} 0.0752{1} 0.1644{6} 0.0981{2} 0.162{5} 0.1071{4} 0.1058{3}

Alexandria Engineering Journal 100 (2024) 15–31
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Table 4
Simulation results at 𝜃= 0.6,𝛼= 0.9and 𝛽= 0.8.
n𝜃 𝛼 𝛽
MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE
250
Mean 0.6303 0.6474 0.6406 0.6589 0.6411 0.6275 1.4249 1.5277 1.4777 1.5442 1.4769 1.3768 0.8405 0.9371 0.8846 0.9501 0.8638 0.8838
Rbias 0.0505{2} 0.079{5} 0.0677{3} 0.0981{6} 0.0686{4} 0.0458{1} 0.5832{2} 0.6975{5} 0.6419{4} 0.7158{6} 0.641{3} 0.5298{1} 0.0506{1} 0.1714{5} 0.1058{4} 0.1876{6} 0.0797{2} 0.1048{3}
MSE 0.0121{1} 0.0198{5} 0.0154{4} 0.0198{6} 0.0143{3} 0.0129{2} 1.3503{1} 1.9563{6} 1.6746{4} 1.9443{5} 1.6486{3} 1.4047{2} 0.9135{3} 1.2247{5} 0.9243{4} 1.2251{6} 0.8072{1} 0.8795{2}
300
Mean 0.6301 0.6560 0.6468 0.6646 0.6461 0.6312 1.4618 1.6133 1.5124 1.6315 1.4793 1.3883 0.7777 0.8916 0.8666 0.9030 0.8798 0.8628
Rbias 0.0501{1} 0.0934{5} 0.0781{4} 0.1077{6} 0.0768{3} 0.0519{2} 0.6242{2} 0.7926{5} 0.6805{4} 0.8127{6} 0.6437{3} 0.5426{1} 0.0279{1} 0.1145{5} 0.0833{3} 0.1288{6} 0.0997{4} 0.0785{2}
MSE 0.0097{1} 0.0177{5} 0.0127{4} 0.0184{6} 0.0117{3} 0.0103{2} 1.3363{2} 2.2127{5} 1.6677{4} 2.2139{6} 1.5208{3} 1.3128{1} 0.6892{1} 1.059{5} 0.8386{3} 1.0809{6} 0.8897{4} 0.787{2}
350
Mean 0.6261 0.6474 0.6399 0.6549 0.6391 0.6279 1.3931 1.5384 1.4981 1.5070 1.4725 1.3599 0.7992 0.8528 0.8052 0.9104 0.8065 0.8533
Rbias 0.0434{1} 0.0791{5} 0.0665{4} 0.0915{6} 0.0651{3} 0.0465{2} 0.5478{2} 0.7093{6} 0.6646{4} 0.6744{5} 0.6362{3} 0.511{1} 0.001{1} 0.066{4} 0.0065{2} 0.138{6} 0.0081{3} 0.0667{5}
MSE 0.008{1} 0.0125{5} 0.0093{4} 0.0129{6} 0.0088{3} 0.0082{2} 1.1757{1} 1.7339{6} 1.474{4} 1.6674{5} 1.4463{3} 1.2364{2} 0.6998{2} 0.9128{5} 0.7453{3} 1.0157{6} 0.6438{1} 0.758{4}
400
Mean 0.6286 0.6468 0.6397 0.6544 0.6392 0.6275 1.4058 1.4540 1.4688 1.4969 1.4342 1.3657 0.7809 0.9062 0.8088 0.8884 0.8269 0.8179
Rbias 0.0476{2} 0.0781{5} 0.0662{4} 0.0906{6} 0.0654{3} 0.0459{1} 0.562{2} 0.6156{4} 0.632{5} 0.6632{6} 0.5936{3} 0.5175{1} 0.0239{3} 0.1328{6} 0.011{1} 0.1105{5} 0.0337{4} 0.0224{2}
MSE 0.0074{1} 0.0127{5} 0.0093{4} 0.0129{6} 0.0088{3} 0.0078{2} 1.1712{2} 1.5038{5} 1.3932{4} 1.5752{6} 1.3277{3} 1.1092{1} 0.575{1} 0.9463{6} 0.6272{3} 0.8853{5} 0.6156{2} 0.647{4}
450
Mean 0.6274 0.6467 0.6392 0.6518 0.6383 0.6298 1.3664 1.4541 1.3587 1.4826 1.4044 1.3547 0.7780 0.8801 0.8822 0.8599 0.8181 0.8185
Rbias 0.0457{1} 0.0778{5} 0.0653{4} 0.0863{6} 0.0639{3} 0.0496{2} 0.5183{3} 0.6156{5} 0.5097{2} 0.6474{6} 0.5604{4} 0.5052{1} 0.0275{3} 0.1001{5} 0.1027{6} 0.0749{4} 0.0227{1} 0.0231{2}
MSE 0.0063{1} 0.0108{5} 0.0079{4} 0.0111{6} 0.0075{3} 0.0067{2} 1.0315{1} 1.4821{5} 1.1401{3} 1.51{6} 1.1857{4} 1.074{2} 0.505{1} 0.8259{6} 0.7094{4} 0.7561{5} 0.5701{2} 0.6044{3}
500
Mean 0.6271 0.6484 0.6405 0.6533 0.6397 0.6284 1.4074 1.4739 1.4245 1.5141 1.4136 1.3342 0.7256 0.8428 0.8059 0.8142 0.8002 0.8053
Rbias 0.0451{1} 0.0806{5} 0.0674{4} 0.0888{6} 0.0661{3} 0.0473{2} 0.5638{2} 0.6377{5} 0.5828{4} 0.6824{6} 0.5707{3} 0.4824{1} 0.093{6} 0.0535{5} 0.0073{3} 0.0177{4} 0.0003{1} 0.0066{2}
MSE 0.0059{1} 0.0097{5} 0.0069{4} 0.0099{6} 0.0065{3} 0.0059{2} 1.0398{2} 1.5{5} 1.1822{4} 1.5249{6} 1.1521{3} 0.9861{1} 0.4549{1} 0.6881{6} 0.5913{4} 0.6296{5} 0.5501{3} 0.5465{2}
550
Mean 0.6267 0.6437 0.6361 0.6480 0.6356 0.6264 1.3920 1.4907 1.3861 1.4683 1.3948 1.3372 0.7513 0.8186 0.8347 0.8486 0.8128 0.8017
Rbias 0.0445{2} 0.0728{5} 0.0601{4} 0.08{6} 0.0594{3} 0.0439{1} 0.5467{3} 0.6564{6} 0.5401{2} 0.6315{5} 0.5498{4} 0.4858{1} 0.0608{6} 0.0233{3} 0.0433{4} 0.0608{5} 0.016{2} 0.0021{1}
MSE 0.0056{1} 0.0099{5} 0.007{4} 0.01{6} 0.0068{3} 0.0058{2} 1.0225{2} 1.4705{6} 1.1168{3} 1.4235{5} 1.1227{4} 0.9604{1} 0.4996{1} 0.7146{5} 0.6451{4} 0.739{6} 0.5743{3} 0.5543{2}
600
Mean 0.6268 0.6454 0.6372 0.6494 0.6366 0.6273 1.3569 1.3976 1.3196 1.4242 1.3545 1.2927 0.7355 0.8550 0.8533 0.8374 0.8078 0.8061
Rbias 0.0446{1} 0.0756{5} 0.0619{4} 0.0823{6} 0.0609{3} 0.0454{2} 0.5077{4} 0.5529{5} 0.4662{2} 0.5824{6} 0.505{3} 0.4364{1} 0.0806{6} 0.0687{5} 0.0666{4} 0.0467{3} 0.0097{2} 0.0076{1}
MSE 0.0046{1} 0.0082{5} 0.0059{4} 0.0085{6} 0.0058{3} 0.0049{2} 0.8632{2} 1.1555{5} 0.9226{3} 1.2071{6} 0.9417{4} 0.8249{1} 0.3665{1} 0.6464{6} 0.5514{4} 0.5709{5} 0.4839{3} 0.4578{2}
650
Mean 0.6235 0.6412 0.6338 0.6451 0.6332 0.6244 1.3701 1.4438 1.3829 1.4526 1.3666 1.2933 0.7015 0.7866 0.7672 0.7846 0.7732 0.7923
Rbias 0.0391{1} 0.0687{5} 0.0563{4} 0.0751{6} 0.0553{3} 0.0406{2} 0.5223{3} 0.6042{5} 0.5366{4} 0.614{6} 0.5184{2} 0.437{1} 0.1231{6} 0.0167{2} 0.041{5} 0.0192{3} 0.0334{4} 0.0096{1}
MSE 0.0047{1} 0.0084{5} 0.0058{4} 0.0084{6} 0.0057{3} 0.005{2} 0.9114{2} 1.2823{5} 1.0187{4} 1.2832{6} 0.9671{3} 0.8437{1} 0.3273{1} 0.5143{6} 0.4359{2} 0.4868{5} 0.4457{3} 0.4795{4}
700
Mean 0.6215 0.6397 0.6316 0.6429 0.6309 0.6227 1.3809 1.4494 1.3595 1.4304 1.3627 1.3070 0.6881 0.7882 0.7882 0.8123 0.7717 0.7763
Rbias 0.0358{1} 0.0662{5} 0.0527{4} 0.0715{6} 0.0516{3} 0.0379{2} 0.5344{4} 0.6104{6} 0.5106{2} 0.5894{5} 0.5141{3} 0.4522{1} 0.1398{6} 0.0147{1} 0.0148{2} 0.0153{3} 0.0353{5} 0.0297{4}
MSE 0.0041{1} 0.0074{5} 0.0051{4} 0.0076{6} 0.0049{3} 0.0045{2} 0.905{2} 1.2819{6} 1.0107{4} 1.2491{5} 0.9802{3} 0.847{1} 0.3146{1} 0.5686{5} 0.4402{3} 0.5789{6} 0.4093{2} 0.4547{4}

Alexandria Engineering Journal 100 (2024) 15–31
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H.E. Semary et al.
Table 5
Simulation results at 𝜃= 1.2,𝛼= 1.2and 𝛽= 0.4.
n𝜃 𝛼 𝛽
MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE
250
Mean 1.2520 1.2765 1.2611 1.3010 1.2624 1.2347 1.6228 1.7529 1.6455 1.7431 1.6094 1.5429 0.4315 0.4593 0.4583 0.4832 0.4707 0.4611
Rbias 0.0434{2} 0.0638{5} 0.0509{3} 0.0842{6} 0.052{4} 0.0289{1} 0.3523{3} 0.4607{6} 0.3713{4} 0.4526{5} 0.3412{2} 0.2858{1} 0.0786{1} 0.1482{3} 0.1457{2} 0.2081{6} 0.1767{5} 0.1528{4}
MSE 0.0575{1} 0.1031{6} 0.0741{4} 0.1026{5} 0.0685{3} 0.0631{2} 1.1846{2} 1.6184{5} 1.3133{4} 1.6267{6} 1.2048{3} 1.0374{1} 0.2588{1} 0.4694{6} 0.3139{3} 0.4234{5} 0.3157{4} 0.3067{2}
300
Mean 1.2571 1.2977 1.2783 1.3171 1.2752 1.2495 1.5833 1.7324 1.6339 1.7125 1.5964 1.5562 0.4173 0.4683 0.4531 0.5036 0.4659 0.4461
Rbias 0.0476{2} 0.0815{5} 0.0653{4} 0.0975{6} 0.0627{3} 0.0412{1} 0.3194{2} 0.4437{6} 0.3616{4} 0.4271{5} 0.3303{3} 0.2968{1} 0.0433{1} 0.1708{5} 0.1328{3} 0.2591{6} 0.1647{4} 0.1153{2}
MSE 0.0534{1} 0.094{5} 0.067{4} 0.0944{6} 0.0648{3} 0.0594{2} 0.9758{1} 1.4307{6} 1.1684{4} 1.4211{5} 1.0948{3} 0.9994{2} 0.1607{1} 0.4673{5} 0.254{2} 0.5024{6} 0.2584{3} 0.2609{4}
350
Mean 1.2500 1.2790 1.2686 1.2963 1.2678 1.2447 1.5964 1.7335 1.6402 1.6970 1.6027 1.5407 0.3901 0.4002 0.4111 0.4491 0.4304 0.4333
Rbias 0.0417{2} 0.0659{5} 0.0572{4} 0.0802{6} 0.0565{3} 0.0372{1} 0.3304{2} 0.4446{6} 0.3669{4} 0.4141{5} 0.3356{3} 0.284{1} 0.0246{2} 0.0004{1} 0.0277{3} 0.1228{6} 0.0761{4} 0.0833{5}
MSE 0.042{1} 0.0724{6} 0.0507{4} 0.0721{5} 0.0481{3} 0.0445{2} 0.9148{2} 1.3368{6} 1.0444{4} 1.2613{5} 0.956{3} 0.8823{1} 0.1342{1} 0.214{3} 0.1775{2} 0.3269{6} 0.2249{4} 0.2305{5}
400
Mean 1.2432 1.2703 1.2610 1.2867 1.2590 1.2366 1.5119 1.6286 1.5280 1.6278 1.5224 1.4979 0.4198 0.4518 0.4687 0.4666 0.4584 0.4300
Rbias 0.036{2} 0.0586{5} 0.0508{4} 0.0723{6} 0.0492{3} 0.0305{1} 0.2599{2} 0.3571{6} 0.2733{4} 0.3565{5} 0.2687{3} 0.2483{1} 0.0494{1} 0.1294{3} 0.1718{6} 0.1665{5} 0.1461{4} 0.0749{2}
MSE 0.0384{1} 0.0717{6} 0.0477{4} 0.071{5} 0.0466{3} 0.0442{2} 0.7841{1} 1.1379{5} 0.9{4} 1.1399{6} 0.8607{3} 0.7988{2} 0.117{1} 0.2669{6} 0.2185{4} 0.2599{5} 0.1881{3} 0.1496{2}
450
Mean 1.2383 1.2793 1.2610 1.2930 1.2579 1.2386 1.4719 1.6150 1.4990 1.6018 1.4837 1.4521 0.4326 0.4583 0.4732 0.4823 0.4734 0.4579
Rbias 0.0319{1} 0.066{5} 0.0509{4} 0.0775{6} 0.0482{3} 0.0321{2} 0.2266{2} 0.3458{6} 0.2491{4} 0.3348{5} 0.2364{3} 0.2101{1} 0.0816{1} 0.1458{3} 0.1831{4} 0.2056{6} 0.1834{5} 0.1447{2}
MSE 0.0329{1} 0.0585{6} 0.0381{4} 0.0582{5} 0.0378{3} 0.035{2} 0.715{1} 1.075{6} 0.8051{4} 1.0644{5} 0.7983{3} 0.7151{2} 0.1322{1} 0.2795{5} 0.2292{4} 0.301{6} 0.1952{2} 0.1986{3}
500
Mean 1.2348 1.2590 1.2491 1.2739 1.2471 1.2312 1.4224 1.5236 1.4451 1.5167 1.4325 1.4069 0.4400 0.4529 0.4625 0.4729 0.4617 0.4559
Rbias 0.029{2} 0.0492{5} 0.0409{4} 0.0616{6} 0.0393{3} 0.026{1} 0.1853{2} 0.2697{6} 0.2042{4} 0.2639{5} 0.1938{3} 0.1724{1} 0.1001{1} 0.1323{2} 0.1562{5} 0.1822{6} 0.1543{4} 0.1397{3}
MSE 0.0315{1} 0.0602{6} 0.0403{4} 0.0585{5} 0.0392{3} 0.035{2} 0.6171{1} 0.873{6} 0.6948{4} 0.8394{5} 0.6857{3} 0.6365{2} 0.1179{1} 0.1789{5} 0.1535{3} 0.2158{6} 0.1268{2} 0.1589{4}
550
Mean 1.2341 1.2600 1.2497 1.2704 1.2484 1.2298 1.4457 1.5691 1.4661 1.5625 1.4577 1.4087 0.4187 0.4248 0.4501 0.4391 0.4467 0.4441
Rbias 0.0284{2} 0.05{5} 0.0415{4} 0.0587{6} 0.0404{3} 0.0249{1} 0.2048{2} 0.3076{6} 0.2218{4} 0.3021{5} 0.2148{3} 0.174{1} 0.0467{1} 0.062{2} 0.1254{6} 0.0976{3} 0.1167{5} 0.1102{4}
MSE 0.0273{1} 0.0532{6} 0.0325{4} 0.0531{5} 0.0317{3} 0.0292{2} 0.5929{2} 0.8954{6} 0.6714{4} 0.8894{5} 0.6415{3} 0.5685{1} 0.0932{1} 0.1736{5} 0.1641{4} 0.1814{6} 0.1547{2} 0.1568{3}
600
Mean 1.2373 1.2533 1.2467 1.2647 1.2445 1.2276 1.4601 1.5207 1.4597 1.5152 1.4442 1.4090 0.4097 0.4349 0.4390 0.4495 0.4407 0.4367
Rbias 0.0311{2} 0.0444{5} 0.0389{4} 0.0539{6} 0.0371{3} 0.023{1} 0.2168{4} 0.2673{6} 0.2164{3} 0.2626{5} 0.2035{2} 0.1742{1} 0.0244{1} 0.0872{2} 0.0975{4} 0.1238{6} 0.1018{5} 0.0918{3}
MSE 0.0263{1} 0.0491{6} 0.0313{4} 0.0482{5} 0.0304{3} 0.0288{2} 0.5946{2} 0.8134{6} 0.6556{4} 0.8043{5} 0.6307{3} 0.5939{1} 0.092{1} 0.1832{5} 0.1336{4} 0.1915{6} 0.1238{2} 0.1238{3}
650
Mean 1.2337 1.2554 1.2472 1.2662 1.2462 1.2291 1.4059 1.5065 1.4135 1.4973 1.4016 1.3688 0.4261 0.4336 0.4607 0.4516 0.4599 0.4461
Rbias 0.028{2} 0.0462{5} 0.0393{4} 0.0552{6} 0.0385{3} 0.0243{1} 0.1716{3} 0.2554{6} 0.1779{4} 0.2477{5} 0.168{2} 0.1407{1} 0.0652{1} 0.0841{2} 0.1518{6} 0.1291{4} 0.1497{5} 0.1151{3}
MSE 0.0234{1} 0.0464{6} 0.0284{4} 0.0455{5} 0.0268{3} 0.0247{2} 0.5126{2} 0.7569{6} 0.5817{4} 0.7497{5} 0.5627{3} 0.4804{1} 0.0841{1} 0.161{5} 0.1593{4} 0.1742{6} 0.1356{3} 0.1218{2}
700
Mean 1.2249 1.2528 1.2413 1.2615 1.2402 1.2229 1.3873 1.4681 1.4127 1.4644 1.3949 1.3531 0.4189 0.4458 0.4378 0.4567 0.4452 0.4399
Rbias 0.0207{2} 0.044{5} 0.0344{4} 0.0512{6} 0.0335{3} 0.0191{1} 0.1561{2} 0.2234{6} 0.1772{4} 0.2203{5} 0.1624{3} 0.1276{1} 0.0473{1} 0.1144{5} 0.0945{2} 0.1417{6} 0.1131{4} 0.0997{3}
MSE 0.0229{1} 0.0422{5} 0.0268{4} 0.0422{6} 0.0256{3} 0.0233{2} 0.4933{2} 0.6903{6} 0.5576{4} 0.6866{5} 0.5326{3} 0.4781{1} 0.075{1} 0.1769{5} 0.104{3} 0.1796{6} 0.1112{4} 0.0957{2}

Alexandria Engineering Journal 100 (2024) 15–31
26
H.E. Semary et al.
Table 6
Simulation results at 𝜃= 1.2,𝛼= 1.2and 𝛽= 0.8.
n𝜃 𝛼 𝛽
MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE
250
Mean 1.2683 1.2902 1.2815 1.3066 1.2820 1.2618 1.9476 2.0544 1.9915 2.0020 1.9616 1.8343 0.7902 0.8627 0.8254 0.9489 0.8313 0.8870
Rbias 0.0569{2} 0.0752{5} 0.0679{3} 0.0889{6} 0.0684{4} 0.0515{1} 0.623{2} 0.712{6} 0.6596{4} 0.6684{5} 0.6347{3} 0.5285{1} 0.0122{1} 0.0783{4} 0.0318{2} 0.1861{6} 0.0391{3} 0.1088{5}
MSE 0.0332{1} 0.0497{5} 0.0387{4} 0.0517{6} 0.0368{3} 0.0343{2} 2.2517{2} 3.1128{6} 2.5677{4} 2.9617{5} 2.5021{3} 2.097{1} 0.9135{2} 1.2403{5} 1.0069{3} 1.4824{6} 0.8926{1} 1.2386{4}
300
Mean 1.2565 1.2869 1.2724 1.3005 1.2719 1.2508 1.9287 2.1068 2.0112 2.0874 1.9670 1.8353 0.7617 0.7791 0.7634 0.8247 0.7867 0.8379
Rbias 0.0471{2} 0.0724{5} 0.0603{4} 0.0837{6} 0.0599{3} 0.0423{1} 0.6073{2} 0.7557{6} 0.676{4} 0.7395{5} 0.6391{3} 0.5294{1} 0.0478{6} 0.0262{2} 0.0458{4} 0.0309{3} 0.0166{1} 0.0473{5}
MSE 0.0285{1} 0.0415{5} 0.0325{4} 0.0429{6} 0.0313{3} 0.0291{2} 2.2343{2} 2.9906{6} 2.5992{4} 2.9651{5} 2.4088{3} 2.1084{1} 0.7105{3} 0.9244{5} 0.637{1} 0.9799{6} 0.7086{2} 0.921{4}
350
Mean 1.2597 1.2839 1.2739 1.2950 1.2733 1.2557 1.8730 2.0062 1.9637 1.9761 1.9085 1.8199 0.7711 0.8053 0.7559 0.8555 0.7911 0.8155
Rbias 0.0497{2} 0.07{5} 0.0616{4} 0.0792{6} 0.0611{3} 0.0464{1} 0.5609{2} 0.6718{6} 0.6364{4} 0.6468{5} 0.5904{3} 0.5166{1} 0.0361{4} 0.0066{1} 0.0551{5} 0.0694{6} 0.0111{2} 0.0193{3}
MSE 0.0249{1} 0.0385{5} 0.0297{4} 0.0399{6} 0.0287{3} 0.0266{2} 1.9186{2} 2.5201{6} 2.177{4} 2.4869{5} 2.0212{3} 1.9116{1} 0.6654{2} 0.935{5} 0.6575{1} 0.9992{6} 0.7296{3} 0.816{4}
400
Mean 1.2514 1.2739 1.2632 1.2837 1.2627 1.2457 1.8460 1.9645 1.8680 1.9392 1.8513 1.7955 0.7649 0.7893 0.7968 0.8318 0.7969 0.7894
Rbias 0.0428{2} 0.0616{5} 0.0527{4} 0.0698{6} 0.0522{3} 0.0381{1} 0.5383{2} 0.6371{6} 0.5566{4} 0.616{5} 0.5427{3} 0.4962{1} 0.0439{6} 0.0134{4} 0.0039{2} 0.0398{5} 0.0039{1} 0.0132{3}
MSE 0.0213{2} 0.0319{5} 0.0239{4} 0.033{6} 0.023{3} 0.0208{1} 1.7825{2} 2.4111{5} 1.9574{4} 2.4363{6} 1.8473{3} 1.7708{1} 0.6653{2} 0.8246{6} 0.715{4} 0.8172{5} 0.7129{3} 0.6559{1}
450
Mean 1.2495 1.2781 1.2659 1.2874 1.2634 1.2469 1.8298 1.9366 1.8877 1.9044 1.8505 1.7552 0.7144 0.7860 0.7368 0.8361 0.7434 0.7758
Rbias 0.0413{2} 0.0651{5} 0.0549{4} 0.0728{6} 0.0528{3} 0.0391{1} 0.5248{2} 0.6138{6} 0.5731{4} 0.587{5} 0.5421{3} 0.4627{1} 0.1069{6} 0.0175{1} 0.079{5} 0.0452{3} 0.0707{4} 0.0302{2}
MSE 0.0202{1} 0.0325{5} 0.0235{4} 0.0334{6} 0.0224{3} 0.0208{2} 1.6508{2} 2.2127{6} 1.8396{4} 2.1758{5} 1.7828{3} 1.5903{1} 0.4142{1} 0.7981{5} 0.5527{3} 0.8717{6} 0.4644{2} 0.5595{4}
500
Mean 1.2440 1.2729 1.2606 1.2808 1.2591 1.2430 1.7203 1.8717 1.7945 1.8522 1.7559 1.6732 0.7598 0.7747 0.7639 0.8058 0.7825 0.8076
Rbias 0.0366{2} 0.0608{5} 0.0505{4} 0.0674{6} 0.0493{3} 0.0358{1} 0.4336{2} 0.5597{6} 0.4955{4} 0.5435{5} 0.4633{3} 0.3943{1} 0.0502{6} 0.0316{4} 0.0451{5} 0.0073{1} 0.0219{3} 0.0095{2}
MSE 0.0166{1} 0.0268{5} 0.0199{4} 0.0277{6} 0.0192{3} 0.0173{2} 1.3345{2} 1.8525{6} 1.523{4} 1.8074{5} 1.4372{3} 1.2925{1} 0.4286{1} 0.6101{5} 0.4732{3} 0.6447{6} 0.4712{2} 0.5931{4}
550
Mean 1.2452 1.2672 1.2575 1.2748 1.2571 1.2435 1.7979 1.8775 1.7969 1.8606 1.7883 1.7291 0.6947 0.7322 0.7455 0.7650 0.7450 0.7632
Rbias 0.0377{2} 0.056{5} 0.0479{4} 0.0624{6} 0.0476{3} 0.0363{1} 0.4983{4} 0.5646{6} 0.4974{3} 0.5505{5} 0.4903{2} 0.4409{1} 0.1316{6} 0.0847{5} 0.0681{3} 0.0437{1} 0.0688{4} 0.046{2}
MSE 0.0152{1} 0.0239{5} 0.0183{4} 0.0248{6} 0.0179{3} 0.0161{2} 1.4249{2} 1.7969{6} 1.4946{4} 1.7945{5} 1.4401{3} 1.3969{1} 0.3206{1} 0.4503{4} 0.4215{2} 0.4769{5} 0.4338{3} 0.4902{6}
600
Mean 1.2462 1.2665 1.2566 1.2734 1.2565 1.2418 1.7552 1.8360 1.7852 1.8261 1.7706 1.6979 0.7316 0.7653 0.7441 0.7885 0.7521 0.7640
Rbias 0.0385{2} 0.0554{5} 0.0471{4} 0.0612{6} 0.0471{3} 0.0348{1} 0.4626{2} 0.53{6} 0.4877{4} 0.5218{5} 0.4755{3} 0.4149{1} 0.0856{6} 0.0434{2} 0.0698{5} 0.0144{1} 0.0598{4} 0.045{3}
MSE 0.0149{1} 0.0236{5} 0.0178{4} 0.0244{6} 0.0175{3} 0.0156{2} 1.3522{2} 1.7725{6} 1.4592{4} 1.7316{5} 1.4144{3} 1.3193{1} 0.4381{1} 0.5541{5} 0.456{2} 0.6074{6} 0.4718{4} 0.4638{3}
650
Mean 1.2401 1.2624 1.2513 1.2691 1.2511 1.2385 1.7040 1.7908 1.7177 1.7737 1.7065 1.6832 0.7104 0.7368 0.7353 0.7669 0.7395 0.7307
Rbias 0.0334{2} 0.052{5} 0.0427{4} 0.0576{6} 0.0425{3} 0.0321{1} 0.42{2} 0.4924{6} 0.4314{4} 0.4781{5} 0.4221{3} 0.4026{1} 0.112{6} 0.0789{3} 0.0809{4} 0.0414{1} 0.0756{2} 0.0867{5}
MSE 0.0119{1} 0.0193{5} 0.0141{4} 0.0198{6} 0.0138{3} 0.0125{2} 1.1004{1} 1.4951{6} 1.2033{4} 1.4618{5} 1.1925{3} 1.1477{2} 0.3595{3} 0.3913{5} 0.3447{2} 0.4403{6} 0.3276{1} 0.3896{4}
700
Mean 1.2412 1.2636 1.2514 1.2694 1.2510 1.2379 1.7157 1.8785 1.7763 1.8846 1.7601 1.6808 0.7292 0.7154 0.7277 0.7203 0.7343 0.7584
Rbias 0.0343{2} 0.053{5} 0.0428{4} 0.0578{6} 0.0425{3} 0.0316{1} 0.4298{2} 0.5654{5} 0.4802{4} 0.5705{6} 0.4668{3} 0.4007{1} 0.0885{3} 0.1058{6} 0.0904{4} 0.0997{5} 0.0821{2} 0.052{1}
MSE 0.0125{1} 0.0213{5} 0.0156{4} 0.0219{6} 0.0153{3} 0.0138{2} 1.1097{1} 1.7594{5} 1.4008{4} 1.7881{6} 1.3399{3} 1.2349{2} 0.4446{5} 0.4548{6} 0.3793{1} 0.4234{4} 0.3964{2} 0.4231{3}

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H.E. Semary et al.
Table 7
Summary of all ranks.
Initial parameters n 𝜃 𝛼 𝛽
MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE MLE LSE WLSE CME ADE RADE
𝜃= 0.6,𝛼= 0.5and 𝛽= 0.4
250 6 2 5 2 2 4 2.5 6 2.5 4.5 4.5 1 3.5 6 1 3.5 2 5
300 6 4.5 1.5 4.5 1.5 3 2 6 4 5 3 1 1 5.5 5.5 3 3 3
350 6 5 2 3.5 1 3.5 1 6 4 5 3 2 1 5.5 3 5.5 3 3
400 6 5 3.5 1.5 1.5 3.5 1 6 4 5 3 2 1 5 6 2 3.5 3.5
450 6 5 3 2 1 4 1 6 4.5 4.5 3 2 3.5 3.5 3.5 3.5 3.5 3.5
500 6 5 3.5 2 1 3.5 1.5 6 4 5 3 1.5 1 3.5 3.5 3.5 6 3.5
550 3.5 6 1.5 5 1.5 3.5 1 6 4 5 3 2 1.5 6 4 1.5 4 4
600 6 5 2 2 2 4 1 6 4 5 3 2 1 3.5 3.5 6 3.5 3.5
650 3.5 6 1.5 5 1.5 3.5 1 6 4 5 3 2 1 6 3.5 5 2 3.5
700 6 5 2 2 2 4 1 6 4 5 2.5 2.5 3.5 6 1.5 5 1.5 3.5
𝜃= 0.6,𝛼= 0.9and 𝛽= 0.4
250 1.5 5 4 6 3 1.5 2 5.5 4 5.5 3 1 1 5.5 2.5 5.5 4 2.5
300 1.5 5 4 6 3 1.5 2 5 4 6 3 1 2 5.5 3 5.5 1 4
350 1 5 4 6 3 2 1.5 6 4 5 3 1.5 1 4.5 2 6 3 4.5
400 1.5 5 4 6 3 1.5 2 6 4 5 3 1 2 4.5 3 6 4.5 1
450 1 5 4 6 3 2 2 5.5 4 5.5 3 1 2 5 3 6 4 1
500 1.5 5.5 4 5.5 3 1.5 2 6 4 5 3 1 3 4 1.5 6 1.5 5
550 1.5 5 4 6 3 1.5 2 6 4 5 3 1 1 3.5 5 6 2 3.5
600 1.5 5 4 6 3 1.5 1.5 6 4 5 3 1.5 2.5 5 1 6 2.5 4
650 1.5 5.5 4 5.5 3 1.5 2 6 4 5 3 1 4 5 2 6 2 2
700 1.5 5.5 4 5.5 3 1.5 2 6 4 5 3 1 2.5 4.5 1 6 4.5 2.5
𝜃= 0.6,𝛼= 0.9and 𝛽= 0.8
250 1.5 5 3.5 6 3.5 1.5 1.5 5.5 4 5.5 3 1.5 2 5 4 6 1 3
300 1 5 4 6 3 2 2 5 4 6 3 1 1 5 3 6 4 2
350 1 5 4 6 3 2 1.5 6 4 5 3 1.5 1 4.5 3 6 2 4.5
400 1.5 5 4 6 3 1.5 2 4.5 4.5 6 3 1 1.5 6 1.5 5 3.5 3.5
450 1 5 4 6 3 2 2 5 3 6 4 1 2 6 5 4 1 3
500 1 5 4 6 3 2 2 5 4 6 3 1 3.5 6 3.5 5 1.5 1.5
550 1.5 5 4 6 3 1.5 2.5 6 2.5 5 4 1 3 4.5 4.5 6 2 1
600 1 5 4 6 3 2 3 5 2 6 4 1 3 6 4.5 4.5 2 1
650 1 5 4 6 3 2 2.5 5 4 6 2.5 1 3 5.5 3 5.5 3 1
700 1 5 4 6 3 2 3 6 3 5 3 1 3.5 2 1 6 3.5 5
𝜃= 1.2,𝛼= 1.2and 𝛽= 0.4
250 1.5 5.5 3.5 5.5 3.5 1.5 2.5 5.5 4 5.5 2.5 1 1 4.5 2 6 4.5 3
300 1.5 5 4 6 3 1.5 1.5 6 4 5 3 1.5 1 5 2 6 4 3
350 1.5 5.5 4 5.5 3 1.5 2 6 4 5 3 1 1 2 3 6 4 5
400 1.5 5.5 4 5.5 3 1.5 1.5 5.5 4 5.5 3 1.5 1 4 5.5 5.5 3 2
450 1 5.5 4 5.5 3 2 1.5 6 4 5 3 1.5 1 4.5 4.5 6 3 2
500 1.5 5.5 4 5.5 3 1.5 1.5 6 4 5 3 1.5 1 3.5 5 6 2 3.5
550 1.5 5.5 4 5.5 3 1.5 2 6 4 5 3 1 1 3 6 5 3 3
600 1.5 5.5 4 5.5 3 1.5 3 6 4 5 2 1 1 3.5 5 6 3.5 2
650 1.5 5.5 4 5.5 3 1.5 2.5 6 4 5 2.5 1 1 3 5.5 5.5 4 2
700 1.5 5 4 6 3 1.5 2 6 4 5 3 1 1 5 2.5 6 4 2.5
𝜃= 1.2,𝛼= 1.2and 𝛽= 0.8
250 1.5 5 3.5 6 3.5 1.5 2 6 4 5 3 1 1 4.5 3 6 2 4.5
300 1.5 5 4 6 3 1.5 2 6 4 5 3 1 5 3 2 5 1 5
350 1.5 5 4 6 3 1.5 2 6 4 5 3 1 3 3 3 6 1 5
400 2 5 4 6 3 1 2 5.5 4 5.5 3 1 4 5.5 3 5.5 1.5 1.5
450 1.5 5 4 6 3 1.5 2 6 4 5 3 1 4 2 5 6 2 2
500 1.5 5 4 6 3 1.5 2 6 4 5 3 1 3.5 6 5 3.5 1 2
550 1.5 5 4 6 3 1.5 3 6 4 5 2 1 3.5 6 1 2 3.5 5
600 1.5 5 4 6 3 1.5 2 6 4 5 3 1 3.5 3.5 3.5 3.5 6 1
650 1.5 5 4 6 3 1.5 1.5 6 4 5 3 1.5 5.5 4 2 3 1 5.5
700 1.5 5 4 6 3 1.5 1.5 5 4 6 3 1.5 4 6 3 5 1.5 1.5
𝑅𝑎𝑛𝑘𝑠 110.5 254 184 264 136.5 101 94 288.5 194 260 150.5 63 109.5 229.5 164 255 140 152.5
Table 8
Some descriptive analysis of all data sets.
Data1 Data2
N 61 100
Mean 22.623 221.980
Median 21.000 195.500
Variance 160.339 20 914.383
Skewness 1.097 1.381
Kurtosis 1.473 3.071
Range 60.000 814.000
Minimum 3.000 15.000
Maximum 63.000 829.000
Sum 1380.000 22198.000
𝜌5and 𝜌6statistics for the datasets. Further, Tables 9 and 11 display
the MLEs and standard errors (SEs) (appear in parentheses) of the
parameters of the ABPE, HLMK_E, MK_E, NAP_E, API_E, PMK_E, E_E,
OLLW, OLLEW and WPo models. In Tables 10 and 12, we compare the
fits of the ABPE, HLMK_E, MK_E, NAP_E, API_E, PMK_E, E_E, OLLW,
OLLEW and WPo models. The figures in these tables indicate that the
ABPE distribution has the lowest values of Lnl, 𝜌1,𝜌2,𝜌3,𝜌5,𝜌6and
largest 𝜌4, among all fitted models. Fig. 5 shows the box plot for the
both datasets. Fig. 6 demonstrates the dataset’s TTT plot. Figs. 7 and
8shows the estimated pdfs, cdfs, complementary cdf (ccdfs) and PP
plots of competitive models for the datasets. The graphs in Figs. 7 and
8demonstrate that the ABPE model fits the both datasets well.
7. Concluding remarks
In this article, we have introduced a new method to generate
new statistical distributions. We have applied the proposed method to
the exponential family of distributions, and a new three parameters
distribution, entitled Alpha Beta power exponential distribution has
been studied in detail. The proposed model have several desirable

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H.E. Semary et al.
Fig. 5. Box plot for the datasets.
Fig. 6. TTT plot for the datasets.
Table 9
Estimates and SEs of the competitive models for the Data1.
Model 
𝜃 𝛼 
𝛽
𝜆 𝑆𝐸 (
𝜃)𝑆𝐸 (𝛼)𝑆 𝐸(
𝛽)𝑆𝐸 (
𝜆)
ABPE 0.100 73.025 70.173 0.011 745.181 730.020
NAP_E 41.683 0.096 33.320 0.009
PMK_E 67.338 0.649 0.020 NaN NaN NaN
E_E 3.721 0.089 0.804 0.011
HLMK_E 0.718 3.336 0.041 0.432 2.584 0.025
MK_E 1.353 0.026 0.137 0.002
API_E 0.008 35.202 0.009 4.318
OLLW 1.647 27.380 1.275 0.798 3.494 0.562
OLLEW 15.582 1.291 2.461 0.942 7.532 0.509 2.052 0.493
WPo 0.956 0.110 6.063 0.265 0.056 3.422
Table 10
Measures of fitting for Data1.
Model -Lnl 𝜌1𝜌2𝜌3𝜌4𝜌5𝜌6
ABPE 233.608 473.216 473.637 0.073 0.902 0.034 0.218
NAP_E 234.736 473.472 473.679 0.084 0.779 0.038 0.230
PMK_E 235.308 476.616 477.037 0.089 0.715 0.053 0.379
E_E 235.758 474.517 474.724 0.093 0.671 0.059 0.326
HLMK_E 234.933 475.865 476.286 0.093 0.669 0.052 0.350
MK_E 237.969 479.937 480.144 0.127 0.281 0.116 0.812
API_E 243.443 490.885 491.092 0.159 0.091 0.289 1.686
OLLW 234.387 474.775 475.196 0.074 0.896 0.035 0.228
OLLEW 234.525 477.051 477.766 0.082 0.812 0.046 0.276
WPo 234.657 475.315 475.736 0.079 0.844 0.039 0.232
statistical properties and is able to model lifetime data with decreasing
and bathtub shaped failure rates. A real data set has been analyzed
and the analytical measures of the ABPE distribution have been com-
pared with other well-known statistical distributions. Although, it is
not guaranteed that the proposed method will always provide best fit.
But, at least in certain situations, the subject model may work better
than the other well-known aging distributions. For future works; We
hope that our new model might attract wider range of applications in
reliability and lifetime data analysis. It will be more interesting to study
the characteristics of bivariate extension of the proposed distribution to
analyze lifetime data and develop inferential procedures. Also, many

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H.E. Semary et al.
Table 11
Estimates and SEs of the competitive models for the Data2.
Model 
𝜃 𝛼 
𝛽
𝜆 𝑆𝐸 (
𝜃)𝑆𝐸 (𝛼)𝑆 𝐸(
𝛽)𝑆𝐸 (
𝜆)
ABPE 0.009 43.201 0.962 0.001 49.753 5.180
WPo 1.093 0.008 3.090 0.190 0.002 1.372
PMK_E 41.470 0.595 0.028 NaN NaN NaN
NAP_E 13.241 0.009 7.843 0.001
OLLW 1.338 259.019 1.280 0.388 22.456 0.325
HLMK_E 0.718 2.507 0.004 0.160 0.717 0.001
E_E 2.386 0.007 0.360 0.001
MK_E 1.125 0.003 0.095 0.000
OLLEW 0.030 0.194 148.846 1.485 0.022 0.020 119.031 0.422
API_E 10 152.030 14.996 NaN 0.000
Table 12
Measures of fitting for Data2.
Model -Lnl 𝜌1𝜌2𝜌3𝜌4𝜌5𝜌6
ABPE 623.727 1253.454 1253.704 0.067 0.669 0.062 0.380
WPo 624.670 1255.339 1255.589 0.072 0.655 0.072 0.396
PMK_E 625.261 1256.522 1256.772 0.076 0.617 0.087 0.516
NAP_E 625.789 1255.577 1255.701 0.077 0.601 0.071 0.418
OLLW 624.556 1255.112 1255.362 0.079 0.566 0.085 0.481
HLMK_E 626.298 1258.596 1258.846 0.081 0.527 0.115 0.670
E_E 625.693 1255.386 1255.510 0.103 0.237 0.146 0.790
MK_E 630.098 1264.197 1264.321 0.113 0.157 0.162 1.026
OLLEW 635.149 1278.297 1278.718 0.144 0.033 0.475 2.607
API_E 651.916 1307.833 1307.957 0.218 0.000 0.704 3.859
Fig. 7. Estimated PDF, CDF, SF and pp plots of Data1.
Fig. 8. Estimated PDF, CDF, SF and pp plots of Data2.

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30
H.E. Semary et al.
authors can used the suggested new family to generate more new
statistical models.
Funding statement
This work was supported and funded by the Deanship of Scientific
Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)
(grant number IMSIU-RPP2023085).
CRediT authorship contribution statement
H.E. Semary: Formal analysis, Data curation. Zawar Hussain:
Methodology. Walaa A. Hamdi: Software, Conceptualization. Maha
A. Aldahlan: Writing – original draft. Ibrahim Elbatal: Writing –
review & editing, Writing – original draft. Vasili B.V. Nagarjuna:
Conceptualization.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
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