https://orcid.org/0000-0002-7772-0386
Figures
Abstract
The Weibull distribution is an important continuous distribution that is cardinal in reliability analysis and lifetime modeling. On the other hand, it has several limitations for practical applications, such as modeling lifetime scenarios with non-monotonic failure rates. However, accurate modeling of non-monotonic failure rates is essential for achieving more accurate predictions, better risk management, and informed decision-making in various domains where reliability and longevity are critical factors. For this reason, we introduce a new three parameter lifetime distribution—the Modified Kies Weibull distribution (MKWD)—that is able to model lifetime scenarios with non-monotonic failure rates. We analyze the statistical features of the MKWD, such as the quantile function, median, moments, mean, variance, skewness, kurtosis, coefficient of variation, index of dispersion, moment generating function, incomplete moments, conditional moments, Bonferroni, Lorenz, and Zenga curves, and order statistics. Various measures of uncertainty for the MKWD such as Rényi entropy, exponential entropy, Havrda and Charvat entropy, Arimoto entropy, Tsallis entropy, extropy, weighted extropy and residual extropy are computed. We discuss eight different parameter estimation methods and conduct a Monte Carlo simulation study to evaluate the performance of these different estimators. The simulation results show that the maximum likelihood method leads to the best results. The effectiveness of the newly suggested model is demonstrated through the examination of two different sets of real data. Regression analysis utilizing survival times data demonstrates that the MKWD model offers a superior match compared to other current distributions and regression models.
Citation: Alahmadi AA, Albalawi O, Khashab RH, Johannssen A, Nasiru S, Almarzouki SM, et al. (2025) Modeling of lifetime scenarios with non-monotonic failure rates. PLoS ONE 20(1): e0314237. https://doi.org/10.1371/journal.pone.0314237
Editor: Umair Khalil, Abdul Wali Khan University Mardan, PAKISTAN
Received: May 10, 2024; Accepted: November 7, 2024; Published: January 22, 2025
Copyright: © 2025 Alahmadi 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 paper.
Funding: We acknowledge nancial support from the Open Access Publication Fund of the University of Hamburg. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Probability distributions form the foundation of parametric statistical analysis, and therefore researchers are constantly developing new distributions and/or modifying existing ones. For instance, the Weibull distribution (WD) [1] is one of the established distributions that has gained considerable attention in reliability analysis and lifetime modeling. The WD is a highly adaptable statistical tool extensively utilized in many sectors like engineering, health, finance, and environmental sciences. It is valued for its effectiveness in accurately representing dependability and failure statistics. Engineering relies on this tool to approximate the duration of product functionality and assess the likelihood of failures. Similarly, medicine uses it in survival analysis to anticipate patient outcomes. Finance uses it to evaluate financial risks, while environmental sciences employ it to simulate data such as rainfall and temperature extremes. The versatility of the WD makes it indispensable for data analysis and forecasting, with several research papers confirming its usefulness in various fields [2, 3]. The cumulative distribution function (cdf) and probability density function (pdf) of the WD are given by
(1)
and
(2) y > 0, where β > 0 and ϑ > 0 are two scale and shape parameters, respectively. The WD is an extreme value distribution and frequently used in modeling extreme events. The practicality of the WD in reliability/survival analysis and in modeling of lifetime data is high due to its property in handling lifetime scenarios with decreasing, constant or increasing failure rates. However, it fails to offer a good fit to lifetime data that exhibits non-monotonic failure rates such as bathtub (for example, human life cycle) or upside-down bathtub (for example, machine life cycle). Due to the drawbacks of the WD, a number of variants of the distribution has been proposed in the literature with the goal of enhancing its modeling capabilities as well as making it suitable for specific modeling lifetime phenomena. There are some recent discussed variants, such as the exponentiated WD [4], transmuted additive WD [5], Kumaraswamy transmuted exponentiated modified WD [6], Topp-Leone Modified WD [7], Burr X exponentiated WD [8], Kavya-Manoharan exponentiated WD [9], Marshall-Olkin power-generalized WD [10], truncated Cauchy power Weibull-G [11], alpha power transformed Weibull-G [12], Weighted WD [13], exponentiated power generalized Weibull power series family [14], exponentiated truncated inverse Weibull-G [15], odd inverse power generalized WD [16], extended inverse WD [17], Weibull WD (WWD) [18], and exponentiated WWD [19].
However, it is still a challenge to develop a distribution that is well suited for modeling lifetime scenarios characterized by non-monotonic failure rates reflecting real-world situations where the failure rate of a system or component may vary over time. To motivate the need for this study, we briefly describe the reasons why it is necessary to have a distribution at hand that can model lifetime scenarios with non-monotonic failure rates in an appropriate way:
- As non-monotonic failure rates occur when the probability of failure changes over the lifetime of a system or component, a suitable distribution would allow for a more accurate representation of the actual failure behavior observed in many systems.
- Understanding the pattern of failure rates over time is crucial for assessing risks associated with a system. If failure rates are non-monotonic, there may be periods of increased risk followed by periods of decreased risk. Properly modeling these fluctuations helps in identifying and managing risks effectively.
- Reliability analysis involves predicting the likelihood of a system operating without failure over a certain period. By using a distribution that accounts for non-monotonic failure rates, engineers and analysts can more precisely estimate the reliability of a system and make informed decisions regarding maintenance, design improvements, or replacement strategies.
- In many industries, decisions regarding maintenance schedules, warranty policies, and product design depend heavily on accurate assessments of failure rates. Using a distribution that can handle non-monotonic failure rates ensures that these decisions are based on realistic expectations and reduce the likelihood of unexpected failures or unnecessary costs.
- Non-monotonic failure rates are common in various fields such as engineering, finance, healthcare, and beyond. Having a distribution that can accommodate these patterns makes it applicable across a wide range of industries and scenarios.
In this study, addressing the above aims, we develop a new three-parameter lifetime distribution, the so called Modified Kies Weibull distribution (MKWD) by utilizing the modified Kies (MK) family of distributions [20]. The cdf and pdf of the MK family of distributions are
(3)
and
(4)
respectively, where g(y; ϖ) and G(y; ϖ) are the parent pdf and cdf for the baseline distribution with set of parameters ϖ, and ζ being the shape parameter of the family. In addition, Al-Babtain et al. [20] utilized the binomial and exponential series to rewrite the pdf as a linear combination of the exponentiated family,
(5)
where
. To determine some measures of uncertainty, we determine the expansion of [f(y; ϖ)]∇ via
(6)
where
.
As a result of the above discussion, this study addresses the following issues:
- We enhance the adaptability of the Weibull model by utilizing the MK family. To be more specific, declining, right-skewed and unimodal forms are denoted for the pdf but the hazard rate function (hrf) can be bathtub, declining, increasing and J-shaped for the MKWD.
- Some statistical properties of the MKWD, such as moments, mean, variance, skewness, kurtosis, coefficient of variation, index of dispersion, moment generating function, incomplete moments, conditional moments, Bonferroni (BON), Lorenz (LOR), and Zenga (ZEN) curves [21, 22], and order statistics (OS), are calculated.
- Various measures of uncertainty for the MKWD such as Rényi entropy (RE) [23], exponential entropy (EE) [24], Havrda and Charvat entropy (HCE) [25], Arimoto entropy (AE) [26], Tsallis entropy (TE) [27], extropy (Ex) [28], weighted extropy (WEx) [29] and residual extropy (REx) [30] are computed.
- Eight different methods are implemented to estimate the parameters β, ϑ and ζ of the MKWD. These methods are maximum likelihood estimation (MLE), Cramer-von-Mises estimation (CME), maximum product of spacings estimation (MPSE), least squares estimation (LSE), weighted least squares estimation (WLSE), minimum spacing absolute-log distance estimation (MSALDE), percentile estimation (PE) and minimum spacing square-log distance estimation (MSSLE).
- We develop a quantile regression to analyze the connections between dependent and independent variables, and illustrate the implementation of our models using survival times data.
This study addresses the following four research questions (RQ) and related hypotheses (H) to contribute to the field of reliability analysis and lifetime modeling:
- RQ1. Can the MKWD effectively model lifetime scenarios characterized by non-monotonic failure rates, which are inadequately handled by the traditional WD?
- H1. The MKWD will provide a more accurate fit for lifetime data with non-monotonic failure rates compared to the existing Weibull variants.
- RQ2. What are the statistical properties of the MKWD, and how do these properties enhance its adaptability in different reliability analysis contexts?
- H2. The MKWD will exhibit diverse statistical characteristics (e.g., moments, entropy measures) that make it suitable for a wide range of applications in reliability and survival analysis.
- RQ3. Which parameter estimation method(s) provide the most reliable estimates for the MKWD parameters in practical applications?
- H3. MLE will outperform other estimation methods in terms of accuracy and reliability.
- RQ4. How does the MKWD perform in real-world data applications?
- H4. The MKWD will demonstrate superior modeling performance when applied to real-world datasets, providing better fits and more accurate predictions than competing distributions.
The subsequent sections of this work are structured in the following manner: the development of the MKWD is described in Section 2. The statistical features of the MKWD are outlined in Section 3. Some measures of entropy and extropy are discussed in Sections 4 and 5, respectively. Section 6 discusses eight approaches to estimate the parameters of the MKWD. Additionally, Monte Carlo simulations are conducted to evaluate the adequacy of these strategies in Section 7. Data analysis using two real data sets is conducted in Section 8. Section 9 presents the formulation of the quantile regression, followed by simulation experiments and applications. The results of the study are summarized in Section 10.
2 Formulation of the Modified Kies Weibull distribution
In this section, we construct the MKWD by inserting (1) and (2) into (3) and (4). Then, the MKWD has the following cdf, pdf, reliability function (rf) and hrf:
(7)
(8)
and
The reversed hrf (rhrf), cumulative hrf (chrf), odd ratio (OR), failure rate average (FRA) and Mills ratio (MR) of the MKWD are
and
respectively. Fig 1 shows that the pdf for the MKWD can have declining, right-skewed and unimodal forms, and the hrf can be bathtub, declining, increasing and J-shaped for the MKWD.
3 Statistical properties
Regarding the statistical characteristics of the MKWD, we discuss them in this section.
3.1 Quantile function
The quantile function is a frequently employed tool in general statistics for determining the mathematical properties of a distribution and percentiles. The quantile function of the MKWD, say Q(u), 0 < u < 1, defined by F(Q(u)) = u, can be computed as follows:
(9)
To investigate the median (m) of the MKWD, we set u = 0.5 in Eq (9) as follows:
3.2 Moments and moment generating function
In statistics, moments of probability distributions are measures related to the structure of the graph. The variance is the second moment around the mean in a probability distribution, whereas the mean value corresponds to the first moment. The ratio of the third mean moment to the variance is the definition of the skewness measure. The ratio of the standard deviation to the fourth power to the fourth moment about the mean is the definition of the kurtosis measure. For any positive integer r, the rth moment of the MKWD can be determined as follows:
(10)
where φ = (β, ϑ, ζ). By inserting (5) into (10), we have
(11)
By using the binomial expansion in the last term in (11), we get
(12)
where
Let z = β(k + 1)yϑ, then
(13)
where Γ(., .) is the gamma function (GFN). The moment generating function of the MKWD is calculated below:
Tables 1 and 2 show the numerical values of the moments ,
,
and
, and the numerical values of the variance (σ2), coefficient of skewness (CS), coefficient of kurtosis (CK) and coefficient of variation (CV) associated with the MKWD. It can be observed from Tables 1 and 2 that as the parameters ϑ and β increase while keeping ζ constant, there is a general trend of decreasing values in the first four moments and σ2, and CS, CK, and CV show an decreasing trend. This pattern indicates a significant sensitivity of the distribution to changes in these parameters of the distribution.
3.3 Incomplete and conditional moments
Incomplete moments are often used to assess inequalities, such as income quantiles, BON, LOR and ZEN curves. The pth incomplete moment of the MKWD is computed as follows:
(14)
where
is the lower incomplete GFN. Conditional moments are essential in many statistical approaches, such as regression and hypothesis testing, since they demonstrate the relationship between variables and their responses in different circumstances. For example, in regression analysis, these moments can predict the dependent variable’s value based on certain independent variable values, providing significant insights into the variable’s behavior under various situations and improving prediction accuracy. The pth conditional moment of the MKWD is computed as follows:
(15)
where
is the upper incomplete GFN.
3.4 Inequality measures
Our primary focus in this subsection is on the LOR, BON, and ZEN curves, which are helpful in demography, econometrics, medicine, survival analysis, and insurance applications. For the MKWD, the LOR, BON, and ZEN curves are provided by
and
respectively, where F(t) and R(t) are the cdf and rf of the MKWD at time t.
3.5 Order statistics
Assume that Y1, Y2, …, Yn are n random samples from the MKWD with pdf (8) and cdf (7). Suppose that Y(1), Y(2), …, Y(n) are the corresponding OS. The pdf of the qth OS is provided as follows:
(16)
By employing (7) and (8) in (16), we obtain the pdf of Y(q) of OS for the MKWD below:
(17)
By setting q = 1 and n in (17), we have the lowest OS and the largest OS for the MKWD as follows:
and
4 Entropy measures
In this section, five different entropy measures for the MKWD, namely, the RE, EE, HCE, AE and TE are computed. Table 3 reports some numerical values of the MKWD’s entropy measures.
4.1 Rényi entropy
The RE of the MKWD can be computed using the following formula
(18)
Now, we want to compute the integral . By inserting (1) and (2) into (6), we get
(19)
By using the binomial expansion for the above Eq (19), we obtain
(20)
where
Let z = (k + ∇) βyϑ, then we have
(21)
and the integral
can be formulated as
(22)
By employing (22) in (18), the RE of the MKWD is given by
(23)
4.2 Exponential entropy
The EE of the MKWD can be computed using the following formula
(24)
By utilizing (22) in (24), the EE of the MKWD is given by
(25)
4.3 Havrda and Charvat entropy
The HCE of the MKWD can be computed using the formula
(26)
By inserting (22) into (26), the HCE of the MKWD is given by
(27)
4.4 Arimoto entropy
The AE of the MKWD can be computed using
(28)
and by employing (22) in (28), the AE of the MKWD is given by
(29)
4.5 Tsallis entropy
The TE of the MKWD can be computed using the formula
(30)
Then, by using (22) in (30), the TE of the MKWD is given by
(31)
In general, Table 3 shows a clear trend where values increase as β rises for both values of ∇. Specifically, the measures RE and EE increase with higher values of β and ∇, while HCE and AE exhibit a similar upward trend, though at a slower rate.
5 Extropy measures
Lad et al. [28] launched Ex in 2015, a new measure of uncertainty. The total log scoring system can be implemented to statistically score forecasting distributions utilizing Ex. For a non-negative random variable Y, the Ex is defined as follows:
(32)
By setting ∇ = 2 in (22) and inserting into (32), the Ex of the MKW is given by:
(33)
The concept of the WEx was introduced in [29] and is defined as follows:
(34)
By setting ∇ = 2 in (22) and substituting the respective expression in (34), the WEx of the MKW is given by:
(35)
Qiu & Jia [30] defined the extropy for residual lifetime Yt as the REx at time t as:
(36)
The REx of the MKWD can thus be expressed as
where Γ(., y) is the upper incomplete GF.
Table 4 shows a few numerical values that are associated with extropy metrics of the MKWD. Overall, Table 4 provides a comprehensive view of how extropy measures vary with different parameter values. The trends suggest that increasing values of ζ and ϑ generally lead to more negative extropy values, indicating increased extropy as these parameters change.
6 Estimation methods
In this section, we implement eight approaches to estimate the parameters β, ϑ and ζ of the MKWD. These methods are MLE, CME, MPSE, LSE, WLSE, MSALDE, PE, and MSSLE.
6.1 Maximum likelihood method
The maximum likelihood method [31, 32] is based on maximizing the (log-)likelihood function to estimate the parameters. This method is one of the most widely used estimation techniques, providing estimators with desirable properties such as consistency and asymptotic efficiency. The log-likelihood function is given by
and the partial derivatives of l are as follows:
To estimate the parameters β, ϑ and ζ it is required to solve the system of equations
,
and
, which has no closed form and therefore is to be solved numerically.
6.2 Cramér-von-Mises method
The Cramér-von-Mises method [33] minimizes the distance between the empirical and theoretical cumulative distribution functions. It is useful when the goal is to achieve a good fit across the entire range of the data, not just the tails. CME depends on minimizing the function
From now on we use the following notations:
(37)
(38)
(39)
and
(40)
The aim is to compute the partial derivatives of C(β, ϑ, ζ) with respect to β, ϑ and ζ, by using Eqs (37), (38), (39) and (40),
Then the system of equations
,
and
is solved using numerical methods.
6.3 Maximum product of spacings method
MPSE is known for providing efficient estimates by maximizing the spacing between ordered statistics. It is particularly useful in continuous distributions. This method needs to maximize the MPS function to estimate the parameters β, ϑ and ζ [34],
where Λi(y(i)) = F(y(i)) − F(y(i−1)), F(y(0)) = 0 and F(y(n+1)) = 1. Using Eqs (37), (38), (39) and (40), the partial derivatives with respect to β, ϑ and ζ are given by
Through solving the equations
,
and
, we get the estimates of β, ϑ and ζ.
6.4 Least squares method
LSE is selected for its straightforward approach, minimizing the squared differences between observed and theoretical quantiles. The LSE depends on minimizing the function [35]
Using Eqs (37), (38), (39) and (40), the partial derivatives of V(β, ϑ, ζ) with respect to the parameters β, ϑ and ζ are
When solving the equations
,
and
, we get the estimates of β, ϑ and ζ.
6.5 Weighted least squares method
WLSE extends LSE by introducing weights, giving more importance to certain data points, typically in the tails. It is selected for its flexibility and ability to provide better estimates in scenarios where some observations are more reliable or important than others. The WLSE method [35] depends on minimizing the function
Again, by using Eqs (37), (38), (39) and (40), the partial derivatives with respect to the parameters β, ϑ and ζ are obtained as:
After solving the system of equations
,
, and
, we get the point estimates of β, ϑ and ζ.
6.6 Minimum spacing absolute-log distance
MSALDE minimizes the absolute logarithmic distance between observed and expected spacings, providing robustness against outliers. It is chosen for its effectiveness in dealing with outliers and small sample sizes. The MSALDE aims to minimize the function
where Λi = F(y(i)) − F(y(i−1)). Following the same steps explained above, we can estimate the parameters β, ϑ and ζ.
6.7 Percentile estimation
PE is a non-parametric method that relies on the percentiles of the distribution, making it less dependent on the underlying distributional assumptions. The PE approach was originally introduced by [36, 37], applied to the Weibull distribution and later on also used for other distributions. The function
is to be minimized to estimate the parameters β, ϑ and ζ by repeating the same steps as discussed above.
6.8 Minimum spacing square-log distance
MSSLE minimizes the square of the logarithmic spacing distances, similar to MSALDE, but with a squared term that can offer different sensitivity to deviations. The MSSLE technique aims to minimize the function
where Λi(yi) = F(yi) − F(yi−1). Following the same steps explained above, we can estimate the parameters β, ϑ and ζ.
7 Simulation
In this section, we employ the software R to conduct a Monte Carlo simulation study, aiming to evaluate the performance of the eight estimation methods outlined in Section 6. Specifically, we calculate the point estimate (mean) and determine the mean square error (MSE) as well as the square root of the mean square error (RMSE) for the parameters β, ϑ, and ζ. These metrics provide a comprehensive assessment of the estimators’ accuracy, bias, and efficiency. To achieve this, we generate random samples of different sizes (n = 20, 50, 100, 200, 320, 450) from the MKWD. Each sample size was chosen to represent different scenarios, from small to large sample conditions, allowing us to evaluate the estimators’ performance across a broad spectrum.
For each combination of parameter values and sample size, we conducted 1000 replications. This number of replications ensures that our simulation results are reliable and provide a stable estimate of the statistical properties of the parameters. The initial values for the parameters β, ϑ, and ζ are {(0.5, 0.5, 0.5), (0.5, 0.5, 0.9), (0.5, 0.9, 0.5), (0.9, 0.5, 0.5), (1.5, 0.5, 0.5), (1.5, 1.5, 0.5)}. They are used in the simulation to cover various distribution shapes and scales, ensuring that our findings are robust and generalized across different scenarios.
The results for the mean, MSE, and RMSE for each estimation method and parameter combination are presented in Tables 5–10. These tables allow for a comprehensive comparison of the performance of the different estimation methods. Additionally, the MSE results in Table 5 are graphically displayed in Figs 2 and 3 to illustrate the trend of the estimators as the sample size increases.
To provide a holistic view, Table 11 summarizes the sum of ranks and overall ranks of the MSE and RMSE across all tables. This comparative analysis highlights the effectiveness of each estimation method. We can deduce the following key findings:
- MSE and RMSE show a decreasing trend with increasing n for all estimation methods. This emphasizes the consistency.
- For all parameters (ϑ, β, ζ), the mean estimates converge towards the initial parameter values as n increases.
- The overall ranks in Table 11 show that MLE leads to the best results when estimating the parameters.
- MLE is the top-performing method, showing the lowest MSE and RMSE values across all parameters and sample sizes.
- MPSE follows closely behind MLE, with competitive MSE and RMSE values. It performs well across different parameter values, especially with larger sample sizes.
- PE is preferred after MLE and MPSE, however, it may require larger sample sizes to achieve optimal performance.
- The other methods according to their ranks (after MLE, MPSE and PE) are MSALDE (which performs well at small sample sizes), MSSLE, CME, LSE and WLSE.
8 Data analysis
The relevance and importance of the MKWD are demonstrated in this section through the use of two real data sets. The data sets are given in Sections 8.1 and 8.2, and their box plots and the total time on test (TTT) plots can be found in Figs 4 and 5. From Fig 4 we note that both data sets are skewed to right, and from Fig 5 we note that the estimated hrf for both data sets are decreasing. The goodness-of-fit criteria, MLEs of parameters and standard errors (SEs) (we focus on ML as it has turned out to be the best estimation method, see Section 7) of the proposed model are compared to those of other competing models. The competing models are modified WD (MWD) [38], exponentiated transmuted generalized Rayleigh distribution (ETGRD) [39], transmuted modified WD (TMWD) [40], new modified WD (NMWD) [41], exponentiated exponential WD (EEWD) [42], transmuted complementary Weibull geometric distribution (TCWGD) [43] and beta WD (BWD) [44]. Based on the results in Tables 12–15, it is clear that the MKWD exhibits the best modeling ability among the models investigated. This is demonstrated by the lowest Akaike information criterion (ζ1), Bayesian information criterion (ζ2), consistency of ζ1 (ζ3), Hannan-Quinn information criterion (ζ4), Kolmogorov-Smirnov test statistic (ζ5), Cramér-von-Mises test statistic (ζ7), Anderson-Darling test statistic (ζ8) values, and largest p-value related to ζ5 (ζ6). Figs 6–11 (estimated pdf, cdf, PP plots for both data sets) also justify this claim, demonstrating the superiority of the MKWD over its competitors.
8.1 Data set 1: proportion of global per capita CO2 emissions in 2020
The first data set shows the proportion of global CO2 emissions per person for 211 nations in 2020. The data set, which was previously utilized in [45], is given by: 0.18, 1.88, 0.58, 3.53, 20.32, 5.39, 7.41, 0.11, 0.68, 2.09, 0.71, 0.26, 0.26, 0.21, 3.8, 0.73, 3.78, 0.99, 0.31, 2.16, 1.76, 5.01, 11.47, 6.53, 0.94, 3.37, 1.93, 6.08, 7.69, 0.67, 5, 0.04, 15.37, 0.56, 4.85, 14, 6.75, 4.66, 9.06, 1.68, 2.62, 2.56, 0.36, 15.52, 1.36, 0.57, 1.75, 0.08, 6.04, 1.75, 3.32, 8.6, 2.5, 2.56, 6.26, 0.92, 0.03, 7.62, 17.97, 0.59, 1.99, 1.53, 1.06, 0.4, 5.63, 5.24, 8.42, 6.94, 0.43, 4.89, 7.09, 3.47, 13.06, 0.64, 8.15, 1.02, 0.13, 3.99, 12.12, 0.43, 5.07, 2.5, 1.14, 0.04, 5.94, 1.06, 4.47, 0.07, 4.99, 1.93, 8.23, 0.38, 1.24, 5.02, 1.47, 6.73, 0.51, 30.45, 0.36, 20.55, 12.17, 0.77, 0.62, 26.98, 2.36, 3.96, 2.38, 4.24, 2.4, 1.56, 3.79, 2.44, 2.98, 7.32, 0.07, 4.65, 3.43, 6.51, 0.2, 3.61, 23.22, 12.49, 0.99, 15.19, 3.83, 0.26, 7.05, 2.77, 14.24, 4.25, 4.94, 2.51, 0.05, 0.98, 0.15, 3.72, 1.55, 7.62, 2.5, 5.07, 0.06, 0.3, 1.24, 6.98, 5.23, 1.55, 10.81, 2.2, 1.77, 0.11, 7.92, 6.4, 2.81, 11.66, 6.03, 2.95, 1.74, 0.56, 1.36, 0.61, 0.74, 0.17, 3.7, 0.99, 0.11, 8.87, 0.21, 2.77, 0.2, 4.52, 25.37, 14.2, 5.24, 20.83, 1.28, 3.69, 0.82, 3.59, 1.78, 8.06, 5.38, 3.73, 8.22, 7.23, 2.5, 3.68, 1.77, 0.33, 0.13, 0.55, 4.52, 0.19, 1.06, 2.61, 4.14, 1.58, 37.02, 8.74, 4.4, 4.61, 7.88, 0.51, 1.75, 10.03, 3.72, 1.94, 0.3, 3.13, 0.26, 7.78, 7.38.
8.2 Data set 2: proportion of global per capita CO2 emissions in 2022
The second data set comprises the proportion of global CO2 emissions per person for 214 nations in 2022. The electronic address from where it is taken is as follows: https://ourworldindata.org/. The data set consists of the following values: 0.295, 1.743, 3.927, 4.617, 0.452, 8.753, 6.422, 4.238, 2.305, 8.133, 14.985, 6.878, 3.675, 5.171, 25.672, 0.596, 4.377, 6.167, 7.688, 1.789, 0.631, 6.937, 1.349, 1.758, 4.083, 6.103, 2.839, 2.245, 5.004, 23.95, 6.804, 0.263, 0.062, 1.19, 0.343, 14.249, 0.959, 0.041, 0.134, 4.304, 7.993, 1.922, 0.493, 1.245, 3.995, 1.523, 0.417, 4.349, 1.866, 9.189, 5.617, 9.336, 0.036, 4.94, 0.404, 2.106, 2.105, 0.499, 2.312, 2.333, 1.217, 3.031, 0.189, 7.776, 1.053, 0.155, 14.085, 1.155, 6.527, 4.604, 2.851, 2.388, 0.285, 2.963, 7.984, 0.622, 5.745, 10.474, 2.713, 1.076, 0.357, 0.155, 4.374, 0.211, 1.07, 4.082, 4.45, 9.5, 1.997, 2.646, 7.799, 4.025, 7.721, 6.209, 5.727, 2.295, 8.502, 2.03, 13.98, 0.46, 0.518, 4.831, 25.578, 1.425, 3.08, 3.562, 4.354, 1.359, 0.165, 9.242, 3.81, 4.606, 11.618, 1.513, 0.149, 0.103, 8.577, 3.248, 0.312, 3.104, 3.635, 0.957, 3.27, 4.015, 1.324, 1.657, 11.151, 3.656, 4.845, 1.826, 0.243, 0.645, 1.54, 4.17, 0.507, 7.137, 17.641, 6.212, 0.799, 0.117, 0.589, 3.873, 1.951, 3.625, 7.509, 15.73, 0.849, 12.124, 0.666, 2.699, 0.771, 1.33, 1.789, 1.301, 8.107, 4.051, 37.601, 3.74, 11.417, 0.112, 3.299, 4.708, 2.615, 10.293, 2.296, 1.122, 0.582, 18.197, 0.674, 6.025, 6.15, 0.131, 8.912, 14.352, 6.052, 5.998, 0.412, 0.037, 6.746, 11.599, 0.168, 5.164, 0.794, 0.47, 5.803, 3.607, 4.048, 1.249, 11.631, 1.006, 0.238, 3.776, 0.291, 1.769, 22.424, 2.879, 5.105, 11.034, 7.637, 1, 0.127, 3.558, 25.833, 4.72, 14.95, 2.306, 3.483, 0.636, 2.717, 3.5, 2.282, 0.337, 0.446, 0.543.
9 Modified Kies Weibull quantile regression
In some lifetime scenarios, we might require to investigate the impact of some exogenous variables on an endogenous variable. The concept of regression analysis offers researchers an alternative way to undertake such investigations. However, the choice of an appropriate regression model is paramount in order to make reliable inference. When the endogenous variable in question is asymmetric, heavy-tailed or contaminated with outliers, the adoption of a robust regression model such as the quantile regression model (QRM) is vital. In the following, we formulate a new QRM based on the MKWD to study the effect of exogenous variables on a response variable defined on the positive real line.
9.1 The quantile regression model
The formulation of the new QRM for modeling a response variable defined on the positive real line is due to its robustness in handling outliers and allowing the estimation of heterogeneous effects of exogenous variables through the evaluation of various quantiles. The MKW QRM is attained by re-parameterization of the distribution in terms of its quantile function (qf) (see [46–48] for more details). Suppose that Y is a random variable that follows the MKWD and η ∈ (0, ∞) is a quantile parameter. Let η = Q(u) in Eq (9), making β the subject from the qf of the MKWD, yields , p ∈ (0, 1). Inserting β into Eq (8) gives us the re-parameterized density in terms of the qf. Thus, the density function of the MKWD is defined in terms of the qf as
(41)
For p = 0.10, 0.25, 0.50, 0.75, 0.90 and 0.95, the density function of the 10th, 25th, 50th, 75th, 90th and 95th percentiles are attained, respectively.
The MKW QRM is formulated using a monotonically increasing and twice differentiable link function to relate the exogenous variables to the conditional quantiles. Hence, we have
where b(⋅) is the link function, ηi is the ith quantile parameter, τ = (τ0, τ1, …, τk)′ is the unknown vector of parameters and
is the unknown ith vector of exogenous variables. Note that the MKW median regression is obtained when p = 0.50. In this paper, the logarithmic link function is used to link the exogenous variables to the conditional quantiles. Hence,
The log-likelihood function for estimating the parameters of the regression model for a sample of size n is obtained by substituting ηi into the re-parameterized density function of the MKWD. Hence, the log-likelihood is given by
(42)
The estimates of the parameters are obtained by directly maximizing the log-likelihood function.
9.2 Model diagnostics
Examining the model adequacy is vital when fitting a model to a data set. The residuals obtained from fitting the model to the data set are often checked to ensure that they behave well and thus the model provides an adequate fit to the data. We employ the randomized quantile residuals (RQR) in this study to examine the adequacy of the model. The RQR are given by:
where Φ−1(⋅) is the qf of the standard normal (SN) distribution. If the model offers adequate fit to the data, the RQR are anticipated to follow the SN distribution [49].
9.3 Simulation experiments for the QRM
In this subsection, simulation experiments are executed utilizing three different scenarios of parameter combinations in order to evaluate the performance of the ML method in estimating the parameters of the QRM. The parameter combination used for scenarios I, II and III are: I: τ0 = 0.6, τ1 = −0.1, ϑ = 0.4, ζ = 0.5, II: τ0 = 4.3, τ1 = 0.3, ϑ = 0.2, ζ = 1.5 and III: τ0 = −0.3, τ1 = −0.1, ϑ = 3.2, ζ = 0.5. The experiments are accomplished using the conditional median regression. The experiments are repeated 1000 times for each sample size n = 25, 100, 250, 500, 800, 1000. During the simulations, we employ the following regression structure:
The exogenous variable, zi1, is generated using the SN distribution and is held fixed in the simulation process. The performance of the ML method is examined using the mean estimate (ME), average absolute bias (AAB), RMSE, coverage probability (CP) of the 95% confidence interval (CI), lower CI (LCI), upper CI (UCI) and average width of the CI (AWCI). From Tables 16–18, it is observed that as the sample size increases, the MEs approaches the true parameter value, the AABs and RMSEs decrease as expected, the 95% CI CPs gets closer to the nominal value of 0.95, the CI becomes narrower and the AWCI decreases. The simulation results suggest that the MLEs are consistent and the ML method is able to estimate the regression parameters well.
9.4 Application
The potential of the MKW QRM is exemplified in this subsection. The QRM model is adopted to study the effect of gender on the survival times (in years) up to the inception of hypertension. The details of the data can be found in Anzagra et al. [46]. Anzagra et al. [46] model the data using the Chen Burr-Hatke exponential (CBHE) QRM and identified the 75th percentile as the best with ζ1 = 1022.7740 and ζ2 = 1033.8910. The data is fitted with the regression structure
where male = 1 and female = 0. Table 19 provides the parameter estimates and information criteria for various quantiles. It shows that “gender” is insignificant. Thus, an individual’s gender has no significant effect on the survival time to the inception of hypertension. The MKW QRM offers a better fit to the data than the CBHE QRM. Among the fitted quantile for the MKW QRM, the 25th percentile yielded the best fit with ζ1 and ζ2 values given as 1013.1460 and 1024.2620, respectively.
The adequacy of the MKW QRM is evaluated by plotting the PP plots of the RQR, see Fig 12, which confirms that the MKW QRM offers reasonable fit to the data.
10 Concluding remarks
This article presented the MKWD, a novel lifetime distribution with three parameters. The statistical properties of the MKWD, including the quantile function, median, moments, mean, variance, skewness, kurtosis, coefficient of variation, moment generating function, incomplete and conditional moments, inequality measures, and order statistics, were computed. Various metrics of entropy and extropy were calculated for the MKWD. The paper examined eight estimation methods to analyze the characteristics of the model parameters for the MKWD. A Monte Carlo simulation was performed to evaluate the effectiveness of these different estimators. The effectiveness of the proposed model was illustrated by analyzing two real data sets. Moreover, the use of survival times data in regression analysis was analyzed by the MKWD, other existing distributions and regression models.
To refer back to the research questions and hypotheses formulated at the beginning:
- (1). The MKWD provides a more accurate fit for lifetime data with non-monotonic failure rates compared to various existing Weibull variants.
- (2). The MKWD exhibits many statistical characteristics that make it suitable for a wide range of applications in reliability and survival analysis.
- (3). MLE demonstrated a better performance than other estimation methods in terms of accuracy and reliability.
- (4). The MKWD showed superior modeling performance when applied to real-world datasets, providing better fits and more accurate predictions than competing distributions.
In summary, we have introduced a distribution that is well suited to model lifetime scenarios with non-monotonic failure rates which, due to its superiority over previously considered distributions, is likely to be used in various domains where understanding the lifespan or durability of objects, systems, or processes is crucial. Thus, it can be applied in engineering and reliability analysis, healthcare and medicine, insurance and actuarial sciences, finance and investment, environmental sciences, quality control and manufacturing, telecommunications and networking, and energy and utilities, just to name a few domains. The limitation of our paper is that we only use the complete samples to estimate the parameters of the MKWD. So, for future works, researchers can use the MKWD to estimate its parameters using different censored schemes.
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