EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5737
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Statistical Inference Based on Censored Data of
Entropy for Lomax Distribution

Samah M. Ahmed1, Gamal M. Ismail2,∗

1 Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
2 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah,
42351, Saudi Arabia

Abstract. The current research centers around entropy. This paper investigates the estimation
of entropy for the Lomax (Lo) distribution using an adaptive progressive Type-II censored data.
The entropy maximum likelihood estimate is computed and the bootstrap confidence intervals
of entropy are displayed, approximate confidence intervals are constructed using the asymptotic
normality of maximum likelihood estimation and the observed Fisher information matrix. The
Bayes entropy estimator is demonstrated using the symmetric and asymmetric loss functions. To
further assess the performance of the entropy estimators, particularly under various loss functions,
such as linear exponential and squared error, the posterior distribution was calculated. Then, using
Monte Carlo simulations, various approaches are compared to identify the believable intervals of
the entropy’s highest posterior density. Lastly, the recommended methods are illustrated using a
numerical example.

2020 Mathematics Subject Classifications: 62F10, 62F15, 62F40

Key Words and Phrases: Lomax model, Entropy; Maximum likelihood estimator, Bootstrap
resampling, Adaptive progressive Type-II censored data, Bayes estimator, symmetric and asym-
metric loss functions

1. Introduction

Entropy is a quantitative measure of uncertainty within a probability distribution.
Entropy is a measure that reflects the expected value of information contained within
a random variable. A higher entropy value signifies a lower information content within
the observed data. The measurement of entropy is a significant issue in numerous areas,
including statistics, economics, information technology, physics, and the analysis of biolog-
ical phenomena. For example, entropy can be used to evaluate the probability distribution
of electric charge among atoms under specific conditions. This wide applicability has led to
extensive research on entropy. The concept of entropy, as a measure of information that

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5737

Email addresses: sam 8711@yahoo.com (S. M. Ahmed), gismail@iu.edu.sa (G. M. Ismail)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 2 of 22

provides a quantitative measure of uncertainty, was introduced by Shannon [1]. Wong
and Chan [2] demonstrated that entropy decreases when data is ordered. AboEleneen
[3] simplified the entropy calculation and derived recurrence relations using progressively
Type-II censored samples. Sunoj et al. [4] introduced a Shannon entropy function based
on quantiles and investigated the properties of the residual entropy function. Seo et al. [5]
obtained an entropy estimator using upper record values from the generalized half-logistic
distribution. Cho et al. [6] discussed an estimation of the entropy for a Rayleigh distri-
bution based on doubly-generalized Type-II hybrid censored samples. The entropy under
Type-II censored data and doubly generalized Type-II hybrid censoring are examined by
Zhao et al. [7]. Cho et al. [8] investigated Bayesian estimators of entropy for the Weibull
distribution using a generalized progressive hybrid censoring scheme. Similarly, Lee [9]
studied maximum likelihood and Bayesian estimators of entropy for the inverse Weibull
distribution under a generalized progressive hybrid censoring scheme. Almohaimeed [10]
derived an exact expression for entropy information within both types of progressively
hybrid censored data, applying it to the exponential distribution. Chacko and Asha [11]
explored entropy estimation for the generalized exponential distribution using record val-
ues. Hassan and Zaky [12] focused on obtaining the maximum likelihood estimator of
Shannon entropy for the inverse Weibull distribution using multiple censored data. Wang
and Gui [13] examined certain entropy inferences based on Type-II progressive for Burr
Type-XI distribution.

The ideal method for gathering data depends on the test’s cost and duration. Common
and basic filtering methods are referred to in the literature as censoring schemes (CSs)
of Type-I and Type-II. The test duration is suggested under Type-I CS, however, there
are an arbitrary number of failures r, 0 < r < n, but the test time is random. However,
the test time T in the Type-II CS is arbitrary and might be T → ∞. Nevertheless, the
removal of units at locations other than the experiment’s termination point is not flexible
enough to be permitted by the traditional Type-I and Type-II censoring procedures. One
of the more general filtering techniques known as progressive CSs that were developed as
a result of a consequence of this inflexibility is the development of the progressive Type-II
censoring scheme. Refer to Balakrishnan and Aggarwala [14]. The progressive Type-II
censoring model’s operation is demonstrated in the following. For each of the n devices that
undergo a life test, write X1, X2, ..., Xn as its corresponding lifetime. The Progressive
Type-II censoring scheme R =R1, R2, ..., Rr, and the number of units observed r (r <
n) are determined before the experiment 0 < Ri, i = 1, 2, ..., r and n = r +

∑r−1
i=1 Ri .

The remaining Ri units are arbitrarily removed from the experiment once the ith failure
has been located. The experiment will continue as long as this guideline is adhered to,
ending when failures are identified. Consequently, Xi: r: n, i = 1, 2, ...., r are the Type-II
progressive right censoring’s observed statistics. Ng et al. [15] propose an adaptive Type-
II progressive censoring to shorten the test duration overall and increase the efficacy of
statistical inference. Here’s how this plan works: Think about conducting a life test on n
identical units. The observed number of failures r (r < n) is predetermined, and the test
length is permitted to exceed the predefined period T . The description of the progressive
censorship R is eliminated from the test once the ith failure is observed during the life



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 3 of 22

test. The r fully observed lifetimes are represented by the notation Xi: r: n, i = 1, 2, ...., r.
If the rth failure time occurs before time T (i.e. Xi: r: r < T ), the test ends at time Xi: r: n

using the same progressive censoring technique (R1, R2, ..., Rr). In this case, Rr = n
−r −

∑r−1
i=1 Ri. If Jth failure time happens before time T , i.e., XJ : r: r < T < XJ+1: r: r,

(1 < J < r−1). In the case of failures, the number of units progressively removed from the
test is then modified by setting RJ+1 = RJ+2 = ... = Rr−1 = 0. At time Rr, any leftover

units Rr are then eliminated, where Rr =n−r −
∑J

i=1Ri. Thus, the progressive censored

scheme that performs best in this situation is (R1, R2, ..., RJ , 0, 0, 0, n −r−
∑J

i=1Ri). We
use Xi in this study rather than Xi:r:n,i = 1, 2, ..., r. The observed data under-considered
censoring strategy may be represented by one of the two scenarios below:

Case 1: (X1, R1), (X2, R2), ..., (Xr, Rr), if Xr < T , where Rr = n −r −
∑J−1

i=1 Ri,
Case 2: (X1, R1), (X2, R2), ..., (XJ , RJ), (XJ+1, 0), ...(Xr−1, 0), (Xr, Rr), if XJ < T <

XJ+1.
It should be noted that Type-II and Type-II progressive censoring systems are based

on the adaptive Type-II censored scheme. Although the adaptive The Type-II censored
scheme is reduced to a Type-II censoring scheme. No units will be removed if T = 0, J = 0,
and if T = ∞, J = m, the adaptive Type-II censored scheme is the same as the Type-II pro-
gressive censored scheme with the pre-fixed progressive censored scheme Ri(i = 1, 2, ...,m),
survival units will be randomly removed during the trial. Discussions about the adaptive
Type-II censored system have been ongoing. As an example, Sobhi and Soliman [16] exam-
ined exponentiated Weibull hazard functions, reliability, and parameter estimation. Using
the Type-II censored adaptive technique, Nassar and Abo-Kasem [17] created Bayes and
machine learning estimations for the inverse Weibull distribution’s unknown parameters.
According to the adaptive Type-II censored scheme, Sewailem and Baklizi [18] looked at
the Bayes and ML estimates for the log-logistic distribution parameters. Xu and Gui [19]
proposed the adaptive Type-II censored technique to calculate entropy estimations for in-
verse Weibull distributions. Panahi and Moradi [20] computed the parameters of an expo-
nentiated inverted Rayleigh model. Chen and Gui’s [21] study focused on Chen’s adaptive
progressive Type-II censoring model. They considered the Kumarswamy-exponential dis-
tribution in Mohan and Chacko’s [22] adaptive progressive Type-II censoring algorithm.
Hora et al. [23] examined the Bayesian and classical outcomes for unknown parameters of
the inverse Lomax distribution under the adaptive progressive Type-II censoring scheme.
Amein et al. [24] investigated several estimation techniques using the adaptive Type-II
progressive censored sample from the Gompertz distribution. By setting Xi = Xr: r: n,
i = 1, ..., r. To keep things simple, the following can be used to express the probability
function adaptive Type-II progressive censored data:

ℓ(x | Ω) = cJ

r∏
i=1

f(xi | Ω)
J∏

i=1

(S(xi | Ω))Ri(S(xr | Ω))R
∗
, (1)

where cJ =
r∏

i=1
[n − i + 1 −

min{i−1,J}∑
k=1

Rk], and the vector of the unknown parameters is

denoted by Ω = (β, ξ).



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 4 of 22

This study’s main goal is to examine the adaptive progressively Type-II from the
entropy of Lo distribution. since there aren’t many relevant works that discuss this subject.
It discussed as follows:

• The entropy function is estimated using the maximum likelihood approach.

• We calculate the maximum likelihood estimator for the parameters and entropy
numerically, which considers the Newton-Rapshon.

• We discuss interval estimation using the bootstrap approach and the approximation
information matrix methods.

• assuming that the scale and shape parameters in the model each follow different
Gamma priors, the Metropolis-Hasting method is then used to get the Bayes esti-
mators and associated credible intervals.

• Lastly, the performance of estimations utilizing Monte Carlo simulation is assessed
using mean squared error. On the other hand, average length and probability cov-
erage are used in interval estimate.

The rest of the document is structured as follows: The model and its underlying
assumptions are presented in Section 2. Section 2 provided the maximum likelihood es-
timate (MLE) and Bayesian analysis applying the Markov Chain Monte Carlo (MCMC),
the squared error (SE) function and the linear-exponential (LINEX) loss function. Section
2, the estimated approximate confidence intervals derived from the MLEs, the bootstrap
interval and the highest posterior density (HPD) credible interval are derived. We do a
simulated analysis and simulate a data set to illustrate the estimate methods covered in
this paper and examine actual data in Section 3. Section 4 contains the final remarks.

2. Methodology

In this particular instance, unit lifetime has a Lo distribution in the model that is
being developed. The point estimates of entropy are created using the Bayesian and MLE
approaches. Additionally, interval estimators are developed using bootstrap methods,
HPD credible intervals, and the asymptotic property of MLEs.

2.1. The Lomax distribution model

If X has the Lo distribution Next, the probability density function (PDF) and cumu-
lative distribution function (CDF) of X are given by:

f(x) = β ξβ(ξ + x)−β−1 , β , ξ, x > 0, (2)

F (x) = 1− ξβ(ξ + x)−β, β , ξ, x > 0, (3)

where the scale parameter is represented by ξ and the shape parameter by β. This
distribution was presented as a model for corporate failure data by Lo [25]. Chahkandi



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 5 of 22

and Ganjali [26] state that it is a member of the class of distributions with declining failure
rates. When the researcher suspects a heavy-tailed distribution in the population, it offers
a very good substitute for popular lifespan distributions including in Gamma Weibull, or
exponential (see [27]. This distribution’s genesis, various features, and application areas
are described in [28, 29]. The literature contains a wide range of applications for the Lo
distribution. For instance, it has been widely utilized for life testing and dependability
modeling; take Balkema and de Haan [30] as an example. Ahsanullah [31] evaluated
the Lo’s record values distribution. Balakrishnan and Ahsanullah [32] proposed some
recurring linkages between record value moments from the Lo distribution. Using both
complete and censored samples, a number of authors have tackled inferential problems
for the Lo distribution. Childs et al. [33] have looked at nonidentical right-truncated
Lo random variables’ order statistics. Howlader and Hossain [34] looked at Bayesian
estimation of the Lo distribution’s survival function. Ghitany et al. [35] took into account
the expanded Lo distribution and Marshall-Olkin method. Elfattah et al. [36] used the
progressive Type-I censoring using the Lo distribution to determine the Bayesian and
non-Baysian estimators for the same sample size. ML and approximation ML predictors
derived from the Pareto distribution using multistage progressive filtering, as well as best
linear unbiased predictors, were among the several failure time predictors covered by
Raqab et al. [37]. Using progressive Type-II censoring, the optimal censoring technique
for determining the parameters of the Lo distribution was investigated by Cramer and
Schmiedt [38]. Asgharzadeh and Valiollahi [39] used progressive Type-II censoring to
create the Bayesian estimator of the scale parameter of the Lo distribution.

2.2. Modeling

The famous Shannon information entropy is defined using the first two moments of a
random variable X, as follows:

H(f) = H(X) = −
∞∫
0

f(x) log f(x)dx, (4)

where f(x) is the PDF of a continuous random variable X.
Theorem
CDF requires that X be a random variable, its entropy would be

H(f) = (β + 1)[log(ξ) +
1

β
]− log(βξβ). (5)

Proof.

H(f) = −
∞∫
0

β ξβ(ξ + x)−β−1[log(β ξβ)− (β + 1) log(ξ + x)]dx

=

∞∫
0

β(β + 1) ξβ(ξ + x)−β−1 log(ξ + x)dx−
∞∫
0

β ξβ(ξ + x)−β−1 log(β ξβ)dx



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 6 of 22

= (β + 1)[log(ξ) +
1

β
]− log(β ξβ), (6)

where
∞∫
0

β ξβ(ξ + x)−β−1 log(β ξβ)dx =

∞∫
0

f(x) log(β ξβ)dx = log(β ξβ). (7)

Using integration by parts, we have:

∞∫
0

β(β + 1) ξβ(ξ + x)−(β+1) log(ξ + x)dx = (β + 1) log(ξ) +

∞∫
0

(β + 1) ξβ(ξ + x)−(β+1)dx

= (β + 1) log(ξ) +

∞∫
0

(β + 1)

β
f(x)dx

= (β + 1) log(ξ) +
(β + 1)

β

= (β + 1)[log(ξ) +
1

β
]. (8)

Consequently, the following expression must be used to represent the Shannon entropy of
the Lo distribution, which depends on the parameters β and ξ.

H(f) = (β + 1)[log(ξ) +
1

β
]− log(β ξβ). (9)

2.3. Point Estimation

2.3.1. Entropy Maximum Likelihood Estimator

To determine the point estimation, for the Lo distribution, let x = (x1:r:n < x2:r:n < ... <
xr:r:n) be an adaptive Type-II progressive censored order statistic. The likelihood function
is provided by Eqs. (1), (2), and (3), which is

ℓ (β, ξ | x) =
r∏

i=1

β ξβ(ξ + xi)
−β−1 ×

J∏
i=1

(
ξβ(ξ + xi)

−β
)Ri

×
(
ξβ(ξ + xr)

−β
)R∗

r
, (10)

where,

R∗
r = n− r −

J∑
i=1

Ri, i = 1, 2, ..., r. (11)

This allows us to express the likelihood function’s logarithm as

L (β, ξ | x) = r log β + (r +
J∑

i=1

Ri)β log ξ − (β + 1)
r∑

i=1

log(ξ + xi)



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 7 of 22

− β
J∑

i=1

Ri log(ξ + xi)− βR∗
r log(ξ + xr). (12)

The likelihood formulas are computed as

∂L (β, ξ | x)
∂β

=
r

β
+(r+

J∑
i=1

Ri) log ξ −
r∑

i=1

log(ξ+xi)−
J∑

i=1

Ri log(ξ+xi)−R∗
r log(ξ+xr) = 0,

(13)

∂L (β, ξ | x)
∂ξ

=

(r +
J∑

i=1
Ri)β

ξ
−(β+1)

r∑
i=1

1

(ξ + xi)
−β

J∑
i=1

Ri

(ξ + xi)
−β

R∗
r

(ξ + xr)
= 0, (14)

The nonlinear Eqs. (13) and (14) Newton’s method can be used to numerically solve it
and find the parameters’ MLEs β and ξ. Thus, the MLE of H(f) equals

Ĥ(f) = − log(β̂ ξ̂β̂) + (β̂ + 1)[log(ξ̂) +
1

β̂
], (15)

where Ĥ(f) as given in Eq. (5) after replacing β and ξ by β̂ and ξ̂, respectively.

2.3.2. Entropy Bayes Estimations Using Markov Chain Monte Carlo

This paper with using an adaptive progressive Type-II censored data computes the Bayesian
estimate of entropy where the informative prior is driven by Gamma priors under both
symmetric and asymmetric loss functions. When computing the Bayes estimators, al-
though any other loss function can be readily included, we typically assume a SE loss
function. Non-informative previous distribution can be used in some circumstances when
we lack adequate prior knowledge.

(1) Prior Distribution and Corresponding Posterior Distribution
This is especially valid for our research. The likelihood function will then determine

the joint posterior density. Independent Gamma priors for the parameters explain the
prior data. Considering the parameters β and ξ to have independent Gamma priors, we
obtain the Bayes estimation of entropy in a manner that

π (β) ∼ βa−1e−bβ and π (ξ) ∼ ξc−1e−dξ, (16)

when we know the hyper-parameters (a, b) and (c, d). According to Eqs. (10) and (16),
the joint posterior density of β and ξ given the data is

π(β, ξ|x) = π (β)π (ξ) ℓ(β, ξ|x)∫
β

∫
ξ π (β)π (ξ) ℓ(β, ξ|x)dβdξ

. (17)

π(β, ξ|x) ∝ βa+r−1ξ
c+β(r+

r∑
i=1

Ri)+R∗
r)−1

e−bβe−dξe
−(β+1)

r∑
i=1

log(ξ+xi)



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 8 of 22

× e
−β

J∑
i=1

Ri log(ξ+xi)
e−βR∗

r log(ξ+xr). (18)

In general, the integration shown by Eq. (18) is more challenging and does not allow for
closed-form formulations, particularly when dealing with high-dimensional causes. Con-
sequently, approximation techniques like numerical integration and the Lindely approx-
imation can be used. However, the key techniques like MCMC rely on building. The
posterior distribution based on empirical data, It can be accomplished instead by model-
ing a large sample of the posterior distribution. With a variety of algorithms, the more
broad Metropolis-Hastings (M-H) algorithm can be used with Gibbs sampling or Gibbs
sampling methods, (see Ahmed [40], Ahmed and Mustafa [41], and Ahmed et al. [42]).
The importance sampling technique is another. The posterior distribution provided by
the following is an expression for Eq. (18):

π(β, ξ|x) ∝ πβ(β|ξ, x)πξ(ξ|β, x), (19)

where

π(β, ξ|x) ∝ βa+r−1ξ
c+β(r+(

r∑
i=1

Ri)+R∗
r)−1

e−bβe−dξ

× e
−(β+1)

r∑
i=1

log(ξ+xi)
e
−β

J∑
i=1

Ri log(ξ+xi)
e−βR∗

r log(ξ+xr), (20)

πβ(β|ξ, x) ∝ βa+n−1e
−β(b−(r+(

r∑
i=1

Ri)+R∗
r) log(ξ)+

r∑
i=1

log(ξ+xi)+
J∑

i=1
Ri log(ξ+xi)+R∗

r log(ξ+xr))
,

(21)

πξ(ξ|β, x) ∝ ξ
c+β(r+(

r∑
i=1

Ri)+R∗
r)−1

e
−

r∑
i=1

log(ξ+xi)−dξ
e
−β(

J∑
i=1

Ri log(ξ+xi)+R∗
r log(ξ+xr)+

r∑
i=1

log(ξ+xi))
.

(22)
(2) Loss Function
Bayesian methods are increasingly popular in reliability research and estimation. Se-

lecting a single value to represent the estimate of an unknown parameter requires defining a
loss function. Bayesian estimation typically employs two types of loss functions: the sym-
metric SE loss function and the asymmetric LINEX loss function, introduced by Varian
[43]). The SE loss function provides the foundation for evaluating estimators’ performance
in a lot of studies.

Ω̂SE =

∫
Ω
Ωπ(β, ξ|x)dβdξ. (23)

Because this function is symmetric, overestimation and underestimation are given equal
weight. An overestimate is far more dangerous than an underestimate, similar to how



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 9 of 22

functions for failure rate and reliability are estimated. Varian invented LINEX loss func-
tion, this is one of the most widely used asymmetric loss functions [43]. This function rises
roughly to zero on one side and is roughly linear on the other. Varian [43] introduced a
representation of the LINEX loss function can be expressed in the manner shown below,
assuming that the minimal loss happens at Ω̂ = Ω

L1(∆) ∝ ec∆ − c∆− 1, c ̸= 0, (24)

where Ω̂ is the estimate of Ω and ∆ = (Ω̂ − Ω). The degree and direction of symmetry
are expressed by the magnitude of c. When c > 0, overestimation is more dangerous than
underestimation, while the opposite is true if c < 0. The LINEX loss reduces to the SE loss
when c is close to zero. The LINEX loss function of Eq. (24) has a posteriors-expectation
of

EΩ(L1(Ω̂− Ω)) ∝ ecΩ̂EΩ(e
(−cΩ))− c(Ω̂− EΩ(Ω)− 1. (25)

The value of Ω̂ under the LINEX loss function, the Bayes estimator is

Ω̂LINEX = −1

c
log(EΩ[e

(−cΩ)]). (26)

In this way EΩ(e
(−cΩ)) exists, after creating β and ξ, the Bayes estimates of entropy

is calculated from the posterior density functions. Given adaptive progressive Type-II
censored data, the MCMC algorithm will execute the subsequent procedures:

Step 1: Choose an initial guess for ( β, ξ), denoted by ( β(0), ξ(0)).

Step 2: Assign w = 1.

Step 3: Using Gamma (r+a, (b− (r+(
r∑

i=1
Ri)+R∗

r) log(ξ)+
r∑

i=1
log(ξ+xi)+

J∑
i=1

Ri log(ξ+

xi) +R∗
n log(ξ + xr)) ) to generate β(w).

Step 4: Applying M-H to πξ(ξ|β, x) to produce ξ(w), with the N(ξ(w−1), var(ξ̂)) proposal
distribution.

M-H algorithm

1. Create ξ(∗) as a starting point, for which πξ

(
ξ(w−1), var(ξ̂)

)
.

2. Determine the likelihood of acceptance

ρξ = min

1, πξ

(
ξ∗, var(ξ̂)

)
πξ

(
ξ(w−1), var(ξ̂)

)
 . (27)

and generate U ∼ U (0, 1) .

3. If the suggestion is accepted by U ≤ ρξ , then set ξ(∗) = ξ(w). If not, turn down the
proposal



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 10 of 22

Step 5: Determine H1(f) using Eq. (5).

Step 6: Assign w = w + 1.

Step 7: Steps 3− 5 N should be repeated

Step 8: Determine R ’s Bayes estimator regarding the SE loss function as

ĤSE(f) =

N∑
i=M+1

H(i)(f)

N −M
, (28)

with respect to the LINEX loss function, H(f)’s Bayes estimator as

ĤLINEX(f) = −1

c
log


N∑

i=M+1

e−cH(i)(f)

N −M

 , (29)

where M is burned in.

2.3.3. Interval Estimation

(1) Asymptotic confidence intervals
For MLE, the asymptotic normality is used to construct the parameters asymptotic

confidence intervals (ACIs). The Fisher information matrix defines the negative expec-
tation of the second derivatives of the log-likelihood function aboutthe the model pa-
rameters. The anticipation of the second derivative is typically more significant in more
circumstances. The observed Fisher information matrix therefore provides an appropriate
approximation that may be utilized to build interval estimation in the manner described
below. The model parameter vector’s second derivative Ω = (β, ξ) log-likelihood function
is I0 with using an adaptive progressive Type-II censored. data:

I0(Ω) =

[
−∂2ℓ (β, ξ | x)

∂β∂ ξ

]
, (30)

∂2L (β, ξ | x)
∂β2

= − r

β2
, (31)

∂2L (β, ξ | x)
∂β∂ξ

=
∂2L (β, ξ | x)

∂ξ∂β
=

(r +
J∑

i=1
Ri)

ξ
−

r∑
i=1

1

(ξ + xi)
−

J∑
i=1

Ri

(ξ + xi)
− R∗

r

(ξ + xi)
,

(32)

∂2L (β, ξ | x)
∂ξ2

=

(r +
J∑

i=1
Ri)β

ξ2
+(β+1)

n∑
i=1

1

(ξ + xi)2
+ β

J∑
i=1

Ri

(ξ + xi)2
+

βR∗
r

(ξ + xi)2
. (33)



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 11 of 22

Next, I0(Ω), the observed information matrix, identifies the vector parameter Ω’s ML
estimate. It is now possible to distribute the MLE asymptotic distribution theory as
I−1
0 (Ω̂) is the variance covariance matrix for the mean Ω of a bivariate normal distribution
satisfies standard regularity requirements let

Ω̂ −→ N(Ω, I−1
0 (Ω̂)).

Using the delta approach, to obtain an approximate estimation of the entropy variance,
allow

A =

(
∂H(f)

∂β
,
∂H(f)

∂ξ

)
, (34)

where the first derivatives of the H(f) in relation to the parameters β and ξ are ∂H(f)
∂β

and ∂H(f)
∂ξ :

H(f) = − log(β ξβ) + (β + 1)[log(ξ) +
1

β
], (35)

∂H(f)

∂β
= − ξβ + βξβ log(ξ)

β ξβ
+ log(ξ) +

1

β
− (β + 1)

β2

= − 1

β
− log(ξ) + log(ξ) +

1

β
− 1

β
− 1

β2

= −β + 1

β2
, (36)

∂H(f)

∂ξ
= −β2ξβ−1

β ξβ
+

(β + 1)

ξ

= − β

ξ
+

(β + 1)

ξ
=

1

ξ
, (37)

ˆH(f)’s estimated asymptotic variance is provided by

var( ˆH(f)) −→
[
AI−1

0 At
]
|(β̂, ξ̂).

where At is the transpose of A, the MLE ˆH(f) of H(f) has an asymptotic distribution.
As a result, the asymptotic 100(1− α)% confidence interval for H(f) is as follows:

ˆH(f)± Zα/2

√
var( ˆH(f)). (38)

(2) Bootstrap confidence intervals
This part uses the percentile interval to create intervals of confidence derived by ap-

plying the parametric bootstrap method to the unknown parameters using an adaptive
progressive Type-II censored data, see Efron [44] for more information about bootstrap
confidence intervals. To get a bootstrap sample, the following algorithm is developed:



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 12 of 22

1. Get MLEs β̂, ξ̂ and Ĥ(f) from the initial two samples {x1, x2, ..., xn}.

2. The bootstrap estimate of β, ξ and H(f) is calculated by generating a bootstrap
sample {x∗1, x∗2, ..., x∗n} using β̂, ξ̂ and Ĥ(f).

3. To acquire the bootstrap samples, repeat steps (1) through (2) N times, making sure
that each estimate is arranged in ascending order: { ĤM+1(f), ĤM+2(f), ..., ĤN−M (f)}

The approximate confidence interval for ˆH(f) at 100(1− α)% is provided by

(Ĥi((N−M)α
2 )
(f), Ĥi((N−M)(1−α

2
))(f)), i = M + 1, .....N.

(3) Credible confidence intervals
For a random quantity, a posterior or 100 (1 − α)% Bayesian credible interval The

interval denoted by Ω is where the posterior probability (1− α) that Ω is situated.
The procedure below is used to get reliable H(f) confidence intervals.
Algorithm (2)
1. In algorithm (1), repeat steps (1) through (5).
2. The Bayesian credible interval for the H(f) is then established using the generated

MCMC samples using the algorithm Chen and Shao [45] developed. The posterior sample
is arranged as follows: ĤM+1(f), ĤM+2(f), ..., ĤN−M (f). The 100(1 − α)% HPD
credible intervals for H(f) are obtained by

(Ĥi((N−M)α
2 )
(f), Ĥi((N−M)(1−α

2
))(f)), i = M + 1, .....N. (39)

where α/2 displays the standard normal values with probability tailed α.

3. Numerical Application

3.1. Simulation study

This subsection compares and evaluates the various estimating techniques using a
Monte Carlo simulation study. The features and effectiveness of each estimate were also
evaluated. In this part, we compare the effectiveness of the different methods considering
different sample sizes and values of parameters using example findings derived from Monte
Carlo simulations. We evaluate how well the Bayes and MLE estimates perform about the
mean SE (MSE). Regarding coverage percentages (CP) and average confidence lengths,
we contrast the confidence intervals derived using the HPD credible intervals, bootstrap,
and MLE using asymptotic distributions. The parameter values that we used were β =
0.8, ξ = 0.3 and β = 1.5, ξ = 0.5. The sample sizes that we utilized were as follows:
(n, r) = (80, 50), (80, 60), (100, 60), (100, 80). For the informative hyper-parameters a =
b = c = d = 1. The LINEX loss function is asymmetric, with the parameter c determining
the direction of this asymmetry. For c > 0, overestimation incurs a higher cost than
underestimation, the opposite is true for c < 0. As c approaches zero, the LINEX loss



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 13 of 22

function exhibits symmetry and approximates the SE loss. We applied the LINEX loss
function with c=±0.05to obtain Bayesian estimates. As outlined in Section 2, posterior
analysis was performed using a hybrid MCMC algorithm employing Gibbs sampling and
M-H. The MCMC algorithm was initialized using the MLEs of the parameters β and ξ.
we provide the average mean, the MSE of the MLE, and Bayes estimates of entropy that
were related to, and τ = 0.3 and τ = 1 and different CS R were taken into consideration,
three different CS were utilized in this study, namely:

scheme I: R1 = n− r, Ri = 0 % for i ̸= 1.
scheme II: R r+1

2
= n− r, Ri = 0 for i ̸= r+1

2 ; if r odd, and

R r
2
= n− r, Ri = 0 for i ̸= r

2 ; if r even.
scheme III: Rr = n− r, Ri = 0 for i̸= r.
Mathematica 10 is used for all computations. Tables 3 and 4 (which provide the

CP and average length of the 95% asymptotic, bootstrap, and HPD credible intervals of
entropy) are determined., whereas Tables 1and 2 display the averaged mean and MSEs
of the estimates in parenthesis. For the MCMC approach, we choose N = 11000 with
M = 1000 as the burn-in time period. The results of the simulation investigation are
shown in Tables 1 - 4. As effect sample size grows, MSEs fall and CP approaches the
suggested value. The Bayes estimation shows lower MSEs with respect to the LINEX loss
function than with respect to the SE loss function. The mean length and MSEs of the
LINEX loss function drop as c increases.

Based on the results presented in Tables 1− 4, we draw the following conclusions:

• Superiority of Bayesian Estimation: Bayesian estimation, implemented through the
MCMC technique, consistently outperforms MLEs and bootstrap estimates in terms
of MSE for the parameter H(f).

• Effect of Sample Size: With fixed values of τ and n, increased effective sample size
leads to a reduction in MSEs and convergence of CP towards the nominal value.

• Impact of Censoring Parameters: When τ is fixed, increasing the values of n and r
generally results in a decrease in the MSEs of all estimators. Conversely, increasing
τ while keeping n and r fixed tends to increase the MSEs in most cases.

• LINEX vs. SE Loss: Bayesian estimation exhibits lower MSEs when using the
LINEX loss function compared to the SE loss function.

• Efficiency of Estimators: All point estimators are highly efficient, as indicated by
their very small average MSEs, which approach zero as n and r increase.

• LINEX Loss Parameter and MSE: The mean length and MSEs of the LINEX loss
function decrease with increasing values of c.

• Overall Performance: The simulation results demonstrate that Bayesian estimations,
particularly those under the LINEX loss function with c = 0.05, achieve the best



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 14 of 22

MSEs for estimating H(f), surpassing other methods. Furthermore, the Bayes cred-
ible intervals exhibit lower widths and higher CP compared to other methods, as
reflected in the mean lengths and CPs presented in Tables 1− 4.

While the Bayesian estimators demonstrate superior performance, the simulations indi-
cate that all point and interval estimation methods considered are efficient. The Bayesian
technique is a particularly advantageous choice when prior knowledge is available.

Table 1: Mean and MSE of the ML, bootstrap and Bayes of the entropy H(f) at τ = 0.3.

β = 0.8, ξ = 0.3

ML Boot Bayes

SE LINEX

c = −0.05 c = 0.05

n r CS mean MSE mean MSE mean MSE mean MSE mean MSE

I 1.8927 0.3037 1.9067 0.2401 1.9249 0.2043 1.9210 0.2010 1.9097 0.1867

80 50 II 1.7892 0.3104 1.9210 0.2548 1.9357 0.2107 1.9657 0.2149 1.9687 0.1987

III 1.9075 0.3110 1.8689 0.2550 1.8590 0.2110 1.9230 0.2207 1.8802 0.2070

I 1.9204 0.3007 1.8832 0.2204 1.9016 0.2108 1.9197 0.1893 1.8940 0.1804

80 60 II 1.8247 0.2970 1.9670 0.2420 1.8999 0.2220 1.8967 0.1801 1.8034 0.1896

III 1.9378 0.3049 1.8349 0.2507 1.9773 0.2247 1.9270 0.1896 1.8034 0.1901

I 1.9107 0.2980 1.9110 0.2701 1.9054 0.2152 1.9130 0.1991 1.9046 0.1846

100 60 II 1.9637 0.2976 1.8369 0.2657 1.9370 0.2210 1.9678 0.1901 1.9078 0.1891

III 1.8934 0.3000 1.9331 0.2789 1.9207 0.2225 1.9875 0.1638 1.9011 0.1920

I 1.9042 0.2807 1.9110 0.2562 1.9049 0.2124 1.8903 0.1937 1.8904 0.1864

100 80 II 1.9678 0.2834 1.9678 0.2489 1.9815 0.2210 1.9680 0.1987 1.8863 0.1899

III 1.8963 0.2975 1.8675 0.2553 1.9822 0.2217 1.8650 0.2018 1.8960 0.2014

β = 1.5, ξ = 0.5

I 1.1007 0.2934 1.0160 0.2347 1.1006 0.2318 0.9987 0.2057 0.9896 0.2012

80 50 II 1.0890 0.3048 1.0789 0.2344 1.0463 0.2486 1.0132 0.2132 0.9681 0.2014

III 0.9982 0.3155 1.0100 0.2355 1.0040 0.24440 0.9861 0.2204 0.9600 0.2093

I 0.9870 0.2807 0.9908 0.2429 1.1003 0.2310 1.0108 0.2027 1.1091 0.2009

80 60 II 0.8670 0.2991 0.9837 0.2341 1.1240 0.2341 1.0855 0.2045 1.0041 0.2014

III 0.9970 0.2998 0.9981 0.2458 1.1104 0.2333 1.0052 0.2109 1.0348 0.2045

I 1.0107 0.2908 1.1084 0.2307 1.0091 0.2019 1.0940 0.2007 1.0207 0.1893

100 60 II 0.9678 0.3045 1.0733 0.2540 0.9840 0.2000 1.0054 0.2041 1.0547 0.1937

III 0.9705 0.3004 1.0145 0.2556 0.9912 0.2104 0.9931 0.2210 1.0024 0.1995

I 0.9987 0.2827 0.9890 0.2320 0.9807 0.2087 0.9937 0.2011 0.9819 0.1708

100 80 II 0.9830 0.2984 0.9930 0.2447 0.9800 0.2104 0.9824 0.2080 0.9870 0.1801

III 0.9378 0.2963 0.9670 0.2555 0.9933 0.2122 0.9687 0.2100 0.9967 0.1833

:



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 15 of 22

Table 2: Mean and MSE of the ML, bootstrap and Bayes of the entropy H(f) at τ = 1.

β = 0.8, ξ = 0.3

ML Boot Bayes
SE LINEX

c = −0.05 c = 0.05

n r CS mean MSE mean MSE mean MSE mean MSE mean MSE

I 1.9245 0.3248 1.9307 0.2528 1.9370 0.2148 1.9325 0.2021 1.9207 0.1933

80 50 II 1.8934 0.3269 1.9427 0.2657 1.9904 0.2117 1.9638 0.2234 1.8965 0.2140

III 1.9250 0.3350 1.9670 0.2751 1.9680 0.2280 1.9740 0.2518 1.9968 0.2284

I 1.9441 0.3087 1.9583 0.2145 1.9529 0.2083 1.9516 0.1932 1.9900 0.1812

80 60 II 1.9801 0.3088 1.9811 0.2734 1.9932 0.2183 1.9924 0.2067 1.9960 0.2154

III 1.9217 0.3157 1.9330 0.2964 1.9821 0.2518 1.9872 0.2146 1.9854 0.2214

I 1.9960 0.3007 1.9423 0.2845 1.9668 0.2008 1.9450 0.1918 1.9484 0.1700

100 60 II 1.9470 0.3096 1.8854 0.2995 1.9993 0.2537 1.9934 0.2056 1.8934 0.1991

III 1.8990 0.3099 1.9127 0.2998 1.9478 0.2850 1.9870 0.2349 1.9956 0.2175

I 1.9807 0.2756 1.9657 0.2450 1.9880 0.2014 1.9704 0.1824 1.9815 0.1634

100 80 II 1.9967 0.2967 1.9218 0.2750 1.9825 0.2527 1.9962 0.2014 1.8863 0.1993

III 1.9670 0.2998 1.8968 0.2934 1.9934 0.2631 1.9278 0.2527 1.8960 0.2427

β = 1.5, ξ = 0.5

I 1.0128 0.3049 0.9907 0.2608 0.9838 0.2546 0.9751 0.2145 0.9910 0.2001

80 50 II 1.0213 0.3157 1.0053 0.2772 1.0041 0.2751 0.9845 0.2548 0.9958 0.2172

III 1.0184 0.3199 1.0048 0.2934 0.9934 0.2281 0.9995 0.2648 0.9938 0.2510

I 0.9967 0.2907 0.9972 0.2560 1.0033 0.2470 1.0087 0.2137 1.0159 0.2001

80 60 II 0.9937 0.3048 0.9990 0.2647 1.0082 0.2510 1.0047 0.2227 1.0051 0.2247

III 0.9853 0.3089 0.9580 0.2780 1.0544 0.2630 1.0208 0.2638 1.0028 0.2599

I 1.0073 0.3007 0.9677 0.2478 1.0408 0.2128 1.0210 0.2084 1.0138 0.1901

100 60 II 1.0128 0.3108 1.0247 0.2529 1.0084 0.2257 1.0249 0.2452 1.0521 0.2400

III 1.0174 0.3127 1.0004 0.2660 1.0899 0.2712 0.9886 0.2234 1.0066 0.2310

I 0.9821 0.2927 0.9933 0.2438 0.9914 0.2123 0.9969 0.2058 0.9916 0.1884

100 80 II 0.9968 0.3046 0.9657 0.2570 0.9937 0.2524 0.9954 0.2634 0.9760 0.2430

III 0.9931 0.3086 0.9980 0.2668 0.9920 0.2554 0.9964 0.2754 0.9931 0.2465



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 16 of 22

Table 3: Mean length and CP of ML, bootstrap and Bayes
of the entropy at τ = 0.3.

β = 0.8, ξ = 0.3

MLE Bootstrap Bayes

n r CS Mean Length CP Mean Length CP Mean Length CP

I 1.5720 0.9460 1.4906 0.9280 1.1185 0.9500

80 50 II 1.5107 0.9320 1.5208 0.9160 1.0982 0.9340

III 1.5073 0.9160 1.5107 0.9080 1.1130 0.9320

I 1.4870 0.9500 1.4458 0.9500 1.0253 0.9640

80 60 II 1.5420 0.9340 1.4283 0.9340 1.2104 0.9560

III 1.4630 0.9280 1.5012 0.9320 1.2113 0.9460

I 1.5004 0.9500 1.5296 0.9520 1.2104 0.9760

100 60 II 1.5133 0.9320 1.5183 0.9460 1.0214 0.9500

III 1.4820 0.9560 1.4170 0.9320 1.3240 0.9320

I 1.3879 0.9520 1.3084 0.9560 1.0245 0.9460

100 80 II 1.4452 0.9340 1.4406 0.9520 1.3201 0.9500

III 1.5480 0.9200 1.4120 0.9500 1.3004 0.9720

β = 1.5, ξ = 0.5

I 0.9140 0.9360 0.9210 0.9360 0.9245 0.9460

80 50 II 0.9128 0.9140 1.0053 0.9040 09543 0.9240

III 0.9348 0.9280 0.9210 0.9160 0.9180 0.9320

I 1.1250 0.9480 0.9207 0.9240 0.9324 0.9500

80 60 II 1.0899 0.9360 0.9990 0.9300 0.9638 0.9340

III 1.0233 0.9280 0.9207 0.9320 0.9109 0.9460

I 0.9967 0.9540 0.9087 0.9460 0.9137 0.9420

100 60 II 1.0018 0.9480 1.0247 0.9300 0.9126 0.9460

III 0.9421 0.9320 0.9087 0.9320 0.9137 0.9500

I 0.9453 0.9280 0.9248 0.9560 0.9218 0.9740

100 80 II 0.9638 0.9160 0.9657 0.9460 0.9875 0.9700

III 0.9453 0.9280 0.9248 0.9340 0.9974 0.9640



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 17 of 22

Table 4: Mean length and CP of ML, bootstrap and Bayes
of the entropy at τ = 1.

β = 0.8, ξ = 0.3

MLE Bootstrap Bayes

n r CS Mean Length CP Mean Length CP Mean Length CP

I 1.4210 0.9520 1.5124 0.9340 1.4563 0.9460

80 50 II 1.4860 0.9460 1.4865 0.9240 1.5429 0.9342

III 1.5240 0.9640 1.5600 0.9160 1.2014 0.9320

I 1.4148 0.9540 1.4458 0.9500 1.4562 0.9640

80 60 II 1.4538 0.9240 1.4283 0.9340 1.2433 0.9600

III 1.5264 0.9320 1.4094 0.9480 1.2363 0.9540

I 1.4984 0.9640 1.5296 0.9520 1.2213 0.9600

100 60 II 1.6433 0.9540 1.5183 0.9460 1.2130 0.9640

III 1.5047 0.9320 1.4190 0.9480 1.2207 0.9500

I 1.4529 0.9140 1.3084 0.9560 1.2001 0.9720

100 80 II 1.4637 0.9240 1.4406 0.9520 1.2540 0.9620

III 1.4258 0.9460 1.2148 0.9500 1.2019 0.9680

β = 1.5, ξ = 0.5

I 0.9120 0.9460 0.9360 0.9460 0.9113 0.9340

80 50 II 0.9248 0.9360 1.1031 0.9340 0.9124 0.9420

III 0.9224 0.9140 0.9138 0.9160 0.9011 0.9420

I 1.1010 0.9480 0.9634 0.9540 0.9541 0.9420

80 60 II 1.1210 0.9460 1.0199 0.9340 0.9124 0.9520

III 1.1022 0.9320 0.9110 0.9460 0.9007 0.9500

I 0.9245 0.9560 0.9324 0.9640 0.9421 0.9620

100 60 II 1.0143 0.9460 1.0245 0.9420 0.9360 0.9540

III 0.9510 0.9460 0.9108 0.9460 0.9016 0.9520

I 0.9123 0.9340 0.9110 0.9640 0.9124 0.9520

100 80 II 0.9112 0.9260 0.9124 0.9340 0.9630 0.9120

III 0.9210 0.9117 0.9560 0.9340 0.8991 0.9640

3.2. Real-life data analysis

The preceding theoretical results are shown in this subsection using real-life data.
According to Nelson [46], the data show how long it requires a fluid to act as an in-
sulator between electrodes to degrade at a voltage of 34 K. This data is 19 is shown as:
{0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.5, 7.35, 8.01, 8.27, 12.06, 31.75, 32.52, 33.91
,36.71, 72.89}, we confirmed that modeling these data with the Lo distribution is feasible,
p-values and the Kolmogorov-Smirnov (K-S) test are employed to provide the empirical
survival functions as well as the fitted ones, as shown in Figure 1. For the data, the K-S
is 0.1479 and p-value is 0.8002. We took r = 12, τ = 33.91, R = {17, 05}, based on R,
the adaptive Type-II progressive censored sample is {0.19, 0.78, 1.31, 3.16, 4.15, 4.67,
4.85, 6.5, 8.01, 8.27, 33.91, 36.71}, based on this data, The MLEs, the Bayes estimates and



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 18 of 22

the 95 % confidence intervals of model parameters and H(f) obtained in Table (4).
Using the first 2000 values as the burn-in, in the Bayes approach for the MCMC

technique, we execute the chan 22000. By depicting the generation outcomes of the entire
conditional distribution as the generation from the posterior distribution and the Bayesian
approach’s convergence under MCMC techniques, Figure 2 and Figure 3 demonstrates the
quality of posterior generation.

Table 5: MLEs, bootstrap, Bayes estimates under SE and LINEX

loss function and 95% interval estimate of the parameters

β ξ H(f)

(.)MLE 1.03095 4.44344 1.86196

ACI (0.2453, 2.1024) (2.3014, 6.1032) (0.78336, 3.5073)

(.)boot 1.02051 4.52180 1.9124

boot CI (3.2563,19.090) (2.3171, 14.8848) (0.82458, 3.4087)

(.) Bayes SE 0.999143 4.60977 2.06395

Bayes credible interval (0.0124, 2.0159) (2.1042, 7.0989) (1.36226, 3.07794)

(.) Bayes LINEX at c = 0.05 0.9926 4.3179 2.0594

(.) Bayes LINEX at c = −0.05 1.0059 4.9495 2.0686

Figure 1: Plots of the fitted CDF function of data and empirical data.

Figure 2: MCMC trace plots of H(f) from data.



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 19 of 22

Figure 3: MCMC histogram of H(f) from data.

4. Conclusions

This paper analyzes the entropy of the Lo distribution offers some statistical con-
clusions and uses adaptive progressively Type-II censoring. The Bayesian approach and
classical frequency theory served as the foundation for the development of point and in-
terval estimators. We have looked at the approximate entropy MLE confidence intervals
and the Bootstrap entropy confidence intervals using the observed Fisher matrix. Next,
we proceed on to the Bayesian approach, which assumes independent Gamma priors and
produces the SE and LINEX loss function-based Bayes estimates of entropy. The posterior
distributions of some unknown parameters reveal that they deviate from known distribu-
tions. To calculate Bayes estimates of entropy with related credible intervals, we use M-H
sampling as a component of the Gibbs sampling steps technique. In a simulated study, the
effectiveness of each of the previously listed methodologies was then directly compared.
The simulation results lead us to the conclusion that when adaptive progressive Type-II
is excluded from independent distributions of Lo, the Bayes technique can be used to esti-
mate and generate approximate confidence intervals for unknown parameters. When the
Lo distribution was used on real industrial data, it was discovered to be able to properly
depict current data to the point where it could be relied upon to analyze comparable
genuine data in those domains.

This study opens several significant avenues for future investigation. Areas of partic-
ular interest include the design of optimal censoring schemes, the statistical prediction of
outcomes under adaptive progressively Type-II censoring, and the extension of inference
methods to handle more complex failure models. Applying data mining methodologies to
these data may also prove insightful, allowing experts to identify differential patient sur-
vival patterns and calculate confidence intervals for survival. These areas warrant further
exploration.

Acknowledgements

The researchers wish to extend their sincere gratitude to the Deanship of Scientific
Research at the Islamic University of Madinah for the support provided to the Post Pub-
lishing Program.



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 20 of 22

References

[1] C E Shannon, A mathematical theory of communication. Bell System Technical Jour-
nal, 27(3): 379-432, 1948.

[2] K M Wong and S Chan, The entropy of ordered sequences and order statistics. IEEE
Transactions on Information Theory, 36(2): 276-284, 1990.

[3] Z A AboEleneen, The entropy of progressively censored samples. Entropy, 13: 437-
449, 2011.

[4] S M Sunoj and P G Sankaran, Quantile based entropy function. Statistics & Proba-
bility Letters, 82:1049-1053, 2012.

[5] J I Seo, H J Lee and S B Kan, Estimation for generalized half logistic distribution
based on records. Journal of the Korean Data & Information Science Society, 23(6):
1249-1257, 2012.

[6] Y Cho, H Sun and K Lee, An estimation of the entropy for a Rayleigh distribution
based on doubly-generalized Type-II hybrid censored samples. Entropy, 16(7): 3655-
69, 2014.

[7] G Q Zhao, W Liang and S Y He, Empirical entropy for right censored data. Acta
Mathematicae Applicatae Sinica, English Series, 31(2): 395-404, 2015.

[8] Y Cho, H Sun and K Lee, Estimating the entropy of a Weibull distribution under
generalized progressive hybrid censoring. Entropy, 17(1): 102-122, 2015.

[9] K Lee, Estimation of entropy of the inverse Weibul ldistribution under generalized
progressive hybrid censored data. Journal of the Korean Data & Information Science
Society, 28(3): 659- 668, 2017.

[10] B Almohaimeed, On the entropy of progressive hybrid censoring schemes. Applied
Mathematics & Information Sciences, 11(6): 1811-1814, 2017.

[11] M Chacko and P S Asha, Estimation of entropy for generalized exponential dis-
tribution based on record values. Journal of the Indian Society for Probability and
Statistics, 19(1): 79-96, 2018.

[12] A S Hassan and A N Zaky, Estimation of entropy for inverse Weibull distribution
under multiple censored data. Journal of Taibah University for Science, 13(1): 331-
337, 2019.

[13] X Wang and W Gui, Bayesian estimation of entropy for burr Type XII distribution
under progressive Type-II censored data. Mathematics, 9(4):313, 2021.

[14] N Balakrishnan and R Aggarwala, Progressive Censoring: Theory, Methods and Ap-
plications, Birkhauser: Boston, M A. (2000).

[15] H K T Ng, D Kundu and P S Chan, Statistical analysis of exponential lifetimes
under an adaptive Type-II progressive censoring scheme. Naval Research Logistics,
56: 687-698, 2009.

[16] M M A Sobhi and A A Soliman, Estimation for the exponentiated Weibull model with
adaptive Type-II progressive censored schemes. Applied Mathematical Modelling, 40:
1180-1192, 2016.

[17] M Nassar and O E Abo-Kasem, Estimation of the inverse Weibull parameters under
adaptive Type-II progressive hybrid censoring scheme. Journal of Computational and



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 21 of 22

Applied Mathematics, 315: 228-239, 2017.
[18] M F Sewailem and A Baklizi, Inference for the log-logistic distribution based on an

adaptive progressive Type-II censoring scheme. Cogent Mathematics & Statistics, 6:
1684228, 2019.

[19] R Xu and W Gui, Entropy Estimation of inverse Weibull distribution under adaptive
Type-II progressive hybrid censoring schemes. Symmetry, 11: 1463, 2016.

[20] H Panahi and N Moradi, Estimation of the inverted exponentiated Rayleigh distri-
bution based on adaptive Type-II progressive hybrid censored sample. Journal of
Computational and Applied Mathematics, 364: 112345, 2020.

[21] S Chen and W Gui, Statistical analysis of a lifetime distribution with a bathtub-
shaped failure rate function under adaptive progressive Type-II censoring. Mathe-
matics, 8: 670, 2020.

[22] R Mohan and M Chacko, Estimation of parameters of Kumaraswamy-exponential
distribution based on adaptive Type-II progressive censored schemes. Journal of Sta-
tistical Computation and Simulation, 91: 81-107, 2021.

[23] R Hora, N C Kabdwal and P Srivastava, Classical and bayesian inference for the
inverse Lomax distribution under adaptive progressive Type-II censored data with
COVID-19. Journal of Reliability and Statistical Studies, 505-534, 2022.

[24] M M Amein, M El-Saady, M M Shrahili and A R Shafay, Statistical inference for
the Gompertz distribution based on adaptive Type-II progressive censoring scheme.
Mathematical Problems in Engineering, 2022, Article ID 1266384, 11, 2022.

[25] K S Lomax, Business failures: Another example of the analysis of failure data. Journal
of the American Statistical Association, 49(268): 847-852, 1954.

[26] M Chahkandi and M Ganjali, On some lifetime distributions with decreasing failure
rate. Computational Statistics & Data Analysis, 53: 4433-4440, 2009.

[27] M C Bryson, Heavy-tailed distributions: properties and tests. Technometrics, 16:
61-68, 1974.

[28] B C Arnold, The Pareto distributions. Fairland, MD: International Co-operative Pub-
lishing House; 1983.

[29] N Johnson and S. Kotz, Balakrishnan N. Continuous Univariate Distribution. New
York: John Wiley and Sons; 1994.

[30] A Balkema and L De Haan, Residual life time at great age. Annals of Probability,
2(1.5): 792-804, 1974.

[31] M Ahsanullah, Record values of Lomax distribution. Statistica Nederlandica, 41(1.1),
21-2, 1991).

[32] N Balakrishnan, M Ahsanullah and P S Chan, Relations for Single and Product
Moments of Record Values from Gumbel Distribution. Statistics & Probability Letters,
15(3): 223-227, 1992.

[33] A Childs, N Balakrishnan and M Moshref, Order statistics from non-identical right-
truncated Lomax random variables with applications. Statistical Papers, 42(1.2): 187-
206, 2001.

[34] H A Howlader and A M Hossain, Bayesian survival estimation of Pareto distribution
of the second kind based on failure censored data. Computational Statistics and Data



S. M. Ahmed, G. M. Ismail / Eur. J. Pure Appl. Math, 18 (1) (2025), 5737 22 of 22

Analysis, 8(1.3): 301-314, 2002.
[35] M E Ghitany, F A Al-Awadhi and L A Alkhalfan, Marshall-Olkin extended lomax dis-

tribution and its application to censored data. Communications in Statistics, 36(2.5):
1855-1866, 2007.

[36] A Elfattah, F Alaboud and A Alharby, On sample size estimation for Lomax distri-
bution. Australian Journal of Basic and Applied Sciences, 4: 373-378, 2007.

[37] M Z Raqab, A Asgharzadeh and R Valiollahi, Prediction for Pareto distribution
based on progressively Type-II censored samples. Computational Statistics and Data
Analysis, 54: 1732-1743, 2010.

[38] E Cramer and A B Schmiedt, Progressively Type-II censored competing risks data
from Lomax distributions. Computational Statistics and Data Analysis, 55(1.3): 1285-
1303, 2011.

[39] A Asgharzadeh and R Valiollahi, Estimation of the scale parameter of the Lomax dis-
tribution under progressive censoring. International Journal of Economics and Statis-
tics, 6: 37-48, 2011.

[40] S M Ahmed, Constant-stress partially accelerated life testing for Weibull inverted
exponential distribution with censored data. Iraqi Journal for Computer Science and
Mathematics, 5(2): 94-111, 2023.

[41] S M Ahmed and A Mustafa, Estimation of the coefficients of variation for in verse
power Lomax distribution, AIMS Mathematics. 9(12): 33423-33441, 2024.

[42] S M Ahmed, M Abdalla and A A Farghal, Statistical inference for competing risks
model with Type-II generalized hybrid censored of inverse Lomax distribution with
applications. Alexandria Engineering Journal, 118:132-146, 2025.

[43] H R Varian, A Bayesian Approach to Real Estate Assessment, North Holland, Ams-
terdam, 195-208, 1975.

[44] B Efron, The Jackknife, the Bootstrap and Other Resampling Plans. CBMS-NSF
Regional Conference Series in Applied Mathematics, Monograph 38, SIAM, Philadel-
phia, 1982.

[45] M H Chen and Q M Shao, Monte Carlo estimation of Bayesian credible and HPD
intervals, Journal of Computational and Graphical Statistics, 8: 69-92, 1999.

[46] W B Nelson, Applied Life Data Analysis. New York: John Wiley & Sons; 1982.