https://orcid.org/0000-0002-6723-6785
Figures
Abstract
In this article, two flexible classes called the modified Kavya–Manoharan-G (MKM-G) and discrete modified Kavya–Manoharan-G (DMKM-G) families are investigated. The two proposed families provide more flexibility for modeling real-lifetime and count data from environmental, medical, engineering, and educational fields. Due to the new extra shape parameter of the two proposed families, their special sub-models are capable of modeling monotonic and non-monotonic hazard rates. The basic properties of the MKM-G family are studied. Eight classical approaches of estimation are used for estimating the MKM-exponential (MKME) parameters. The performances of the estimators are explored using simulation results. Additionally, the DMKM-exponential (DMKME) distribution is defined. Finally, the importance and flexibility of the MKME and DMKME distributions are addressed by fitting seven real-lifetime and count data from aforementioned applied fields. The real data analysis shows that the special models of the two classes are good candidates and can provide close fit as compared to well-known competing continuous and discrete distributions.
Citation: Afify AZ, Helmi MM, Aljohani HM, Alsheikh SMA, Mahran HA (2025) Modeling lifetime and count data using a unified flexible family: Its discrete counterpart, properties, and inference. PLoS ONE 20(4): e0319091. https://doi.org/10.1371/journal.pone.0319091
Editor: Shaiful Anuar Abu Bakar,, University of Malaya, MALAYSIA
Received: July 22, 2024; Accepted: January 27, 2025; Published: April 17, 2025
Copyright: © 2025 Afify et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All data used in this study are included in the article.
Funding: The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-162).
Competing interests: The authors of this paper declare no conflict of interest.
1 Introduction
Recently, several attempts to develop generalized distributions have been made to model lifetime and count data in different applied fields. These generalized models have several applications including medicine, economics, biological studies, engineering, finance, and environmental sciences, among others. However, there is a clear need for more flexible distributions, which are capable of modeling different shapes of aging and failure criteria.
Some notable families are the Marshall–Olkin-G [1], Kumaraswamy-G [2], Weibull-G [3], Kumaraswamy transmuted-G [4], odd Dagum-G [5], log–logistic tan-G [6], modified generalized-G [7], logarithmic-U [8], Lambert-G [9], new exponential-H [10], and alpha beta-power-F [11], among many others.
One of the recent classes is called the Kavya–Manoharan-G (KM-G) family [12], which is used to introduce some generalized distributions include the KM Kumaraswamy distribution [13], KM log-logistic distribution [14], KM Kumaraswamy exponential distribution [15], KM Burr X distribution [16], KM power Lomax distribution [17].
In this article, we propose a flexible extension of the KM-G class by adding an extra shape parameter. The newly constructed class is called the modified Kavya–Manoharan-G (MKM-G) family, which increases the flexibility of the generated models. Furthermore, the discrete counterpart of the new MKM-G family is proposed. The discrete counterpart class is called the discrete modified Kavya–Manoharan-G (DMKM-G) family. The DMKM-G family is derived using the survival discretization (SD) approach.
The objectives of the current article are five-fold: (i) to propose two new flexible continuous and discrete families; (ii) to present four special lifetime models and a discrete model as special cases of the two proposed classes, with a more detailed analysis of the MKM-exponential (MKME) and DMKM-exponential (DMKME) sub-models; (iii) to address the mathematical characteristics of the MKM-G family; (iv) to discuss the estimation of MKME parameters using eight classical estimation approaches; and (v) to explore the empirical importance of the MKME and DMKME models through analysis of seven real-life datasets, including two count datasets.
We are motivated to introduce the MKM-G and DMKM-G families for several reasons: (i) The MKM-G special models can represent reversed-J shaped, right-skewed, and unimodal densities, as well as bathtub, increasing, modified bathtub, decreasing, and unimodal hazard rate (HR) shapes; (ii) These sub-models generalize several published lifetime models such as the KM Burr X and KM exponential models; (iii) The DMKME distribution exhibits unimodal, reversed-J, increasing, bathtub, and decreasing discrete HR shapes, making it more versatile than other count distributions, which typically only exhibit increasing or decreasing shapes; (iv) The special sub-models of both proposed classes are well-suited for modeling asymmetric lifetime and count data across various applied fields, including insurance, biology, medicine, engineering, and life testing; and (v) The empirical importance of the MKME and DMKME models is explored using seven real-lifetime and count datasets, showing that the two proposed distributions outperform several well-known lifetime and discrete distributions in modeling real-world data; and (vi) Finally, the new families offer simple analytical forms and exceptional flexibility, which may lead to wider applications in engineering, reliability, environmental sciences, insurance, medicine, and economics.
The rest of the paper is outlined in the following eight sections. In Section 2, the MKM-G and DMKM-G families are defined. Five special sub-models of the MKM-G and DMKM-G classes are presented in Section 3. In Section 4, some mathematical properties of the MKM-G class are derived. Estimation methods of the MKME parameters are discussed in Section 5. Detailed simulation results are presented in Section 6. Section 7 provides seven applications for real-lifetime and count data to show the flexibility of the MKME and DMKME distributions. Finally, some concluding remarks are explored in Section 8.
2 Synthesis of the MKM-G family and its discrete counterpart
If X is a random variable (RV) following the KM-G family with parameters α and θ (see [12]), then it has the following cumulative distribution function (CDF)
where and
is the baseline CDF with a vector of parameters ϑ.
The corresponding probability density function (PDF) of (1) reduces to
Definition 1. A RV X is said to follow the MKM-G family, denoted by MKM-G (
), if its CDF has the form
where θ is a shape parameter.
The corresponding PDF of Equation (3) has the form
The HR function (HRF) of the MKM-G family becomes
The added shape parameter θ allows us to explore the tail behavior of the density (4) and provides more flexibility as we can see in Section 3. Additionally, the importance of the MKM-G family follows from its ability to generate new flexible distributions which exhibit monotone and non-monotone failure rates.
Remark: The KM-G class follows as a special case by setting in Equation (3).
The SD approach is used to discretize the MKM-G family of distributions. The SD technique depends on the survival function (SF), say, and
, which can be adopted to define the probability mass function (PMF) as follows
The SF of the MKM-G family has the form
Applying the SD approach in (5), the PMF of the DMKM-G family is given by
where
The CDF and SF of the DMKM-G family is given by
and
3 Five special lifetime and discrete models
This section presents four special sub-models of the MKM-G family. These sub-models provide flexible forms of some baseline distributions namely the exponential (E), Burr X (Bx), Burr XII (BXII), and log-logistic (LL) distributions. The special sub-models of the MKM-G class are called the MKME, MKM-Burr X (MKMBX), MKM-Burr XII (MKMBXII), and MKM-log logistic (MKMLL) distributions. The four special distributions provide decreasing, bathtub, reversed J shaped, increasing, unimodal, and modified bathtub shapes. Additionally, their densities provide right-skewed, symmetrical, and reversed-J shapes as displayed in Figs 1-4. Furthermore, the DMKME model is defined in Section 3.5. Figs 5 and 6 display the PMF and HRF plots of the DMKME model for different values of its parameters θ and λ. The HRF of the DMKME distribution provides unimodal, bathtub, increasing, reversed-J, and decreasing discrete failure rates.
3.1 The MKME distribution
The CDF of the MKME distribution follows by setting the E CDF, , in Equation (3). Then, the CDF of the MKME distribution reduces to
The corresponding PDF and HRF of the MKME distribution have the forms
and
The RV with PDF (7) is denoted by . For
, the MKME distribution reduces to the KME distribution [12]. Fig 1 gives some possible shapes of the density and HR functions of the MKME distribution.
3.2 The MKMBXII Distribution
By taking the CDF of the BXII distribution (for and
), say,
, as a baseline CDF in (3), the MKMBXII CDF follows as
The PDF of the MKMBXII distribution reduces to
For , the MKMBXII distribution reduces to the KMBXII distribution. Fig 2 gives some shapes of the PDF and HRF of the MKMBXII distribution for different parametric values
and λ.
3.3 The MKMBx distribution
Consider the CDF of the Bx distribution (for and
), say,
. By inserting the Bx CDF in Equation (3), the CDF of the MKMBx distribution follows as
where .
The PDF of the MKMBx model reduces to
By setting in the above equation, the KMBx distribution [16] follows as a special case. The MKM-Rayleigh distribution is obtained when
. The MKMBx model reduces to the KM-Rayleigh distribution for
Some shapes of the density and failure rate functions of the MKMBx distribution are given in Fig 3, for different values of η and λ.
3.4 The MKMLL distribution
Consider the LL CDF (for and
), say,
. By inserting the LL CDF in Equation (3), we obtain the CDF of the MKMLL distribution (for
)
The MKMLL density takes the form
The MKMLL distribution reduces to the KMLL model for . Fig 4 provides the shapes of the density and hazard functions of the MKMLL distribution.
3.5 The DMKME distribution
The DMKME distribution is derived here by substituting the CDF of the E distribution in Equation (6), then the PMF of the DMKME distribution follows as
where . Then, the corresponding SF and CDF of (8) reduce to
and
The HRF of the DMKME takes the form
The plots of the PMF and HRF of the DMKME model are presented in Figs 5 and 6, for some choices of the parameters θ and λ. The DMKME HRF can be unimodal, bathtub, increasing and decreasing discrete HRF.
4 Properties of the MKM-G family
Some general mathematical properties of the MKM-G family are provided in this section.
4.1 Linear representation
Important and simple mixture representations for the CDF and PDF of the MKM-G family in terms of exponentiated-G (Exp-G) density are provided in this section.
Consider the exponential series, which is given by
Applying (9) to Equation (3), we obtain
Hence, the MKM-G CDF is expressed as
where And
is the Exp-G CDF with power parameter
. By differentiating Equation (10), the PDF of the MKM-G family reduces to
where and
is the Exp-G PDF with
. Thus, many mathematical characteristics of the MKM-G family follow simply from those of the Exp-G family.
4.2 Quantile function and moments
The quantile function (QF) of the MKM-G family, say, , follows by inverting (3). Then, the MKM-G QF takes the form
where ais the baseline QF and
.
Hereafter, let denotes the Exp-G RV with positive power parameter k. Hence, based on Equation (11), the rth moment of X has the form
Setting in Equation (12) gives the mean of X (
).
The moment generating function (MGF) of the RV X is defined by . Hence, the MGF of the MKM-G family can be derived from (11) in two different forms. The first one follows as
where is the MGF of
. Then,
of the MKM-G family is calculated based on the MGF of the Exp-G class.
The second formula for takes the form
where .
The qth incomplete moment of X is expressed based on (11) as
The first incomplete moment (FIM), say, , follows from (13) when
. It also can be calculated by two formulae. The first formula for
follows from (13) as
where is the FIM of the Exp-G class. The second formula for
reduces to
where , where
is computed numerically and
is the baseline QF.
Bonferroni () and Lorenz
curves are useful applications for
because they can be calculated, for a given probability τ, by
and
, where
is the mean of X and
is the QF of X at τ. The two curves have important applications in demography, economics, insurance, reliability, and medicine. Additionally, the mean deviations about the mean
and about the median
of X, where
is the median and
is simply evaluated from (3).
4.3 Entropies
The variation of the uncertainty can be measured by the Rényi entropy, say , which is given by
Using the MKM-G density (4), we obtain
By applying the power series in Equation (9), we have
Then, reduces to
Hence, follows as
where
Then, the Rényi entropy of the MKM-G family reduces to
The Shannon entropy follows from the Rényi entropy when ϕ tends to 1.
4.4 Mean residual life and mean inactivity time
The expected additional life length for a unit, which is alive at age t can be expressed by the mean residual life (MRL). The MRL of a RV X is defined (for ) by
. The MRL of X reduces to
where is the CDF of the MKM-G family. Inserting (14) in Equation (15) gives
The waiting time elapsed since the failure of an item on condition that this failure had occurred in is expressed as the mean inactivity time (MIT). The MIT of X, say
follows as
Combining the last equation and Equation (14), the MIT of X takes the form
4.5 Probability weighted moments
The expectation of a certain function of a RV whose mean exists is known as the probability weighted moments (PWMs). The th PWM of X has the form
Using the CDF and PDF of the MKM-G family, we can write
Applying the exponential and binomial series, the above equation reduces to
Equivalently, we can write
where
Then, the PWM of the MKM-G family follows as
4.6 Order statistics
Let be a random sample from the MKM-G family. The density of the ith order statistic, say
, has the form
where is the beta function.
Using Equations (3) and (4) of the MKM-G family, we obtain
Based on Equation (16), the last equation has the form
where is the Exp-G PDF with parameter
and
Then, the PDF of is expressed by
Hence, the PDF of the MKM-G order statistic is a linear combination of Exp-G densities. Equation (19) illustrates that the properties of can be derived from those properties of
.
The rth moment of follows as
4.7 Moments of the MKME distribution
This subsection provides a simple expression for the rth moment of the MKME model.
Based on Equation (11), the PDF of the MKME distribution reduces to
Applying the binomial expansion to the last term, the above equation becomes
where
and is the PDF of the E model with scale parameter
. Then, the MKME PDF is expressed as a single linear combination of E PDFs.
The rth moment of the MKME distribution is obtained from Equation (20) as follows
The mean of say,
, follows from (21) with
Furthermore, the R software is used to obtain some numerical values for the
, variance
, skewness
, and kurtosis
measures of the MKME distribution. The values of the given measures are reported in Table 1 It is noted that that the spread for its
is much larger ranging from 6.6978 to 18.6635, whereas the
of the MKME distribution can range in the interval (1.4482, 3.2662).
Additionally, Table 2 provides the numerical values of for the MKME model based on the numerical integration (NUI) and summation (SUM) formula for several values of λ and θ at truncated M terms, where L is the truncated terms from this summation. Table 2 shows that the summation in (21) converges to the NUI of
for all values of λ and θ when L reaches to 50.
The QF of the MKME distribution is
5 Estimation approaches
This section discusses the estimation of the MKME parameters using some classical methods called the maximum likelihood (ML), least-squares (LS), weighted least-squares (WLS), maximum product of spacing (MPS), percentiles (PC), Cramér–von Mises (CM), Anderson–Darling (AD), and right-tail AD (RTAD) estimators.
Let be a random sample from the MKME distribution and
be their order statistics. The log-likelihood function, say, l, reduces to
where
The ML estimators (MLEs) of θ and λ are determined by maximizing the above equation with respect to the parameters θ and λ, or by solving the following two equations
and
The MLEs are also calculated by using different statistical programs such as Mathematica, Mathcad, and R (optim function), among others.
Swain et al. [18] estimated the parameters of the beta distribution using the LS estimators (LSEs) and WLS estimators (WLSEs). The LSEs and WLSEs of the MKME parameters θ and λ are determined by minimizing
where for the LS approach,
for the WLS approach and
Furthermore, the LSEs and WLSEs can be obtained by solving the following nonlinear equations
where
and
Cheng and Amin [19, 20] pioneered the MPS approach as an alternative method to estimate the parameters of different continuous univariate models. The uniform spacings, say , of a random sample of size n from the MKME distribution are defined by
where and
The MPS estimators (MPSEs) of the MKME parameters can be determined by maximizing
with respect to θ and λ.
Additionally, the MPSEs of the MKME parameters are also calculated by solving
where are defined in Equations (22) and (23) for
.
The PC estimators (PCEs) are proposed by [21] to estimate the model parameters by equating the sample PC points with the population PC points. If is an unbiased estimator of
, then the PCEs of the parameters of the MKME distribution follow by minimizing
with respect to θ and λ.
Cramér [22] and Von Mises [23] introduced the CVM estimators (CVMEs) which are obtained as the difference between the estimated CDF and empirical CDF.
The CVMEs of the MKME parameters can be determined by minimizing
Furthermore, the CVMEs are also obtained by solving the following nonlinear equations
The AD estimators (ADEs) are an important type of minimum distance estimators.
The ADEs of the MKME parameters can be determined by minimizing
with respect to θ and λ. The ADEs are also calculated by solving the following equations
The RTAD estimators (RTADEs) of θ and λ are given by minimizing
6 Simulation results
This section explores the performance of several estimators of the MKME parameters by using detailed simulation studies. We generate 5000 samples from the MKME distribution based on some sample sizes and some parametric values of the two parameters
and
. We calculate the average estimates (AEs) of the parameters and mean square errors (MSEs) for each sample size using the eight estimators.
The performance of the studied estimators is evaluated using MSEs. The AEs and MSEs (in parentheses) for different estimators are presented in Tables 3–6. It is observed that as the sample size n increases, the AEs converge to the true parameter values, and the MSEs decrease towards zero, demonstrating that the estimators are asymptotically unbiased. Overall, the numerical simulations show that all eight estimation methods perform excellently with respect to the MSEs.
7 Lifetime and count data analysis in applied sciences
This section is devoted to showing the empirical flexibility of the special models of the two proposed families using some real-life data, including both continuous and discrete data.
7.1 Applications for the MKME model
In this subsection, we analyze five real-life datasets from applied fields including environmental sciences, medicine, and engineering to explore the flexibility of the MKME model. The first dataset refers to waiting times (in minutes) before the service of 100 bank customers. This dataset is studied by [24]. The data observations are: 3.2, 0.8, 4.6, 1.9, 0.8, 3.3, 6.2, 4.7, 6.2, 8, 7.7, 9.7, 12.5, 9.8, 12.9, 18.1, 17.3, 27, 1.3, 31.6, 3.5, 6.2, 4.7, 8.2, 13, 10.7, 18.2, 13, 1.5, 33.1, 1.8, 3.6, 1.9, 4, 2.6, 4.8, 4.1, 4.9, 6.3, 4.9, 6.7, 4.2, 8.6, 11, 6.9, 8.6, 11.2, 10.9, 8.6, 11, 13.7, 11.2, 13.6, 13.3, 18.4, 19, 18.9, 38.5, 2.1, 4.2, 5, 4.3, 5.3, 7.1, 5.5, 7.1, 8.8, 7.1, 8.9, 11.1, 8.8, 13.9, 19.9, 14.1, 20.6, 2.7, 21.3, 4.4, 2.9, 4.3, 3.1, 21.9, 4.4, 5.7, 9.6, 7.1, 6.1, 7.4, 8.9, 7.6, 5.7, 11.5, 9.5, 11.9, 15.4, 12.4, 17.3, 15.4, 23, 21.4.
The second dataset is discussed [25], and it contains annual maximum flood levels over a 20-year period of the Susquehanna River at Harrisburg, Pennsylvania. These flood levels are measured in millions cubic of feet per second. The data observations are: 0.494, 0.654, 0.315, 0.297, 0.449, 0.379, 0.402, 0.379, 0.423, 0.324, 0.740, 0.269, 0.418, 0.416, 0.412, 0.338, 0.484, 0.392, 0.265, 0.613.
The third dataset is analyzed by [26], and it refers to remission times (in months) for 128 bladder cancer patients. This dataset contains the following observations: 3.48, 0.08, 2.09, 4.87, 8.66, 6.94, 13.11, 0.20, 23.63, 2.23, 4.98, 3.52, 6.97, 13.29, 9.02, 0.40, 3.57, 2.26, 5.06, 9.22, 7.09, 13.80, 0.50, 25.74, 2.46, 5.09, 3.64, 7.26, 14.24, 9.47, 25.82, 2.54, 0.51, 3.70, 7.28, 5.17, 9.74, 26.31, 14.76, 0.81, 3.82, 2.62, 5.32, 10.06, 7.32, 14.77, 2.64, 32.15, 3.88, 7.39, 5.32, 10.34, 34.26, 14.83, 0.90, 4.18, 2.69, 5.34, 10.66, 7.59, 15.96, 1.05, 36.66, 2.69, 5.41, 4.23, 7.62, 16.62, 10.75, 43.01, 2.75, 1.19, 4.26, 7.63, 5.41, 17.12, 12.63, 1.26, 46.12, 2.83, 5.49, 4.33, 7.66, 17.14, 11.25, 79.05, 2.87, 1.35, 5.62, 11.64, 7.87, 17.36, 3.02, 1.40, 4.34, 7.93, 5.71, 11.79, 1.46, 18.10, 4.40, 8.26, 5.85, 11.98, 1.76, 19.13, 3.25, 6.25, 4.50, 8.37, 2.02, 22.69, 12.02, 3.31, 6.54, 4.51, 8.53, 20.28, 12.03, 2.02, 6.76, 3.36, 12.07, 2.07, 21.73, 3.36, 8.65, 6.93.
The fourth dataset is studied by [27], and it refers to the number of vehicle fatalities for 39 counties in South Carolina in the year 2012. The data observations are: 50, 22, 13, 17, 26, 4, 9, 27, 9, 48, 31, 20, 6, 12, 5, 9, 14, 16, 33, 9, 68, 3, 20, 51, 4, 13, 17, 2, 16, 52, 6, 48, 12, 23, 10, 8, 13, 1, 15.
The fifth dataset is considered by [28]. This dataset refers to the time between failures for 30 repairable items, and its observations are: 0.70, 0.11, 1.43, 0.71, 2.63, 0.77, 1.49, 2.46, 3.46, 0.59, 1.23, 0.74, 1.17, 0.94, 0.40, 4.36, 1.74, 2.23, 4.73, 0.45, 1.46, 1.06, 0.30, 2.37, 1.82, 0.63, 1.24, 1.23, 1.86, 1.97.
The ML method is used to estimate the model parameters from each dataset, and the R program is used to obtain different computations. The fitting performance of the MKME distribution is compared to other competing E models including the generalized E (GE) [29], Marshall–Olkin E (MOE), alpha-power E (APE) [30], generalized DUS E (GDUSE) [31], generalized inverted E (GIE) [32], KME, and E distributions.
The fitting performance of the proposed MKME model and other competing distributions is explored using some goodness-of-fit measures including the Anderson–Darling Cramér–von Mises
and Kolmogorov–Smirnov (KS) statistics with its associated p-value.
The values of the four measures, ,
, and KS as well as the p-value, of the fitted models are given in Tables Tables 7–11 for the five datasets, respectively. These tables also display the ML estimates and standard errors (SEs) of the parameters of the MKME distribution and other rival models. It is shown that, the MKME distribution has the lowest values of
,
, and KS statistics and largest p-value, showing its close fit to the five analyzed datasets. Furthermore, the fitting performance of the MKME model is explored visually through the plots of the PDF, CDF, SF, and probability-probability (PP) for all datasets. The plots are presented in Figs Fig 8, Fig 9, Fig 10. The plots indicate that the MKME provides the best fit to all datasets.
7.2 Applications of the DMKME Model
In this subsection, we use two real count datasets to illustrate the flexibility of the DMKME distribution as compared to the existing discrete models including the transmuted record type geometric (TRTG) [33], exponentiated discrete Lindley (EDL) [34], natural discrete Lindley (NDL) [35], discrete Lindley (DL) [36], geometric (Gc), discrete Ramos-Louzada (DRL) [37], and discrete Poisson-Lindley (DPL) [38] distributions.
The first dataset represents the final marks of mathematics examination for 48 slow space students. This exam was done in the Indian Institute of Technology at Kanpur as discussed in [39]. The data are: 60, 29, 25, 50, 15, 13, 39, 27, 14, 15, 40, 18, 7, 7, 8, 19, 37, 12, 18, 5, 21, 15, 44, 86, 15, 14, 15, 70, 6, 50, 23, 58, 19, 23, 11, 6, 34, 18, 34, 12, 4, 28, 20, 23, 65, 19, 31, 21.
The second dataset represents the infant mortality rate per 1000 live births for some nations in 2021. The dataset is reported at https://data.worldbank.org/ indicator/SP.DYN.IMRT.IN. The data are: 12, 56, 10, 3, 22, 69, 7, 6, 11, 19, 44, 27, 13, 7, 12, 4, 3, 11, 27, 84, 25, 35, 6, 14, 6, 11.
Table 12 shows the ML estimates of the parameters of the discrete models and their corresponding SEs (in parenthesis) for the two count datasets. Furthermore, Table 12 provides the goodness-of-fit measures for all discrete models.
The values in Table 12 indicate that the DMKME distribution outperforms all competing discrete models, with the lowest values for , f
and KS measures and the highest p-value, for the two count datasets. The PP plots for the two count datasets are displayed in Figs 11 and 12, respectively. These plots support the superior fit of the DMKME model, which provides a closer fit for both datasets as compared to other discrete distributions.
8 Some conclusions
We proposed two new classes of distributions called the modified Kavya-Manoharan-G (MKM-G) and discrete modified Kavya-Manoharan-G (DMKM-G) families. The MKM-G class extends and improves the flexibility of the Kavya-Manoharan (KM) family. Five special models of the MKM-G and DMKM-G families are provided. These special models have the advantage of being capable of modeling different shapes of aging and failure criteria. Basic mathematical properties of the MKM-G family are explored. We discuss eight approaches for estimating the parameters of the MKM-exponential (MKME) distribution, evaluating their performance through numerical simulations. We demonstrate that the MKME distribution provides a superior fit to five real-life datasets from the fields of reliability, medicine, engineering, and environmental science, outperforming several existing exponential models. Additionally, two count datasets are analyzed to illustrate the flexibility of the DMKM-exponential model. The newly discrete model provides better fits as compared to other important discrete distributions. Overall, the proposed models provide flexible and effective alternatives to existing distributions for modeling both lifetime and count data across diverse applied fields.
Future work on the MKM-G and DMKM-G families could include extending these models to bivariate and multivariate distributions for modeling dependent data in complex applications, as well as further generalizing them to incorporate more flexible hazard rate functions. Advanced estimation techniques, such as Bayesian or machine learning approaches, could be explored to improve accuracy in large datasets. The families could also be tested on big data from industries like finance, healthcare, and telecommunications. A comparative study with other flexible distribution families, the development of user-friendly software for implementation, and the extension to handle censored or truncated data are additional areas for future research.
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