Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3545 
https://internationalpubls.com 

Classical and Bayesian Estimation for a New Long Tailed Distribution: 

Modi-Lomax Distribution 

Surinder Kumar1, Rahul Shukla1, *, Bhupendra Meena1 

1Babasaheb Bhimrao Ambedkar University, Vidya Vihar, Raebareli Road, Lucknow – 226025, India 

Email: surinderntls@gmail.com1, rahul.shukla.stats@gmail.com1, *(corresponding author), bhupendrakm57@gmail.com1  

 

Article History: 

Received: 19-09-2024 

Revised: 24-10-2024 

Accepted: 11-11-2025 

Abstract:  

This paper presents the new Modi-Lomax distribution (MOLD), a three-parameter 

distribution obtained by using the Modi family generator to include one extra shape 

parameter in the standard two-parameter Lomax distribution. We establish extensive 

statistical properties such as survival function, hazard rate function, moment 

generating function, quantile function with quartile representations, mean residual 

lifetime, order statistics, and stress-strength reliability. For parameter estimation, we 

use maximum likelihood estimation and Bayesian methods with Gamma priors under 

squared error loss functions, comparing their performances based on bias and mean 

square error criteria. The practical usefulness of MOLD is shown through the 

analysis of one real data set of bladder cancer patients and a comparative analysis 

with a number of alternative distributions. Results validate the superior flexibility 

and goodness-of-fit of the suggested distribution under different test criteria, 

emphasizing its flexibility to model data with diverse failure rate shapes and aging 

behaviours. This paper makes MOLD a great contribution to lifetime distributions 

with prominent practice in reliability engineering and survival analysis. 

Keywords: Lomax Distribution, Modi Family, Maximum Likelihood Estimation, 

Bayesian Estimation, Stress-Strength Model 

1. Introduction 

Probability distributions serve as fundamental tools in statistical modelling across various scientific 

fields. With the growing complexity of observed phenomena, statistical research has increasingly 

focused on developing more adaptable distributions through mathematical transformations, weighting 

schemes, and mixture techniques. In recent years, the generalization of probability models has become 

particularly prominent, with various approaches available, such as Alpha Power Transformation 

(APT), exponentiation, mixture and weighted techniques, power transformations, among others.  

The Lomax distribution, or Lomax-Pareto Type II distribution, has found its place as a useful statistical 

model in reliability studies and as well as distribution theory. The distribution gained popularity 

because it is easily flexible in representing and modelling heavy-tailed behaviours in various fields of 

mathematics [(Harris, 1968); (Bryson, 1974)]. 

Initial theoretical contributions by (Balkema & De Haan, 1974) laid the foundation of the Lomax 

distribution's application in reliability modelling and life testing. (Arnold, 1983) further developed a 

detailed framework that linked the Lomax distribution to the rest of the Pareto family. (Johnson et al., 

1994) continued from such foundations, providing elaborate details about the properties and features 

of the distribution that cemented its place in statistical literature. 

mailto:surinderntls@gmail.com
mailto:rahul.shukla.stats@gmail.com1
mailto:bhupendrakm57@gmail.com


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3546 
https://internationalpubls.com 

The statistical properties of Lomax distribution have been well explored. (Ahsanullah, 1991) made 

noteworthy contributions by studying record values that came from Lomax-distributed variables. 

(Balakrishnan & Ahsanullah, 1994) derived significant recurrence relations among moments of record 

values from the Lomax distribution. (Atkinson & Harrison, 1978) illustrated the Lomax distribution's 

suitability in modelling income and wealth data. (Tadikamalla, 1980) formulated significant 

relationships among the Lomax and Burr family of distributions. The failure rate characteristics of 

Lomax model were also comprehensively studied by (Chahkandi et al., 2009), who identified that it 

comes under the class of decreasing failure rate distributions. 

The literature has seen significant development in extensions of the standard Lomax distribution. 

(Ashour & Eltehiwy, 2013) presented Transmuted Exponential Lomax (TEL) model. (Rady et al., 2016) 

used the power transformation on Lomax distribution (PoL). (Shabbir et al., 2018) contributed to 

Lomax family by creating Rayleigh Lomax distribution (RaL). (Joshi & Kumar, 2021) came up with 

the Poisson Inverted Lomax (PoIL) distribution. Another four-parameter distribution is Exponentiated 

odd Lomax, developed by (Dhungana & Kumar, 2022). There is another four-parameter distribution 

competitive to our study is Power XLindley (PXLin) by (Elgarhy et al., 2025) Some of the recent 

contributions to the Lomax family is Log-Lomax (LoL) by (Ishaq et al., 2025). 

(Modi et al., 2020) introduced the Modi family of distributions, demonstrating its adaptability for 

modelling diverse phenomena in engineering, economics, and finance. Subsequent studies have 

expanded this family - (Kumawat et al., 2024) developed the Modi-Weibull distribution, examining 

its statistical properties, while (Ndayisaba et al., 2023) proposed the Modi exponentiated exponential 

distribution, analysing its mathematical features and practical applications. In a recent study, (Kumar 

et al., 2025) proposed a new Modi-Rayleigh distribution, demonstrating its superior performance 

relative to existing distributions generated through the identical Modi transformation approach. These 

methods enable statisticians to refine existing distributions to better represent complex data 

characteristics, such as heavier tails, greater skewness, or unique hazard rate patterns, which 

conventional distributions may fail to capture adequately. 

This manuscript introduces the Modi-Lomax distribution (MOLD) using the Modi-family generator 

method. Our primary motivation stems from adding one new shape parameters to an existing long-

tailed distribution, enhancing its ability to fit failure time data containing extremely small values. We 

compare our proposed model with well-established lifetime distributions to demonstrate its flexibility 

in fitting real-world datasets. The article employs both classical and Bayesian estimation procedures 

to estimate the parameters of our proposed model. For classical estimation, we utilize the maximum 

likelihood method, while our Bayesian approach incorporates gamma priors and square error loss 

function (SELF). 

The manuscript is organized as follows: Section 2 introduces the Modi-Lomax (MOLD) distribution. 

Section 3 presents reliability analysis of MOLD. Section 4 covers the statistical properties, moment 

generating function, quantile function and order statistics, respectively. Section 6 covered the detailed 

parameter estimation using both maximum likelihood and Bayesian approaches with gamma priors 

and SELF. Section 7 presents simulation studies, Section 8 demonstrates applications to real-world 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3547 
https://internationalpubls.com 

datasets with comparative analysis, and Section 9 concludes with a summary of findings. In section 

10, we discussed the future scope of the article. 

2. Modi-Lomax Distribution 

The Modi family generator (Modi et al., 2020) is widely employed across various fields due to its 

capacity to introduce two additional shape parameters into existing lifetime models, thereby enhancing 

their flexibility for modelling real-world datasets. If f(t) and F(t) represent the probability density 

function (pdf) and cumulative distribution function (cdf), respectively, of an existing lifetime 

distribution, then the pdf g(.) and cdf G(.) of the new Modi family distribution are given by: 

 

𝑔(𝑥) =
𝛼𝛽(1+𝛼𝛽)𝑓(𝑥)

(𝛼𝛽+𝐹(𝑥))
2 ;  𝑥 > 0, 𝛼 > 0, 𝛽 > 0     ( 1) 

𝐺(𝑥) =
(1+𝛼𝛽)𝐹(𝑥)

𝛼𝛽+𝐹(𝑥)
;  𝑥 > 0, 𝛼 > 0, 𝛽 > 0     ( 2) 

In the Modi family generator, there are two shape parameters viz. α and β. We combine these two 

shape parameters and introduce single shape parameter ξ = αβ. The motivation behind this substitution 

is that, we want to check the joint effect of these two-shape parameter after the fusion with Lomax 

distribution. 

If t follows the Lomax distribution, so the pdf and cdf of t are as follows: 

𝑓(𝑡, 𝜆, 𝜃) =
𝜆

𝜃
(1 +

𝑡

𝜃
)

−(𝜆+1)

;  𝑡 > 0, 𝜆 > 0, 𝜃 > 0    ( 3) 

𝐹(𝑡, 𝜆, 𝜃) = 1 − (1 +
𝑡

𝜃
)

−𝜆

;  𝑡 > 0, 𝜆 > 0, 𝜃 > 0    ( 4) 

The MOLD is created from the Lomax distribution by putting Eq. (3) and Eq. (4), respectively, into 

Eq. (1) and Eq. (2). As a result, the MOLD's pdf is ascertained as 

𝑔(𝑡) =
𝜉(1+𝜉)𝜆𝜃−𝜆(𝜃+𝑡)−(𝜆+1)

[(1+𝜉)𝜃−𝜆−(𝜃+𝑡)−𝜆]
2 ; 𝑡 > 0, 𝜉 > 0, 𝜆 > 0, 𝜃 > 0   ( 5) 

The corresponding cdf of MOLD is given as 

𝐺(𝑡) =
(1+𝜉)[𝜃−𝜆−(𝜃+𝑡)−𝜆]

[(1+𝜉)𝜃−𝜆−(𝜃+𝑡)−𝜆]
; 𝑡 > 0, 𝜉 > 0, 𝜆 > 0, 𝜃 > 0   ( 6) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3548 
https://internationalpubls.com 

 

Figure 1. PDF plots of MOLD for different parametric values of ξ, λ and θ. 

 

Figure 2. 3D PDF plots of MOLD for varying ξ, λ and θ values. 

3. Reliability Characteristics of MOLD 

This section presents the derivation of essential reliability characteristics for the MOLD distribution, 

specifically addressing survival function, hazard rate, reverse hazard rate, cumulative failure rate, and 

the Mills ratio. 

3.1. Survival Function 

The survival function S(t) is one of important reliability measure in the reliability realm. It is defined 

as 

S(t)  =  1  −  G(t) 

where G(t) is the cumulative density function. 

From Eq. (6), we get the survival function for MOLD as following 

𝑆(𝑡) =
𝜉(𝜃+𝑡)−𝜆

𝜃−𝜆(1+𝜉)−(𝜃+𝑡)−𝜆     ( 7) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3549 
https://internationalpubls.com 

 

Figure 3. Survival function plots for MOLD distribution. 

3.2. Hazard Rate and Inverse Hazard Rate Function 

The hazard rate function H(t)  =  
g(t)

S(t)
 and the reversed hazard rate function H′(t) =

g(t)

G(t)
  are important 

quantities characterizing life time phenomena. For the MOLD distribution, they are given as 

 

𝐻(𝑡) =
𝜆(1+𝜉)

(𝜃+𝑡)(1+𝜉)−𝜃𝜆(𝜃+𝑡)1−𝜆
      ( 8) 

and 

𝐻′(𝑡) =
𝜉𝜆𝜃−𝜆(𝜃+𝑡)−(𝜆+1)

[(1+𝜉)𝜃−𝜆−(𝜃+𝑡)−𝜆][𝜃−𝜆−(𝜃+𝑡)−𝜆]
    ( 9) 

 

3.3. Cumulative Hazard Rate 

The cumulative hazard rate, measures the total risk that an event will occur by some specified time. 

The function is actually the integration of the hazard rate over time intervals, which illustrates how 

probability of the event occurrence increases with time. It is defined as 

Δ(𝑡) = − log(𝑆(𝑡)) 

Hence for the MOLD, it is given as 

Δ(𝑡) = − log [
ξ(θ + 𝑡)−λ

θ−λ(1 + ξ) − (θ + 𝑡)−λ
] 

𝛥(𝑡) = − 𝑙𝑜𝑔(𝜉) + 𝜆 𝑙𝑜𝑔(𝜃 + 𝑡) + 𝑙𝑜𝑔[𝜃−𝜆(1 + 𝜉) − (𝜃 + 𝑡)−𝜆]  ( 10) 

3.4. Mills Ratio 

The Mills ratio (M.R) for the MOLD is obtained as 

M.R =
𝐺(𝑡)

𝑆(𝑡)
 

M.R = (1 + 𝜉) [(1 +
𝑡

𝜃
)

𝜆

− 1]    ( 11) 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3550 
https://internationalpubls.com 

 

Figure 4. Hazard function plots for MOLD distribution 

3.5. Cumulative Hazard Rate 

The mean residual lifetime (MRL) represents the expected remaining lifespan of a subject given that 

it has survived to a particular time point. This measure provides valuable insight into the conditional 

life expectancy and is particularly useful for understanding survival patterns in different phases of the 

study period. 

MRL = E[(𝑥 − 𝑡)|𝑥 > 𝑡] 

=
∫ (𝑥 − 𝑡)𝑓(𝑥)𝑑𝑥

∞

𝑡

𝑆(𝑡)
 

For the MOLD, MRL is derived as 

MRL =
(θ + 𝑡)

λ − 1
 ; λ > 1 

This shows that the MRL for the MOLD increases linearly with t, with the rate determined by the 

parameter λ. This is a characteristic of distributions with the lack of memory property similar to the 

exponential distribution, but with a different structure. 

4. Statistical Properties of MOLD 

4.1. Moment Generating Function 

Theorem 1. If X is the MOLD variable having pdf given in Eq. (5), then its moment generating function 

𝑀𝑥(𝑠) is given by 

𝑀𝑥(𝑠) = 𝜉(1 + 𝜉)𝜆𝜃−𝜆𝑒−𝑠𝜃 ∫
𝑒𝑠𝑢𝑢−(𝜆+1)

[(1+𝜉)𝜃−𝜆−𝑢−𝜆]2 𝑑𝑢
∞

𝜃
    ( 12) 

Proof. The MGF of MOLD is obtained from 

𝑀𝑥(𝑠) = 𝐸[𝑒𝑠𝑥] = ∫ 𝑒𝑠𝑡
∞

0

𝑔(𝑡)𝑑𝑡 

If we define L[g(t)](s)as the Laplace transformation of g(t), then 

𝑀𝑥(𝑠) = 𝐿[𝑔(𝑡)](−𝑠) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3551 
https://internationalpubls.com 

where 𝐿[𝑔(𝑡)](𝑠) = ∫ 𝑒−𝑠𝑡∞

0
𝑔(𝑡)𝑑𝑡 

Hence, we get 

𝑀𝑥(𝑠) = ξ(1 + ξ)λθ−λ𝑒−𝑠θ ∫
𝑒𝑠𝑢𝑢−(𝜆+1)

[(1 + 𝜉)𝜃−𝜆 − 𝑢−𝜆]2
𝑑𝑢

∞

𝜃

 

Theorem 2. The rth moment about origin of MOLD is defined as 

μ𝑟
′ = 𝐸(𝑡𝑟) =

θ𝑟+λ

λ
∑ (

𝑟

𝑘
) (−1)𝑘B (

1 − (𝑟 − 𝑘)

λ
+ 1,1) (1 + ξ)1−(𝑟−𝑘)/λ−2

𝑟

𝑘=0

 

where, B(x,y) is the beta function. 

Proof. The rth moment about origin for the MOLD is obtained from the following expression using 

Eq.(5) 

μ𝑟
′ = 𝐸(𝑡𝑟) = ∫ 𝑡𝑟𝑔(𝑡)𝑑𝑡

∞

0

 

Let 𝑢 =  θ + 𝑡 

𝐸(𝑡𝑟) = ∫
(𝑢 − θ)𝑟ξλθ−λ(𝑢)−(λ+1)(1 + ξ)

[(1 + ξ)θ−λ − 𝑢−λ]2
𝑑𝑢

∞

θ

 

Now we use second substitution 𝑣 = (
θ

𝑢
)

λ

 

𝐸(𝑡𝑟) =
θ𝑟+λ

λ
∫

(𝑣−1/λ − 1)
𝑟

𝑣1/λ

(1 + ξ − 𝑣)2
𝑑𝑣

1

0

 

Now expanding (𝑣−1/λ − 1)
𝑟

= ∑ (𝑟
𝑘

)(𝑣−1/λ)
𝑟−𝑘

(−1)𝑘𝑟
𝑘=0  using binomial expression, we get 

𝐸(𝑡𝑟) =
θ𝑟+λ

λ
∑ (

𝑟

𝑘
) (−1)𝑘B (

1 − (𝑟 − 𝑘)

λ
+ 1,1) (1 + ξ)1−(𝑟−𝑘)/λ−2

𝑟

𝑘=0

 

4.2 Quantile Function 

Let 𝑡 ∼ MOLD(ξ, λ, θ) and G(t) = p be the cdf of t, so the quantile function is defined as 

𝑄(𝑝) = 𝐺−1(𝑝),  for 0 < 𝑝 < 1 

Hence the quantile function of MOLD is obtained as follows 

𝑄(𝑝) = 𝜃 [(
1+𝜉−𝑝

(1+𝜉)(1−𝑝)
)

1/𝜆

− 1] ,  0 < 𝑝 < (1 + 𝜉)  ( 13) 

The first, second and third quartile can be obtained from Eq. (13) by putting p =  1/4, 1/2 and 3/4, 

respectively. 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3552 
https://internationalpubls.com 

The first quartile 

𝑄1 = 𝜃 [(
3+4𝜉

3(1+𝜉)
)

1/𝜆

− 1]    ( 14) 

The second quartile 

𝑄2 = 𝜃 [(
1+2𝜉

(1+𝜉)
)

1/𝜆

− 1]    ( 15) 

The third quartile 

𝑄3 = 𝜃 [(
1+4𝜉

(1+𝜉)
)

1/𝜆

− 1]    ( 16) 

4.3 Order Statistics 

Consider a finite random sample consisting of observations t1, t2, … , tn drawn from the MOLD with 

pdf g(t) and cdf G(t). When these sample values are rearranged in non-decreasing sequence, the 

resulting ordered observations t1, t2, … , tn are called the order statistics of the sample, where: 

𝑡(1) ≤ 𝑡(2) ≤ ⋯ ≤ 𝑡(𝑛) 

The order statistic 𝑡(𝑟) corresponds to the 𝑟𝑡ℎ smallest observation among the sorted sample values. 

Specifically, 𝑡(1) is the minimum value, 𝑡(2) is the second minimum, and this pattern continues until 

𝑡(𝑛) which represents the maximum value in the sample. For any r where 1 ≤ 𝑟 ≤ 𝑛 the pdf of the 

𝑟𝑡ℎ order statistic is expressed as: 

𝑞(𝑟)(𝑡) =
𝑛!

(𝑟−1)!(𝑛−𝑟)!
𝑔(𝑡)[𝐺(𝑡)]𝑟−1[1 − 𝐺(𝑡)]𝑛−𝑟    ( 17) 

Putting the expression of pdf and cdf of MOLD from Eq. (5) and Eq. (6) respectively in Eq.(17), 

𝑞(𝑟)(𝑡) = 𝑚
𝜆𝜉(1+𝑛−𝑟)(1+𝜉)𝑟(𝜃+𝑡)𝜆(𝑟+𝑛−1)−1𝜃−𝜆𝑟[1−(1+

𝑡

𝜃
)

−𝜆
]

𝑟−1

[(1+𝜉)𝜃−𝜆−(𝜃+𝑡)−𝜆]
𝑛+1    ( 18) 

where, 𝑚 =
𝑛!

(𝑟−1)!(𝑛−𝑟)!
 

The pdf for first order statistics of MOLD, when the value of r is 1, and given by 

𝑞(1)(𝑡) =
𝑛𝜆𝜉𝑛(1+𝜉)(𝜃+𝑡)−(𝜆𝑛+1)𝜃−𝜆

[(1+𝜉)𝜃−𝜆−(𝜃+𝑡)−𝜆]
𝑛+1     ( 19) 

 

The pdf of nth order statistics of the MOLD when the value of r is n, and is given by 

𝑞(𝑛)(𝑡) =
𝑛𝜆𝜉(1+𝜉)𝑛(𝜃+𝑡)−(𝜆+1)𝜃−𝜆𝑛[1−(1+

𝑡

𝜃
)

−𝜆
]

𝑛−1

[(1+𝜉)𝜃−𝜆−(𝜃+𝑡)−𝜆]
𝑛+1    ( 20) 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3553 
https://internationalpubls.com 

4.4 Stress-Strength Reliability 

The reliability model based on stress-strength analysis is expressed as R = P(X > Y), where X denotes 

the random strength capacity of a system or its component, while Y represents the random stress 

applied to the system or component at any given moment. In this study, we compute the stress-strength 

reliability by assuming that the stress Y follows an exponential distribution and the strength X follows 

a MOLD distribution. The rationale for modeling Y with an exponential distribution is that in many 

practical engineering scenarios, stress tends to grow exponentially over time. 

Let 𝑋 ∼ 𝑀𝑂𝐿𝐷(ξ1, λ, θ) and 𝑌 ∼ 𝑀𝑂𝐿𝐷(ξ2, λ, θ) with the pdf and cdf function given in Eq. (5) and 

(6) respectively. 

Hence 

𝑅 = 𝑃(𝑋 > 𝑌) 

𝑅 = ∫ 𝑃
∞

0

(𝑋 > 𝑌)𝑓(𝑥)𝑑𝑥 

𝑅 = ∫
(1 + ξ2)[θ−λ − (θ + 𝑥)−λ]

[(1 + ξ2)θ−λ − (θ + 𝑥)−λ]

∞

0

.
ξ1(1 + ξ1)λθ−λ(θ + 𝑥)−(λ+1)

[(1 + ξ1)θ−λ − (θ + 𝑥)−λ]2
𝑑𝑥 

Putting two substitutions in sequence, u = (θ + x)−λ and 𝑣 = 𝑢/θ−λ, we get 

𝑅 = ξ1(1 + ξ1)(1 + ξ2) ∫
(1 − 𝑣)

[(1 + ξ1) − 𝑣]2[(1 + ξ2) − 𝑣]
𝑑𝑣

1

0

 

After using By-Part method from calculus, we get 

𝑅 =
𝜉1(1+𝜉2)

(𝜉1−𝜉2)
+

𝜉1𝜉2(1+𝜉1)(1+𝜉2))

(𝜉1−𝜉2)2 𝑙𝑜𝑔 ([
(1+𝜉1)𝜉2

(1+𝜉2)𝜉1
])    ( 21) 

6. Parameter Estimation 

6.1 Classical estimation procedure 

The method of maximum likelihood estimation (MLE) is a key classical statistical methodology for 

the inference of parameters of probability distributions. The method operates by identifying the set of 

parameters which maximizes the value of the likelihood function, which is defined as the probability 

of the observed data under a specified model. For a series of i.i.d. observations t1, t2, … , tn  constituting 

a sample of size n from the MOLD (ξ, λ, θ) distribution, we can write the likelihood function as: 

𝐿 (𝑡, 𝜉, 𝜆, 𝜃) = 𝜉𝑛𝜆𝑛𝜃−𝑛𝜆(1 + 𝜉)𝑛 ∏ (𝜃+𝑡𝑖)−(𝜆+1)𝑛
𝑖=1

∏ [(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆]
2𝑛

𝑖=1

   ( 22) 

Using the Eq. (22) the log-likelihood function is given as 

log(𝐿) = 𝑛 log(ξ) + 𝑛 log(λ) − (λ + 1) ∑ log(θ + 𝑡𝑖)

𝑛

𝑖=1

− 𝑛λ log(θ) 

+𝑛 𝑙𝑜𝑔(1 + 𝜉) − 2 ∑ 𝑙𝑜𝑔 ((1 + 𝜉)𝜃−𝜆 − (𝜃 + 𝑡𝑖)
−𝜆)𝑛

𝑖=1    ( 23) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3554 
https://internationalpubls.com 

By partially differentiating log likelihood in Eq. (23) with respect to each parameter, we get the 

following closed form equations for the MLE for the parameters ξ, λ and θ, respectively as 

𝜕 𝑙𝑜𝑔(𝐿)

𝜕𝜉
=

𝑛

𝜉
+

𝑛

(1+𝜉)
− 2 ∑

𝜃−𝜆

(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆
𝑛
𝑖=1     ( 24) 

 

𝜕 𝑙𝑜𝑔(𝐿)

𝜕𝜆
=

𝑛

𝜆
− 𝑛 𝑙𝑜𝑔(𝜃) − ∑ 𝑙𝑜𝑔(𝜃 + 𝑡𝑖)

𝑛
𝑖=1 + 2 ∑

(1+𝜉)𝜃−𝜆 𝑙𝑜𝑔(𝜃)−(𝜃+𝑡𝑖)−𝜆 𝑙𝑜𝑔(𝜃+𝑡𝑖)

(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆
𝑛
𝑖=1  ( 25) 

𝜕 𝑙𝑜𝑔(𝐿)

𝜕𝜃
=

−𝑛𝜆

𝜃
−

(𝜆+1)

𝜃
∑

1

1+𝑡𝑖/𝜃

𝑛
𝑖=1 + 2 ∑

𝜆(1+𝜉)𝜃−(𝜆+1)−𝜆(𝜃+𝑡𝑖)−(𝜆+1)

(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆
𝑛
𝑖=1   ( 26) 

The above non-linear equations cannot be solved analytically so, the MLE of the parameters ξ, λ and 

θ can be obtained as the numerical solution of the Eq. (24), (25) and (26), respectively. 

6.2 Bayesian estimation 

In this section we used Bayesian parameter estimation methods for the Modi-Lomax distribution's 

parameters. Bayesian inference demands appropriate prior distribution selection to incorporate 

available information or parameter assumptions. Here, we employ gamma prior distributions for 

estimating parameters, which are suitable due to positive parameter space restrictions. The parameters 

are assumed to be independent and have gamma prior distributions such that ξ ∼ Gamma(a1, b1) , λ ∼

Gamma(a2, b2) and θ ∼ Gamma(a3, b3). 

Hence the joint prior density of ξ, λ and θ is given by 

𝜓(𝜉, 𝜆, 𝜃) ∝ 𝜉𝑎1−1𝑒−𝑏1𝜉𝜆𝑎2−1𝑒−𝑏2𝜆𝜃𝑎3−1𝑒−𝑏3𝜃   ( 27) 

Where a1, b1, a2, b2, a3  and b3  are the hyper parameters. The hyper-parameters in the prior 

distributions are assumed to be known. 

The choice of loss function is pivotal in Bayesian decision theory, as it quantifies the penalty for 

estimation errors. Here, we adopt the squared error loss function (SELF), defined as: 

𝐿𝑠(α, α̂) = (α, α̂)2 

where α is the true parameter and α̂ is its estimator. The SELF is symmetric and penalizes over and 

under estimation equally, yielding the posterior mean as the optimal estimator. This approach balances 

computational tractability with theoretical rigor, making it widely used in reliability and survival 

analysis. 

By employing the likelihood function in Eq. (22) with the joint prior distribution in Eq. (28), the joint 

posterior distribution of ξ, λ and θ can be expressed in the following form: 

 

π(ξ, λ, θ) ∝ ξ𝑎1+𝑛−1λ𝑎2+𝑛−1θ𝑎3−𝑛λ−1𝑒−𝑏1ξ−𝑏2λ−𝑏3θ(1 + ξ)𝑛  

∏ (𝜃+𝑡𝑖)−(𝜆+1)𝑛
𝑖=1

∏ [(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆]
2𝑛

𝑖=1

     ( 28) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3555 
https://internationalpubls.com 

 

The Bayes estimators under the SELF loss function is the posterior mean of the posterior distribution, 

are given as, 

ξ̂ =
1

δ
∫ ∫ ∫ξ𝑎1+𝑛

θλξ

λ𝑎2+𝑛−1θ𝑎3−𝑛λ−1𝑒−𝑏1ξ−𝑏2λ−𝑏3θ(1 + ξ)𝑛 

∏ (𝜃+𝑡𝑖)−(𝜆+1)𝑛
𝑖=1

∏ [(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆]
2𝑛

𝑖=1

 𝑑𝜉 𝑑𝜆 𝑑    ( 29) 

λ̂ =
1

δ
∫ ∫ ∫ξ𝑎1+𝑛−1

θλξ

λ𝑎2+𝑛θ𝑎3−𝑛λ−1𝑒−𝑏1ξ−𝑏2λ−𝑏3θ(1 + ξ)𝑛 

∏ (𝜃+𝑡𝑖)−(𝜆+1)𝑛
𝑖=1

∏ [(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆]
2𝑛

𝑖=1

 𝑑𝜉 𝑑𝜆 𝑑    ( 30) 

and 

θ̂ =
1

δ
∫ ∫ ∫ξ𝑎1+𝑛−1

θλξ

λ𝑎2+𝑛−1θ𝑎3−𝑛λ𝑒−𝑏1ξ−𝑏2λ−𝑏3θ(1 + ξ)𝑛 

∏ (𝜃+𝑡𝑖)−(𝜆+1)𝑛
𝑖=1

∏ [(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆]
2𝑛

𝑖=1

 𝑑𝜉 𝑑𝜆 𝑑    ( 31) 

respectively, where 

δ = ∫ ∫ ∫ξ𝑎1+𝑛−1

θλξ

λ𝑎2+𝑛−1θ𝑎3−𝑛λ−1𝑒−𝑏1ξ−𝑏2λ−𝑏3θ(1 + ξ)𝑛 

∏ (𝜃+𝑡𝑖)−(𝜆+1)𝑛
𝑖=1

∏ [(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆]
2𝑛

𝑖=1

 𝑑𝜉 𝑑𝜆 𝑑     

The mathematical intricacy of posterior expectations from mathematically formulated expressions 

inherently renders it difficult to obtain closed-form analytical solutions. To avoid this computational 

hurdle, researchers can utilize a number of Bayesian approximation techniques, including Lindley's 

approximation, Tierney-Kadane's method, or Markov Chain Monte Carlo (MCMC) techniques. In our 

study, we use MCMC sampling to approximate the posterior expectations, as described in the next 

section. 

6.3 Markov Chain Monte Carlo (MCMC) method 

We must first determine the complete conditional distributions of the ξ, λ, and θ in order to apply the 

MCMC approach, as indicated below. 

𝜋∗(𝜉|𝜆, 𝜃, 𝑑𝑎𝑡𝑎) ∝ 𝜉𝑎1+𝑛−1𝑒−𝑏1𝜉(1 + 𝜉)𝑛 1

∏ [(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆]
2𝑛

𝑖=1

    ( 32) 

𝜋∗(𝜆|𝜉, 𝜃, 𝑑𝑎𝑡𝑎) ∝ 𝜆𝑎2+𝑛−1𝜃−𝑛𝜆𝑒−𝑏2𝜆 ∏ (𝜃+𝑡𝑖)−(𝜆+1)𝑛
𝑖=1

∏ [(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆]
2𝑛

𝑖=1

   ( 33) 

and 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3556 
https://internationalpubls.com 

𝜋∗(𝜃|𝜉, 𝜆, 𝑑𝑎𝑡𝑎) ∝ 𝜃𝑎3−𝑛𝜆−1𝑒−𝑏3𝜃 ∏ (𝜃+𝑡𝑖)−(𝜆+1)𝑛
𝑖=1

∏ [(1+𝜉)𝜃−𝜆−(𝜃+𝑡𝑖)−𝜆]
2𝑛

𝑖=1

   ( 34) 

 

The conditional posterior distributions in Eq. (32), (33) and (34) of the parameters under the Bayesian 

framework do not follow any standard or closed-form distribution. This non-standard nature arises due 

to the complexity of our likelihood function combined with the Gamma prior distributions, which 

prevents the use of direct sampling methods. As a result, conventional analytical approaches become 

impractical. To address this, the Metropolis-Hastings (MH) algorithm is employed as a flexible and 

efficient Markov Chain Monte Carlo (MCMC) technique to generate samples from the intractable 

conditional distributions and obtain the Bayesian estimates of the parameters. MH algorithm is carried 

out in following steps: 

1. Set the starting value ξ(0) = ξ̂, λ(0) = λ̂ and θ(0) = θ̂ through MLE. 

2. Set 𝑖 = 1. 

3. Create xi∗, λ∗ and θ∗ from N(ξ̂, σ2
ξ̂), N(λ̂, σ2

λ̂) and N(θ̂, σ2
θ̂), respectively.  

4. Find 𝐴𝜉 ,Aλ and A𝜃 as 

 

𝐴ξ = 𝑚𝑖𝑛 {1,
π∗(ξ∗|λ(𝑖−1), θ(𝑖−1), 𝑑𝑎𝑡𝑎)

π∗(ξ(𝑖−1)|λ(𝑖−1), θ(𝑖−1), 𝑑𝑎𝑡𝑎)
}, 

𝐴λ = 𝑚𝑖𝑛 {1,
π∗(λ∗|ξ(𝑖−1), θ(𝑖−1), 𝑑𝑎𝑡𝑎)

π∗(λ(𝑖−1)|ξ(𝑖−1), θ(𝑖−1), 𝑑𝑎𝑡𝑎)
} 

and 

𝐴θ = 𝑚𝑖𝑛 {1,
π∗(θ∗|ξ(𝑖−1), λ(𝑖−1), 𝑑𝑎𝑡𝑎)

π∗(θ(𝑖−1)|ξ(𝑖−1), λ(𝑖−1), 𝑑𝑎𝑡𝑎)
} 

5. Utilizing the uniform U (0,1) distribution, generate samples u1, u2 and u3 respectively for 

ξ, λ and θ. 

6. If u1, u2 and u3 are less than Aξ, Aλ and Aθ, respectively than set ξ(i) = ξ∗, λ(i) = λ∗ and 

θ(i) = θ∗, respectively. Otherwise, set ξ(i) = ξi−1, λ(i) = λi−1 and θ(i) = θi−1, respectively. 

7. Set i = i + 1 

8. Redo steps 3 - 7, L times to get ξ(i), λ(i) and θ(i)for i = 1,2, − − −, L. 

9. After a suitable burn-in phase, say D, obtain the Bayes estimate of the parameter ξ, λ and 

θ as an example, as 

ξ̃ =
1

𝐿 − 𝐷
∑ ξ(𝑖)

𝐿

𝑖=𝐷+1

, λ̃ =
1

𝐿 − 𝐷
∑ λ(𝑖)

𝐿

𝑖=𝐷+1

, θ̃ =
1

𝐿 − 𝐷
∑ θ(𝑖)

𝐿

𝑖=𝐷+1

. 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3557 
https://internationalpubls.com 

7. Simulation Study 

In this section, to determine the effectiveness of MOLD estimators, we conduct a Monte Carlo 

simulation. Using specific selected variable values and the MOLD distribution, a number of random 

numbers are generated. The following procedures to investigate the new model are now presented: 

[Step:1] Fix the values of MOLD (ξ, λ, θ) as 

 Set-1 MOLD (9, 5, 0.25) and  

 Set-2 MOLD (0.02, 0.05, 10). 

 

[Step:2] Fix the sample sizes n as 

 n = 10, 30, 50, 200. 

[Step:3] Fix the number of replications as 1000. 

[Step:4] Generate the random samples using Quantile function given in Eq. (13). 

𝑥𝑖 = θ [(
1 + ξ − 𝑢𝑖

(1 + ξ)(1 − 𝑢𝑖)
)

1/λ

− 1] ,where 𝑢𝑖 ∼ 𝑈(0,1), 𝑖 = 1,2, … , 𝑛. 

[Step:5] The MLE of each parameter are obtained using the optim function in R software using the 

log-likelihood function from Eq. (23). 

[Step:6] Bayes estimates of the parameters are obtained based on two informative prior sets as 

 Prior-1 (P-1) (a1, a2, a4) = (18,10,0.5) and bi = 2 for i = 1, 2, 3. 

 Prior-2 (P-2) (a1, a2, a4) = (0.08,0.20,40) and bi = 4 for i = 1, 2, 3. 

[Step:7] The performance of the point estimates is assessed using two criteria: mean squared error 

(MSE) and absolute bias (AB). 

𝑀𝑆𝐸 =
1

𝑁
∑(θ̂ − θ)

2
𝑁

𝑖=1

 

𝐴𝐵 =  |(𝜃 − θ)| 

where, θ= Parameter true value, θ̂ = Estimated value of the parameter. 

Table 1: Mean Estimate, Absolute Bias (AB) and Mean Squared Error (MSE) of the parameters of 

MOLD under MLE and Bayesian Method (BM) for Case-1 

𝑛 Properties MLE BM 

    ξ = 9 λ = 5 𝜃 = 0.25 ξ = 9 λ = 5 𝜃 = 0.25 

  Mean Estimate 8.35711 7.12690 0.26704 8.96335 4.97687 0.25111 

10 AB 0.64289 2.12690 0.01704 0.03665 0.02313 0.00111 

  MSE 10.91290 16.11558 0.01407 0.34820 0.50040 0.00636 

 Mean Estimate 8.33814 7.05030 0.25669 9.08599 4.98748 0.25240 

30 AB 0.66187 2.05030 0.00669 0.08599 0.01252 0.00240 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3558 
https://internationalpubls.com 

  MSE 9.48808 12.27414 0.00397 0.55615 0.50785 0.00424 

 Mean Estimate 8.20359 6.85089 0.25723 9.04125 5.11707 0.25892 

50 AB 0.79641 1.85089 0.00723 0.04125 0.11707 0.00892 

  MSE 9.52439 9.91493 0.00253 0.42477 0.64209 0.00343 

 Mean Estimate 8.81283 6.18065 0.25527 9.02399 5.00547 0.25215 

200 AB 0.18717 1.18065 0.00527 0.02399 0.00547 0.00215 

  MSE 3.23149 3.64873 0.00051 0.74400 0.69949 0.00272 

 

Table 2: Mean Estimate, Absolute Bias (AB) and Mean Squared Error (MSE) of the parameters of 

MOLD under MLE and Bayesian method (BM) for Case-2 

𝑛 Properties MLE BM 

    ξ = 0.02 λ = 0.05 𝜃 = 10 ξ = 0.02 λ = 0.05 𝜃 = 10 

  Mean Estimate 0.09588 0.02575 9.99064 0.05352 0.09236 10.04951 

10 AB 0.07588 0.02425 0.00936 0.03352 0.04236 0.04951 

  MSE 0.01284 0.00188 0.08164 0.00241 0.00580 2.46946 

  Mean Estimate 0.06400 0.03510 9.99969 0.05335 0.10310 9.92382 

30 AB 0.04400 0.01490 0.00031 0.03335 0.05310 0.07618 

  MSE 0.00439 0.00141 0.00000 0.00344 0.01054 2.12657 

  Mean Estimate 0.05590 0.03890 9.99959 0.05349 0.09945 9.82858 

50 AB 0.03590 0.01110 0.00041 0.03349 0.04945 0.17142 

  MSE 0.00321 0.00128 0.00001 0.00476 0.01133 1.42323 

  Mean Estimate 0.02828 0.05120 9.97097 0.02491 0.05684 9.84556 

200 AB 0.00828 0.00120 0.02904 0.00491 0.00684 0.15444 

  MSE 0.00063 0.00071 0.10122 0.00038 0.00140 0.43880 

 

Tables (1) and (2) present the parameter estimates for the MOLD distribution obtained through 

maximum likelihood estimation and Bayesian methodology, respectively. A comparative analysis of 

the results indicates that the Bayesian approach yields superior parameter estimates relative to the 

maximum likelihood method, consistent with findings reported in the existing literature. 

8. Application 

To demonstrate the practical applicability of the proposed MOLD distribution, we analyse a medical 

dataset comprising remission times (measured in months) for 137 bladder cancer patients. This dataset 

Table (3), originally reported by (Lee & Wang, 2003), provides survival times that exhibit the 

characteristics commonly observed in oncological studies. The observed remission times range from 

0.08 to 79.05 months, with the complete dataset presented as follows: 

Table 3: Remission time (in month) of 137 cancer patients 

4.5 32.15 3.88 13.8 19.13 4.87 3.02 5.85 14.24 5.71 

19.36 7.09 7.87 7.59 20.28 5.32 5.49 3.02 46.12 4.33 

2.02 4.51 5.17 2.83 9.22 1.05 0.2 8.37 3.82 9.47 

36.66 14.77 26.31 79.05 10.06 8.53 4.65 2.02 4.98 11.98 

2.62 4.26 5.06 1.76 0.9 11.25 16.62 4.4 21.73 10.34 

12.07 34.26 0.87 10.66 6.97 2.07 0.51 12.03 0.08 17.12 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3559 
https://internationalpubls.com 

3.36 2.64 1.4 12.63 43.01 14.76 2.75 7.66 0.81 1.19 

7.32 4.18 3.36 8.66 1.26 13.29 1.46 14.83 6.76 23.63 

24.8 5.62 8.6 3.25 10.86 18.1 7.62 7.63 17.14 25.74 

3.52 2.87 15.96 17.36 9.74 3.31 7.28 1.35 0.4 2.26 

4.33 9.02 5.41 2.69 22.69 6.94 2.54 11.79 2.46 7.26 

2.69 5.34 3.48 4.7 8.26 6.93 4.23 3.7 0.5 10.75 

6.54 3.64 5.32 13.11 8.65 3.57 5.09 7.39 5.41 11.64 

2.09 2.23 6.25 7.93 4.34 25.82 12.02    

 

We applied MLE and Bayesian method to fit the MOLD parameters and conducted a comparative 

study against seven competing distributions from the Lomax family. The comparative models included 

Log-Lomax (LoL) by (Ishaq et al., 2025), Power Lomax (PoL) by (Rady et al., 2016), Rayleigh Lomax 

(RaL) by (Shabbir et al., 2018), Power X-Lindey (PXLin) by (Elgarhy et al., 2025), Poisson Inverted 

Lomax (PoIL) by (Joshi & Kumar, 2021), Exponentiated odd Lomax exponential distribution (EOLE) 

by (Dhungana & Kumar, 2022) and Transmuted Exponential Lomax (TEL) by (Ashour & Eltehiwy, 

2013). Distribution comparison is performed using a comprehensive set of information criteria, 

specifically AIC, BIC, CAIC, and HQIC, representing the Akaike, Bayesian, Consistent Akaike, and 

Hannan-Quinn information criteria, respectively. We also used the Kolmogorov-Smirnov (K-S) test to 

better check how well each distribution worked. We picked the best distribution by looking for the one 

with the highest log-likelihood value, the best p-value from the K-S test, and the lowest AIC, BIC, and 

CAIC scores. 

Table 4: Goodness of fit tests for cancer patient dataset 

Distributions -2logL AIC BIC HQIC CAIC K-S p-value 

MOLDx 885.4448 891.4448 900.2047 895.0046 891.6253 0.17 0.000726 

RaL 885.8124 891.8124 900.5723 895.3722 891.9929 0.1961 5.33e-05 

LoL 888.3446 894.3446 903.1045 897.9044 894.5251 0.2103 1.09e-05 

PXLin 912.2588 918.2588 927.0187 921.8186 918.4393 0.2052 1.94e-05 

PoIL 989.5734 995.5734 1004.333 999.1332 995.7539 0.3391 4.18e-14 

EOLE 997.8764 997.5734 997.5734 1009.253 1002.32 0.1955 5.64e-05 

TEL 1017.408 1025.408 1037.087 1030.154 1025.711 0.3384 4.72e-14 

PoL 1111.972 1117.972 1126.732 1121.531 1118.152 0.2587 2.17e-08 

 

Table (4) presents a comparative analysis of the MOLD against seven alternative distribution functions 

using goodness-of-fit measures. The results demonstrate that MOLD exhibits superior performance, 

achieving the minimum values for AIC, BIC, HQIC, and CAIC relative to the seven competing 

distributions. Additionally, we computed the Kolmogorov Smirnov test statistic alongside its 

associated p-value. Optimal model fit is characterized by minimal K-S values and maximal p-values. 

The MOLD yielded a K-S statistic of 0.17 with a corresponding p-value of 0.000726, representing the 

most favourable performance among all competing models, thereby establishing MOLD as the most 

appropriate distribution for practical applications, particularly in modelling cancer patient remission 

times. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3560 
https://internationalpubls.com 

 

Figure 5. Plots for the Cancer patient’s dataset 

Figure (5) displays three visual diagnostic tools—probability density function histogram, empirical 

cumulative distribution function (ECDF), and probability-probability plot utilized to assess model 

adequacy across various distributions for the given dataset. Throughout all three evaluation methods, 

MOLD exhibits optimal fitting characteristics, as evidenced by its curves closely following the 

observed data patterns. 

The newly developed MOLD distribution achieved superior performance when compared to existing 

competitive models, showing better goodness-of-fit measures and diminished information criteria 

scores. These empirical outcomes emphasize its capability in modelling non-symmetric and 

sophisticated data configurations. The findings of this investigation present the MOLD as a significant 

analytical resource for reliability engineering, survival research, and risk management applications. 

9. Conclusion 

This research introduces the MOLD, a new three-parameter extension of the standard Lomax 

distribution, created to overcome challenges in studying complex survival and reliability data. The 

distribution features heavy tails and a decreasing hazard rate, making it suitable for long-term survival 

analysis. By combining the flexible properties of the Modi family with the basic structure of the Lomax 

distribution, the MOLD shows improved adaptability through its hazard function behaviour, quantile 

features, and tail properties. 

We developed a complete mathematical framework for the distribution, including moments, survival 

functions, and entropy measures, which confirms its theoretical strength. For parameter estimation, we 

applied both classical maximum likelihood estimation (MLE) and Bayesian methods using gamma 

priors. Monte Carlo simulations validated that the MLE estimators are consistent and asymptotically 

efficient. Our simulation studies examined estimator performance using mean squared error, showing 

that estimation accuracy improves as sample sizes increase. 

To test practical usefulness, we applied the MOLD to a bladder cancer patient dataset and compared 

it with existing distributions using standard measures like AIC, BIC, CAIC, log-likelihood, and the 

Kolmogorov-Smirnov test. The MOLD consistently performed better than competing models, showing 

superior fit quality and lower information criteria values. These results demonstrate its effectiveness 

in modelling skewed and complex data patterns. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3561 
https://internationalpubls.com 

The study establishes the MOLD as a useful tool for reliability analysis, survival studies, and risk 

assessment, particularly when dealing with heavy-tailed data that exhibits decreasing failure rates over 

time. 

10. Future Scope 

Future work could develop the MOLD for multiple variables and use Bayesian approaches for 

complicated data analysis. Researchers could also improve how it handles incomplete data and 

combine it with artificial intelligence for better predictions. Testing the MOLD against newer 

statistical models and adapting it for specific fields like weather studies or smart device networks 

would show how widely it can be used and help advance survival analysis and risk assessment. 

References 

[1] Ahsanullah, M. (1991). “Record values of the Lomax distribution”. Statistica Neerlandica, 45(1), 21–29. 

[2] Arnold, B. C. (1983). Pareto Distribution. John Wiley & Sons, Ltd. 

[3] Ashour, S. K., & Eltehiwy, M. A. (2013). “Transmuted exponentiated Lomax distribution”. Australian 

Journal of Basic and Applied Sciences, 7(7), 658–667. 

[4] Atkinson, A. B., & Harrison, A. J. (1978). Distribution of personal wealth in Britain. Cambridge, UK: 

Cambridge University Press. 

[5] Balakrishnan, N., & Ahsanullah, M. (1994). “Relations for single and product moments of record values 

from Lomax distribution”. Sankhyā: The Indian Journal of Statistics, Series B, 140–146. 

[6] Balkema, A. A., & De Haan, L. (1974). “Residual life time at great age”. The Annals of Probability, 2(5), 

792–804. 

[7] Bryson, M. C. (1974). “Heavy-tailed distributions: properties and tests”. Technometrics, 16(1), 61–68. 

[8] Chahkandi, M., Farsi, M., & Ganjali, M. (2009). “On the renewal function of Poisson shock models with 

Lomax distribution as the time between the shocks”. Journal of the Iranian Statistical Society, 8(1–2), 63–

75. 

[9] Dhungana, G. P., & Kumar, V. (2022). “Exponentiated Odd Lomax Exponential distribution with 

application to COVID-19 death cases of Nepal”. PloS One, 17(6), e0269450. 

[10] Elgarhy, M., Hassan, A. S., Alsadat, N., Balogun, O. S., Shawki, A. W., & Ragab, I. E. (2025). “A Heavy 

Tailed Model Based on Power XLindley Distribution with Actuarial Data Applications”. Computer 

Modeling in Engineering & Sciences (CMES), 142(3). 

[11] Harris, C. M. (1968). “The Pareto distribution as a queue service discipline”. Operations Research, 16(2), 

307–313. 

[12] Ishaq, A. I., Usman, A. U., Alqifari, H. N., Almohaimeed, A., Daud, H., Abba, S. I., & Suleiman, A. A. 

(2025). “A new Log-Lomax distribution, properties, stock price, and heart attack predictions using 

machine learning techniques”. AIMS Mathematics, 10(5), 12761–12807. 

[13] Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). Continuous univariate distributions, volume 2 (Vol. 

2). John Wiley & Sons. 

[14] Joshi, R. K., & Kumar, V. (2021). “Poisson inverted Lomax distribution: Properties and applications”. 

International Journal of Research in Engineering and Science (IJRES), 9(1), 48–57. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

3562 
https://internationalpubls.com 

[15] Kumar, S., Meena, B., Shukla, R., & Singh, S. P. (2025). “Modi rayleigh distribution and its application to 

survival time data sets”. Life Cycle Reliability and Safety Engineering, 1–10. 

[16] Kumawat, H., Modi, K., & Nagar, P. (2024). “Modi-Weibull Distribution: Inferential and Simulation 

Study”. Journal Annals of Data Science, 11(06), 1975–1999. 

[17] Lee, E. T., & Wang, J. (2003). Statistical methods for survival data analysis (Vol. 476). John Wiley & 

Sons. 

[18] Modi, K., Kumar, D., & Singh, Y. (2020). “A new family of distribution with application on two real 

datasets on survival problem”. Journal Science & Technology Asia, 1–10. 

[19] Ndayisaba, A. D., Odongo, L. O., & Ngunyi, A. (2023). “The Modi Exponentiated Exponential 

Distribution”. J Mod Appl Stat Methods, 11, 341–-359. 

[20] Rady, E.-H. A., Hassanein, W. A., & Elhaddad, T. A. (2016). “The power Lomax distribution with an 

application to bladder cancer data”. SpringerPlus, 5(1), 1838. 

[21] Shabbir, M., Riaz, A., & Gull, H. (2018). “Rayleigh Lomax distribution”. The Journal of Middle East and 

North Africa Sciences, 4(12), 1–4. 

[22] Tadikamalla, P. R. (1980). “A look at the Burr and related distributions”. International Statistical 

Review/Revue Internationale de Statistique, 337–344.