https://orcid.org/0000-0001-8070-956X
Figures
Abstract
As the amount and complexity of engineering data that need to be analyzed and interpreted continue to increase, the development of new distributions with outstanding adaptability is necessary. The aim of this work is to improve the precision of data modeling, particularly with respect to reliability and lifetime analyses. In this regard, a novel distribution called the Lomax Kavya Manoharan exponential (LKME) distribution derived from the exponential form of a hazard rate function is proposed. The introduction of the Kavya Manoharan exponential distribution with the properties of the Lomax distribution promotes the adaptability to capture different patterns of failure rates, thereby providing a better fit for lifetime data. The LKME distribution is highly flexible and accommodates almost all possible forms of densities, including symmetric, skewed, and inverted J-shaped, as well as diverse shapes of the hazard rate function. This ensures its suitability for modeling various applications in engineering and other fields. Monte Carlo simulations are performed to examine the performance of several classical estimation methods according to benchmarks, such as absolute bias and mean squared error. Furthermore, five engineering datasets are analyzed using the novel LKME distribution, which provides a better fit than comparison distributions, as demonstrated by different goodness-of-fit metrics.
Citation: Alnashri H, Baaqeel H, Alsulami D, Baharith L (2025) Analysis of engineering data with an innovative generalization of the Lomax distribution. PLoS One 20(10): e0334323. https://doi.org/10.1371/journal.pone.0334323
Editor: Mostafa Tamandi, Rafsanjan University of Vali Asr, IRAN, ISLAMIC REPUBLIC OF
Received: May 11, 2025; Accepted: September 25, 2025; Published: October 27, 2025
Copyright: © 2025 Alnashri 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 relevant data are within the manuscript.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Understanding the random and complex behavior in engineering applications and other real-world phenomena through statistical distributions is crucial in data analysis and interpretation. The first essential step to be taken for any statistical analysis is to choose a distribution that precisely models the data, as choosing an incorrect or unsuitable distribution can lead to misleading reliability and risk predictions. As real-world data become increasingly complex, general-purpose distributions with fixed structures can no longer capture this complex behavior, creating the need for more flexible and adaptable distributions.
Recently, several researchers have attempted to construct new families of statistical distributions using strategies meant to generalize classical distributions and improve their fit to complex real data. Following the T-X-transformed family introduced in [1], applying a suitable transformation function to any probability distribution has become a well-known approach for developing new distributions. The T-X-transformed family has contributed to the development of several distribution families, such as the new Weibull-X family proposed by [2], the new Kumaraswamy generalized family of distributions proposed by [3], the new modified Kies family proposed by [4], and the exponential T-X family proposed by [5]. These families, among others, have been used by researchers to develop new distributions. Some examples include the exponentiated odd Lomax exponential distribution proposed by [6], a new extended Gumbel distribution proposed by [7], a new modified Weibull distribution proposed by [8], the Burr XII-moment exponential distribution proposed by [9], the odd Lindley Weibull distribution proposed by [10], and the half logistic-truncated exponential distribution proposed by [11], all of which expand the toolbox for data modeling.
Recently, the Kavya Manoharan (KM) transformation proposed by [12] has been employed in the formation of new distributions. This transformation does not add extra parameters to the base distribution; thus, it is considered a parameter-parsimonious method. Accordingly, various lifetime distributions based on the KM transformation, including the AKM exponential distribution proposed by [13], the KM generalized exponential distribution proposed by [14], the KM inverse length bias distribution proposed by [15], the KM logistic distribution proposed by [16], and the KM Burr X distribution proposed by [17], have been established. Additionally, [12] proposed the KM exponential distribution characterized by the following probability density function (PDF) and cumulative distribution function (CDF):
Moreover, [18] introduced the Lomax-G family with the following PDF and CDF:
where , λ is a location parameter, G(t) and g(t) represent the CDF and PDF of the baseline distribution, respectively.
In this study, we present a new probability distribution termed Lomax Kavya Manoharan exponential, a new member of the aforementioned Lomax-G family. The introduction of the LKME distribution is motivated by the following considerations:
- Formulate a novel distribution by extending the KME distribution using the favorable features of the Lomax-G family.
- Offer a distribution with exceptional data-modeling versatility and accuracy in engineering and several other areas based on lifetime and reliability data analyses.
- Cover a wide range of density shapes, including symmetric and asymmetric densities, and diverse hazard rate function shapes.
- Present a robust distribution as an alternative to existing distributions for fitting and modeling different engineering data types.
The remainder of this article is organized as follows. In Sect 2, we introduce the LKME distribution. Some statistical properties of the LKME distribution are described in Sect 3. The parameter estimation methods for the LKME distribution are detailed in Sect 4. Simulations to assess the performance of the estimation methods are presented in Sect 5. The LKME distribution is applied to five real-world datasets in Sect 6 to assess its performance. Sect 7 provides some concluding remarks.
2 The Lomax Kavya-Manoharan exponential distribution
In this section, we present the LKME distribution. Its CDF and PDF can be obtained by inserting Eqs (1) and (2) into Eqs (3) and (4) as follows:
where λ is a location parameter.
The survival function, S(x), for the LKME is given by
and the hazard rate function, H(x), is expressed as follows:
The density function and hazard function H(x) under different parameter settings are shown in Figs 1 and 2, respectively. The shape of the density function can be symmetric, right-skewed, left-skewed or even an inverted J-shape, as seen in Fig 1. Similarly, the hazard function H(x) in Fig 2 can take various forms: it can increase monotonically, decrease monotonically, be J-shaped, or be reversed J-shaped. These variations highlight the flexibility of the LKME model in effectively capturing a wide range of data distribution behaviors.
2.1 Special cases of the LKME distribution
- The Kavya-Manoharan exponential (KME) distribution is obtained when
in Eq (6).
- The Kavya-Manoharan Weibull (KMW) distribution is obtained when
in Eq (6).
- Kavya-Manoharan Lomax (KM-Lomax) distribution with
is obtained when
in Eq (6).
- The generalized Kavya-Manoharan exponential (GKME) distribution is obtained by substituting
into Eq (6) and setting
.
- The DUS-Lomax distribution is obtained by substituting
in Eq (6) and setting
.
2.2 Linear representations of the LKME density
This subsection provides a detailed expansion of the LKME PDF function in (6), Using the following exponential expansion and binomial expansions.
results in a linear representation of the LKME density as follows:
3 Some statistical properties of LKME
3.1 Quantile function
The quantile function of the LKME model can be obtained by inverting the distribution function given in (6).
The median can be determined by substituting p = 0.5 into Eq (12), as shown below.
3.2 The Galton skewness and Moors Kurtosis
The Galton skewness (GSk) is a measure of distribution symmetry [19] and is defined as follows:
The Moors kurtosis (MKur) [20], derived from octiles, is defined as follows:
where x(.) denotes the quantile function of the LKME model in Eq (12).
Fig 3 shows the plots of the Moors kurtosis and Galton skewness as functions of λ and θ. The Moors kurtosis plot indicates that increasing λ and decreasing θ result in higher Moors kurtosis and a sharper distribution. In contrast, the Galton skewness plot reveals a strong correlation between Galton skewness and small values θ, which are associated with variations in distribution asymmetry. These results illustrate the sensitivity of the factors used as descriptors of the distribution properties.
3.3 Moments
If , then the rth moment of X can be expressed as follows.
By substituting , we obtain
Next, we use the following series expansion.
Thus, the rth moment of the LKME distribution can be expressed as follows:
Consequently, the mean of the LKME distribution can be expressed as follows:
Therefore, the variance of the LKME distribution is given by
3.4 Moment generation function
The moment generating function (MGF) of the LKME distribution can be obtained as follows:
Therefore,
3.5 Characteristic function
The characteristic function of the LKME distribution is expressed as follows:
3.6 Probability weighted moment
The probability weighted moment (PWM) of the LKME distribution is defined as follows:
The series expansion in Eq (18) and the following exponential series are employed.
Thus, we obtain
By substituting , we obtain the following.
Using the series expansion in Eq (18), the PWM of the LKME distribution can be expressed as follows:
3.7 Order statistics
Let represent a random sample of size n drawn from the LKME distribution, where xr:n denotes the rth order statistic. The PDF of xr:n can be expressed as follows:
By substituting Eqs (5) and (11) into Eq (32) and then applying the series expansions in Eqs (18) and (28), we obtain the rth order statistic of the LKME distribution as follows:
Proof. See S1 Appendix
3.8 R’enyi entropy
The R’enyi entropy of order u is given by
Thus, the R’enyi entropy of the LKME distribution can be expressed as follows:
Proof. See S2 Appendix.
3.9 Shannon entropy
If a random variable X follows the LKME distribution with the PDF given by Eq (6), then the Shannon entropy, SEX, is expressed as follows:
Using the following sequence expansions, along with the sequences in Eqs (9) and (18),
Therefore, the Shannon entropy of the LKME distribution is given by
Proof. See S1 Appendix.
4 Estimation methods
To evaluate the performance of the LKME distribution, we employed five different estimation techniques, namely, maximum likelihood estimation (MLE), percentile estimation (PE), least squares estimation (LS), weighted least squares estimation (WLS), and the Cramer von Mises method (CVM), to assess the fit of the LKME distribution and accurately quantify its parameters.
4.1 Maximum likelihood method
If a random sample of size n is drawn from LKME distribution, then the log-likelihood function (l) for the parameters
is expressed as follows:
The MLE of the parameters is obtained by computing the partial derivatives of Eq (39) with respect to each parameter, equating the resultant equations to zero, and solving for the parameters, as follows:
The system of Eqs (40)–(43) can be solved using some iterative optimization techniques such as R packages [21].
4.2 Percentiles method
Suppose that is a random sample of size n from the LKME distribution. Then, the percentile estimation (PE) method is
where xi is the ith order statistics of the sample,
The parameter estimates of using the PE method are obtained by minimizing the sum of squares of the difference between the observed sample order statistics and the expected values of the fitted distribution in the corresponding percentiles.
4.3 Ordinary least squares estimators
Let be a random sample of size n from the LKME distribution. The ordinary least squares estimation (LSE) method for estimating the LKME parameters is as follows:
That is, the parameters are optimized by minimizing the sum of the squared differences. Thus, the LSE estimator for the unknown parameters of the LKME distribution is obtained by minimizing the following expression.
where xi is the ith order statistics of the sample.
4.4 Weighted least squares estimators
Let be a random sample from the LKME distribution. Then, the weighted least squares estimator (WLS) is given by
The WLS minimizes the following expression to estimate the LKME parameters.
where xi is the ith order statistics of the sample.
4.5 Cramer-von Mises minimum distance method
The Cramer-von Mises (CVM) method works by minimizing the difference between the estimated CDF and empirical CDF. Specifically, the aim of this method is to determine the parameter values that best align the theoretical distribution with the observed distribution.
By incorporating Eq (5) with Eq (49), the CVM estimator for the LKME parameters is obtained by minimizing the following.
5 Simulation study
This section presents the numerical results of a Monte Carlo simulation to assess the performance of five estimation methods: MLE, PE, LSE, WLS, and CVM. Data were generated from the LKME distribution using Eq (12), where p was sampled from Uniform(0,1). We consider six sample sizes and five sets of parameter values specified as follows:
- Set I:
- Set II:
- Set III:
- Set IV:
- Set V:
Parameter estimates are obtained using the “optim” function in R statistical software [21]. In addition, the absolute bias (Bias) and mean squared error (MSE) are calculated as follows:
where is the true parameter value,
is the corresponding estimate, and n is the sample size.
The average estimates of the parameters based on 1,000 replications using various approaches, with the corresponding and MSE, are displayed in Tables 1–5.
The values in Tables 1–5 demonstrate that the average estimates of the LKME parameters approach their actual values as the sample size increases. This finding indicates that the estimation becomes more accurate as the dataset size increases. Furthermore, Figs 4–8 show comparative results of the different estimation methods, demonstrating their mean squared errors (MSE) under large sample sizes (n=200, 300, and 500). As indicated by these figures, compared with all the other estimation methods, the ML method exhibits superior performance and provides the most accurate estimates. This is succeeded by the PE and WLE methods for most circumstances. Moreover, except for parameters α and λ in case 3, it is clear that the LSE and CVM are the least precise methods among all the examined methods, as they yield the highest MSEs. All the statistical analyses were performed in R Statistical Software version 4.3.2 (R Core Team, 2023) [21].
6 Applications
The LKME model was used to fit real-world datasets, and it outperformed the comparison distributions on the five datasets. Here is a brief overview of the dataset.
Service Time Dataset
The first dataset is extracted from [22], and contains 63 recorded service times for the aircraft windshields.
0.046, 1.436, 2.592, 0.140, 1.492, 2.600, 0.150, 1.580, 2.670, 0.248, 1.719, 2.717, 0.280, 1.794, 2.819, 0.313, 1.915, 2.820, 0.389, 1.920, 2.878, 0.487, 1.963, 2.950, 0.622, 1.978, 3.003, 0.900, 2.053, 3.102, 0.952, 2.065, 3.304, 0.996, 2.117, 3.483, 1.003, 2.137, 3.500, 1.010, 2.141, 3.622, 1.085, 2.163, 3.665, 1.092, 2.183, 3.695, 1.152, 2.240, 4.015, 1.183, 2.341, 4.628, 1.244, 2.435, 4.806, 1.249, 2.464, 4.881, 1.262, 2.543, 5.140.
Breaking stress of carbon fibers.
The second dataset includes 100 observations and was derived from [23],
3.70,2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19,3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39,2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83,1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2.00,1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.70, 2.03, 1.80, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65.
The tensile strength.
The third dataset, containing 69 observations, is obtained from [24] and is given by:
0.312, 0.314, 0.479, 0.552, 0.700, 0.803, 0.861, 0.865, 0.944, 0.958, 0.966, 0.997, 1.006, 1.021, 1.027, 1.055, 1.063, 1.098, 1.140, 1.179, 1.224, 1.240, 1.253, 1.270, 1.272, 1.274, 1.301, 1.301, 1.359, 1.382, 1.382, 1.426, 1.434, 1.435, 1.478, 1.490, 1.511, 1.514, 1.535, 1.554, 1.566, 1.570, 1.586, 1.629, 1.633, 1.642, 1.648, 1.684, 1.697, 1.726, 1.770, 1.773, 1.800, 1.809, 1.818, 1.821, 1.848, 1.880, 1.954, 2.012, 2.067, 2.084, 2.090, 2.096, 2.128, 2.233, 2.433, 2.585, 2.585.
Single fibers with gauge lengths of 20mm.
The fourth dataset is obtained from [13] and contains 69 observations:
1.312, 1.314, 1.479, 1.552, 1.7, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 2.14, 2.179, 2.224, 2.24, 2.253, 2.27, 2.272, 2.274, 2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.49, 2.511, 2.514, 2.535, 2.554, 2.566, 2.57, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.77, 2.773, 2.8, 2.809, 2.818, 2.821, 2.848, 2.88, 2.954, 3.012, 3.067, 3.084, 3.09, 3.096, 3.128, 3.233, 3.433, 3.585, 3.858.
Single fibers with gauge lengths of 50mm.
The fifth dataset was sourced from [13], and contains 69 observations:
0.562, 0.564, 0.729, 0.802, 0.95, 1.053, 1.111, 1.115, 1.194, 1.208, 1.216, 1.247, 1.256, 1.271, 1.277, 1.305, 1.313, 1.348, 1.39, 1.429, 1.474, 1.49, 1.503, 1.52, 1.522, 1.524, 1.551, 1.551, 1.609, 1.632, 1.632, 1.676, 1.684, 1.685, 1.728, 1.74, 1.761, 1.764, 1.785, 1.804, 1.816, 1.824, 1.836, 1.879, 1.883, 1.892, 1.898, 1.934, 1.947, 1.976, 2.02, 2.023, 2.05, 2.059, 2.068, 2.071, 2.098, 2.13, 2.204, 2.317, 2.334, 2.34, 2.346, 2.378, 2.483, 2.683, 2.835, 2.835, 2.262.
The fitting performance of the LKME model across the five datasets is assessed by comparison with the fits of the following distributions.
- Lomax distribution with the CDF function:
- The Kavya-Manoharan exponential (KME) distribution with the CDF function is given in (3).
- The Kavya–Manoharan generalized exponential distribution (KMGE) by [14]
- The Kavya–Manoharan transformation inverse length biased exponential (KMILBE) by [15]
- The alpha power transformed generalized Lomax (APTGL) distribution, by [25]
where
We computed several goodness-of-fit (GoF) metrics, namely, the Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), Hannan Quinn information criterion (HQIC), Anderson Darling (A-D) test and Kolmogorov Smirnov (KS) statistic, with their corresponding p values to compare these distributions with those of the LKME model.
Tables 6–10 demonstrate that the fit of the LKME model is superior to that of the other distributions for the five datasets. Compared with the other fitted models, the LKME model has the lowest GoF metric values and the highest p value. This finding verifies that the LKME model outperforms rival distributions and demonstrates its efficacy in appropriately describing the given datasets.
Furthermore, Figs 9–13 show the fitted PDF and CDF for each dataset. These plots indicate that compared with the other models, the LKME model is the most effective at capturing the data skewness.
7 Conclusions
In this paper, we present a new member of the Lomax-G family called the Lomax Kavya Manoharan exponential distribution. The LKME density can be left-skewed, right-skewed, symmetric, semi-symmetric, or inverted J-shaped. In addition, the hazard function can exhibit a broad spectrum of asymmetric patterns, including increasing, decreasing, J shape, and inverted J shape with different tail behaviors, allowing applications to real-world data. The statistical properties of the LKME distribution, such as quantiles, medians, moments, characteristic functions, order statistics, and R’enyi and Shannon entropies, were calculated. The LKME model’s parameters were estimated using five distinct estimation techniques: MLE, PE, LSE, WLS, and CVM. Monte Carlo simulations were performed to assess the accuracy and reliability of the parameter estimations, which revealed that the ML approach outperforms all the other estimation methods, yielding the most precise values. The superior fit of the LKME model over five real-world datasets compared with that of competing distributions demonstrates its applicability to real-world scenarios. These results highlight the remarkable ability of the LKME distribution to model and understand complex phenomena across multiple domains.
Acknowledgments
We acknowledge the editor in chief and the reviewers for their valuable suggestions to improve the quality of this manuscript.
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