Extended odd Weibull–Lindley distribution

In various practical fields, such as medicine, engineering, and finance, it is essential to create models and analyze data related to the lifetime or longevity of objects or processes. Over the past two decades, researchers have introduced and investigated various probability distributions for the purpose of modeling data related to lifetime in various practical domains, including engineering, biological study, environmental sciences, economics, and medical sciences. Some of the original probability distributions have the drawback of having a monotonic hazard rate function; therefore, many of the generalized families have been introduced by several researchers to generate flexible and useful distributions that can capture non-monotonic hazard rate curves. Some of these families of distributions in this context include the exponentiated-G (E-G) family,¹ beta-G (B-G) family,² Marshall and Olkin-G (MO-G) family,³ log-logistic-G family,⁴ Kumaraswamy-G (K-G) family,⁵ gamma-G (G-G) family,⁶ exponentiated generalized class family,⁷ Weibull-G family,⁸ Lindley-H family,⁹ and alpha power transformed-G (APT-G) family.¹⁰ For more details, see Refs. 11 and 12.

The exponential (Exp.) distribution finds wide use in analyzing real-world data, mostly in fields such as reliability and survival analysis. In the field of lifetime analysis, it was the initial model for which significant statistical methods were systematically developed. A novel distribution known as the Lindley (L) distribution¹³ is suggested as an alternative to the Exp. and gamma (Ga) distributions. Subsequently, Ghitany et al.¹⁴ conducted an examination of its characteristics and demonstrated that the L distribution outperforms the Exp. and Ga distributions in modeling certain situations. The L distribution is usually used for data with a steadily increasing failure rate. Several authors introduced and suggested many extensions of the L distribution, for example power Lindley distribution,¹⁵ generalized Lindley distribution,¹⁶ beta-Lindley distribution,¹⁷ the Kumaraswamy–Lindley distribution,¹⁸ exponentiated generalized Lindley distribution,¹⁹ alpha-power transformed Lindley distribution,²⁰ and alpha power transformed extended power Lindley distribution.²¹ 

Numerous probability distributions have been developed to address the complexity of modeling natural phenomena, which often prove challenging to represent accurately using conventional distributions. Nevertheless, established probability distributions still fall short in accurately modeling certain aspects of natural phenomena data. This has necessitated the expansion and adaptation of generalized probability families of distributions. The introduction of new parameters into existing probability families of distributions has enhanced their suitability for representing data related to natural phenomena, resulting in an improved accuracy in describing the distribution’s tail shape, for example, extended generalized log-logistic-G family,²² Harris family of survival functions,^23,24 extended wrapped Cauchy-G family,²⁵ exponentiated-G (E-G) family,⁷ new exponentiated extended-G family,²⁶ extended Cordeiro and de Castro-G (ECC-G) family,²⁷ generalized odd log-logistic-G family,²⁸ new generalized odd log-logistic-G family,²⁹ new extended alpha power transformed (APT-G) family,³⁰ and the odd log-logistic Lindley-G (OLLLi-G) family.³¹ Based on these families, some extended Lindley distributions were suggested, such as odd log-logistic Marshall–Olkin–Lindley distribution³² and new odd log-logistic Lindley (NOLL-L) distribution.³³ 

Using this family, many authors proposed some distribution extensions, for example extended odd Weibull–Weibull (EOWW) distribution, extended odd Weibull–normal (EOWN) distribution,³⁴ extended odd Weibull–Rayleigh (EOWR) distribution,³⁵ extended odd Weibull–inverse Nadarajah–Haghighi (EOWINH) distribution,³⁶ and extended odd Weibull–Pareto (EOWP) distribution.³⁷ 

Now, motivated by Refs. 34–37, we introduce the extended odd Weibull–Lindley (EOWL) distribution as an extension of the Lindley distribution by combining the EOW-G family with the Lindley distribution. Furthermore, the introduction of the new extension is motivated by its capacity and adaptability to model various datasets. Evidently, this extension demonstrates superior fitting capabilities when compared to alternative distributions.

The rest of this paper is organized as follows: the EOWL distribution is defined and its properties are derived in Sec. II. In Sec. III, three estimation methods are established, and a simulation study is applied to compare the performance of estimation methods. The applications of different data types are provided in Sec. IV to show the ability and efficiency of the EOWL distribution. Finally, findings and conclusions are provided in Sec. V.

Hence, we can say that the RV X follows the EOWL distribution, denoted by X ∼ EOWL(ν, ω, γ), if its PDF is defined by (6).

The behaviors of the PDF and HRF of the EOWL distribution for different values of parameters are shown in Fig. 1. In Fig. 1(a), we observe that the PDF can be decreasing, symmetric, and right-skewed shaped. Figure 1(b) highlights that the HRF can exhibit diverse shapes, including increasing, decreasing, inverse-J, and concave shapes.

Numerical results for quartiles, skewness, and kurtosis of the EOWL distribution for different values of parameters γ, ω, and ν are presented in Table I.

To facilitate method comparison, Monte Carlo simulations are conducted within this section. The estimations for the parameters of the EOWL lifetime distribution are computed using ML, MPS, and LS estimations in the R programming language. We utilize a dataset of 10 000 randomly generated samples from the EOWL distribution, where X follows the EOWL distribution with varying parameter values and sample sizes (20, 40, 80, 160, and 320). The quality of the estimations is assessed based on criteria such as mean squared error (MSE), root mean squared error (RMSE), and bias. The results of simulations are briefed in Tables II–V. The findings indicate that, with increasing sample size, each of MSE, RMSE, and bias decreases, and the most effective estimation method is MPSE followed by MLE.

In this section, we present some real-life datasets from different fields, to evaluate the fitness of applying the EOWL distribution. We conduct a comparative analysis between the EOWL distribution and other competitive models, specifically the EOWR and EOWW distributions. The essential statistical metrics, such as negative log-likelihood (−ℓ), Akaike information criterion (AIC), corrected Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan–Quinn information criterion (HQIC), and the Anderson–Darling (A*), Cramer–von Mises (W*), and Kolmogorov–Smirnov (KS) statistics with their P-value, were applied to verify which distribution better fits the real-life datasets.

The data considered pertain to the endurance of Kevlar 373/epoxy composites under a constant pressure at 90% of their stress capacity until failure occurs. This was reported by Ref. 38 and studied by Ref. 39. The descriptive statistics for this dataset are n = 76, min = 0.0251, Q1 = 0.9048, median = 1.7362, mean = 1.9595, Q3 = 2.2959, max = 9.0960, skewness = 1.9402, and kurtosis = 4.9466. Based on these results, these data are right-skewed and leptokurtic. Tables VI and VII give the maximum likelihood estimates (MLEs) with standard errors (SEs) and the goodness-of-fit (GOF) metrics, respectively. The fitted PDF, CDF, and P–P plot of the EOWL and other distributions are displayed in Figs. 2 and 3.

Based on the findings in Table VII, it is obvious that the EOWL distribution demonstrates a superior fitting performance compared to alternative models. In addition, Fig. 2(a) provides additional insights by presenting a histogram of the data alongside the fitted PDFs for all distributions. This graphical representation underscores the EOWL distribution’s exceptional suitability for accommodating skewed data. Furthermore, the conclusions drawn from the results in Table VII are further substantiated by Figs. 2 and 3.

The second dataset consists of recorded precipitation levels in Minneapolis–St. Paul for the month of March, covering the period of 30 years. It is reported by Ref. 40. The descriptive statistics for this dataset are n = 30, min = 0.320, Q1 = 0.915, median = 1.470, mean = 1.675, Q3 = 2.087, max = 4.750, skewness = 1.033, and kurtosis = 0.931. Based on these results, the shape of dataset 2 is right-skewed and platykurtic. Tables VIII and IX give the MLEs with SEs and the GOF metrics, respectively. The fitted PDF, CDF, and P–P plots of the EOWL and other distributions are shown in Figs. 4 and 5.

Based on the findings in Table IX, it is obvious that the EOWL distribution demonstrates a superior fitting performance compared to alternative models. In addition, Fig. 4(a) provides additional insights by presenting a histogram of the data alongside the fitted PDFs for all distributions. This graphical representation underscores the EOWL distribution’s exceptional suitability for accommodating skewed data. Furthermore, the conclusions drawn from the results in Table IX are further substantiated by Figs. 4 and 5.

Total Factor Productivity (TFP) is a metric used to assess the agricultural output generated from the collective utilization of resources such as land, labor, capital, and materials in the process of farm production. An increase in total factor productivity (TFP) is indicated when the rate of growth in overall output surpasses that of the total resources expended. Alyami et al.⁴¹ studied the TFP growth data for 37 African countries from 2001 to 2010. The descriptive statistics are n = 37, min = 4.566, Q1 = 6.571, median = 10.966, mean = 10.649, Q3 = 13.805, max = 17.830, skewness = 0.113, and kurtosis = −1.307. Therefore, the TFP data are right-skewed and platykurtic. Tables X and XI give the MLEs with SEs and the GOF measures, respectively. The fitted PDF, CDF, and P–P plots of the EOWL and other distributions are shown in Figs. 6 and 7.

Based on the findings in Table XI, it is obvious that the EOWL distribution demonstrates a superior fitting performance compared to alternative models. In addition, Fig. 6(a) provides additional insights by presenting a histogram of the data alongside the fitted PDFs for all distributions. This graphical representation underscores the EOWL distribution’s exceptional suitability for accommodating skewed data. Furthermore, the conclusions drawn from the results in Table XI are further substantiated by Figs. 6 and 7.

In this research, a novel extension of the Lindley and Weibull distributions was proposed, which is called the extended odd Weibull–Lindley (EOWL) distribution. The statistical properties of the EOWL distribution were derived, and the linear representation of its PDF was obtained. The linear representation helps to derive functions for moments, quantiles, and moment generation. Three methods for estimating the parameters of the EOWL distribution were studied, including MLE, MPSE, and LSE. To evaluate the performance of these estimation methods, a thorough comparative analysis was conducted using simulation techniques with the R package. To further validate the findings, three real-world datasets were applied and examined. The results demonstrated that the EOWL distribution provided a better fit than competing distributions.

We extend our sincere gratitude to the reviewers and editor for their invaluable contributions to this work. Their insightful comments and constructive feedback have greatly enhanced the quality and clarity of our manuscript. Their dedication and expertise have been instrumental in shaping the final outcome of this research. We deeply appreciate their time and effort invested in evaluating our work and providing valuable guidance throughout the review process.

The authors have no conflicts to disclose.

Fatehi Yahya Eissa: Conceptualization (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Chhaya Dhanraj Sonar: Writing – review & editing (equal).

The data that support the findings of this study are available within the article.