Non-monotonic Hazard Rate Research Articles

Novel family of probability generating distributions: Properties and data analysis

Abstract Current probability distributions often fall short when modeling lifespan data with non-monotonic hazard rate shapes, highlighting a gap in probability theory. To address this issue, this study introduces the novel family of probability-generating distributions (NP-G) as a solution to capture complex data patterns better. This research aims to develop and validate the NP-G distributions, which include a two-parameter continuous model based on the Weibull distribution. The study extensively explores the mathematical properties of the NP-G distributions, including series expansions of probability density functions, and evaluates various probabilistic measures such as moments, stress strength, mean deviations, entropy measures, and order statistics. Using the maximum likelihood method for parameter estimation and conducting a numerical simulation analysis, the study assesses the performance of the proposed extended exponentiated Weibull model. The findings demonstrate that this new model offers superior fit and accuracy compared to existing distributions when applied to two referenced datasets. This advancement is significant as it provides a more effective tool for analyzing lifespan data with non-monotonic hazard rates, enhancing both theoretical understanding and practical applications in fields such as reliability engineering and survival analysis.

A Novel Distribution in the Family of Lifetime Distributions for Enhancing Predictive Modeling for Medical and Engineering Data

In nearly all scientific disciplines, the statistical inference about the population rely on handling the sampled data. In the time to event analysis, there are many lifetime distributions to model variation of the lifetime observations based on the shape of hazard rate of the data. In the literature, it has been observed that for non-monotonic hazard the existing distributions do not provide good fits. Practically it is not possible for a distribution to fit any kind of data. Therefore, in this study, a new lifetime distribution is suggested called Flexible Exponentiated Weibull distribution (FEW) to model monotonic and non-monotonic hazard rate data. Maximum likelihood estimation approach is used to estimate the model parameters. In addition to these some prominent statistical properties like, reliability function, moments, hazard function, order statistics, quantile function and entropy measure are obtained. Two real data sets were taken to compare the proposed distribution with existing distributions, and the results showed that the proposed distribution is more flexible than other existing lifetime distributions. Furthermore, simulation study is carried out to check the consistency of model parameters that showed that the parameters are consistent when the sample size increases. These results establish a foundational rationale for selecting the suggested distribution as a model for such a data type. It shows that this distribution is more flexible and suitable for the data studied, making a strong case for choosing it over other options. These findings not only boost trust in the chosen model but also help in deciding how to model similar data in the future.

A robust multi-risk model and its reliability relevance: A Bayes study with Hamiltonian Monte Carlo methodology

This paper proposes a robust model that describes complex time-to-failure and competing risk datasets found with bathtub curve hazard rate (BCHR). The resulting model, named the flexible Dhillon–Weibull competing risk (FDWCR) model, hybridized the Weibull and flexible Dhillon distributions. It is built with the assumptions of competing risk data characterized by monotone and non-monotone hazard rates. In contrast to most recent additive or competing risk models, the FDWCR model can analyze not only time-to-failure datasets identified with increasing hazard rates and BCHR but also effectively describe time-to-failure data associated with decreasing, upside-down bathtub and modified bathtub hazard rates. Several reliability and structural properties of the model are discussed. We present a complete Bayesian paradigm for FDWCR and suggest employing the Hamiltonian Monte Carlo (HMC) algorithm to enhance precision and expedite inference. The maximum likelihood estimators are also presented. We evaluate the appropriateness of the proposed estimators via simulation experiments, considering both techniques. We offer three case studies to demonstrate the practical utility of the proposed model. The results outperform those produced by other alternative models, indicating a reliable approach for modeling diverse competing risk problems, particularly in the context of engineering reliability studies or other domains of quantitative research.

Secant Kumaraswamy Family of Distributions: Properties, Regression Model, and Applications

In this study, Secant Kumaraswamy family of distributions is proposed and studied. This is motivated by the fact that no one distribution can model all types of data from different fields. Therefore, there is the need to develop distributions with desirable properties and flexible enough for modelling data exhibiting different characteristics. Some properties of the new family of distributions, including the quantile function, moments, moment generating function, and mean residual life function, are derived. Five special cases of the family of distributions are presented, and their flexibility is shown by the varying degrees of skewness and kurtosis and nonmonotonic hazard rates. The maximum likelihood estimation method is used to obtain estimators of the family of distributions. Two location-scale regression models are developed for the Secant Kumaraswamy Weibull distribution, which is a special case of the family of distributions. Six different real datasets are used to demonstrate the usefulness of the family of distributions and the regression models. The results show that the family of distributions can be used to model real datasets.

Statistical analysis of Saudi Arabia and UK Covid-19 data using a new generalized distribution

In this article, we extended the Topp–Leone distribution to create the Odd-Topp–Leone–Gompertz-G distribution, a new family of distributions. We were motivated by the limitations of classical models in modeling real life datasets hence, the creation of a new flexible distribution. Expansion of density, distribution of order statistics, Rényi entropy, moments, quantile, generating functions, residual life function and reverse residual life function, and reliability model are some of its structural aspects that are obtained. We looked at some special cases of the new distribution and concluded that our new model can handle both highly skewed data and non-monotonic hazard rate functions. The maximum likelihood estimation technique was used to estimate the model parameters. A simulation exercise was carried out to evaluate the performance of the proposed family of distribution. The article used the Saudi Arabia and United Kingdom dataset to demonstrate the utility of the new model and it was found that the new model performs better in modeling these datasets as compared to non-nested models presented in this article.

The Gull Alpha Power Lomax distributions: Properties, simulation, and applications to modeling COVID-19 mortality rates.

The Gull Alpha Power Lomax distribution is a new extension of the Lomax distribution that we developed in this paper (GAPL). The proposed distribution's appropriateness stems from its usefulness to model both monotonic and non-monotonic hazard rate functions, which are widely used in reliability engineering and survival analysis. In addition to their special cases, many statistical features were determined. The maximum likelihood method is used to estimate the model's unknown parameters. Furthermore, the proposed distribution's usefulness is demonstrated using two medical data sets dealing with COVID-19 patients' mortality rates, as well as extensive simulated data applied to assess the performance of the estimators of the proposed distribution.

The Gompertz-Topp Leone-G Family of Distributions with Applications

We introduce a new family of distributions, referred to as the Gompertz-Topp-Leone-G (Gom-TL-G) distribution. The new family of distributions can be expressed as an infinite linear combination of the Exp-G distributions. It can handle extremely tailed data and has both monotonic and non-monotonic hazard rate functions. A simulation study is used to assess the estimation method. Some real data examples are analyzed for illustrative purposes.

The Type II Topp-Leone-G Power Series Distribution with Applications on Bladder Cancer

Statistical distributions are important in modeling the real life of an item and therefore proper distributions that provide useful information for sound conclusions and decisions are needed. For that reason, the demand for developing new generalized distributions have become appropriate for data that have both monotonic and non-monotonic hazard rate functions. In this paper, we develop a new distribution called the Type II Topp-Leone-G Power Series (TIITLGPS) distribution by compounding the Type II Topp-Leone-G (TIITLG) distribution with the power series distribution. Statistical properties of the TIITLGPS distribution are obtained. A variety of shapes for the densities and hazard rate are presented of the considered special case. A simulation study to examine the efficiency of the maximum likelihood estimates is also conducted. Finally, the bladder cancer data example is analyzed for illustrative purposes, it is displayed that the introduced distribution provides better fit when compared to other non-nested distributions considered in this work.

The Topp-Leone Gompertz-G family of distributions with applications

A new extension of the Topp-Leone distribution is developed to produce a new family of distributions, namely Topp-Leone Gompertz-G distribution. Some of its structural properties including expansion of density, distribution of order statistics, Renyi entropy, moments, quantile, and generating functions are derived. We considered four special cases of the new distribution and we conclude that the distribution can handle heavy tailed data and non-monotonic hazard rate functions. A simulation study was conducted to examine the performance of the special case of the proposed family of distributions. Two lifetime data examples are included in this article to illustrate the applicability of the proposed model in comparison with other competing non-nested models.

Additive Trinomial Fréchet distribution with practical application

This article presents an innovative model called Additive Trinomial Fré chet (ATF) distribution using six parameters. The indicated model is worthy of modeling survival data with a non-monotonic hazard rate. The statistical characteristics of ATF model such as probability generating function, Renyi, Shannon, Tsallis and Mathai–Houbold entropy, quantile function, order statistics, maximum likelihood estimation, factorial and characteristic function, moment generating function, Stress-Strength analysis are thoroughly discussed. The effectiveness of suggested model is demonstrated by the use of a data set from real life. The suggested model has demonstrated better performance and fits the data used superior than other significant counterparts.

The Marshall-Olkin Odd Exponential Half Logistic-G Family of Distributions: Properties and Applications

We develop a new family of distributions, referred to as the Marshall-Olkin odd exponential half logistic-G, which is a linear combination of the exponential-G family of distributions. The family of distributions can handle heavy-tailed data and has non-monotonic hazard rate functions. We also conducted a simulation study to assess the performance of the proposed model. Real data examples are provided to demonstrate the usefulness of the proposed model in comparison with several other existing models.

New Weighted Lomax (NWL) Distribution with Applications to Real and Simulated Data

The rationale of the paper is to present a new probability distribution that can model both the monotonic and nonmonotonic hazard rate shapes and to increase their flexibility among other probability distributions available in the literature. The proposed probability distribution is called the New Weighted Lomax (NWL) distribution. Various statistical properties have been studied including with the estimation of the unknown parameters. To achieve the basic objectives, applications of NWL are presented by means of two real-life data sets as well as a simulated data. It is verified that NWL performs well in both monotonic and nonmonotonic hazard rate function than the Lomax (L), Power Lomax (PL), Exponential Lomax (EL), and Weibull Lomax (WL) distribution.

Alpha-Power Exponentiated Inverse Rayleigh distribution and its applications to real and simulated data.

The main goal of the current paper is to contribute to the existing literature of probability distributions. In this paper, a new probability distribution is generated by using the Alpha Power Family of distributions with the aim to model the data with non-monotonic failure rates and provides a better fit. The proposed distribution is called Alpha Power Exponentiated Inverse Rayleigh or in short APEIR distribution. Various statistical properties have been investigated including they are the order statistics, moments, residual life function, mean waiting time, quantiles, entropy, and stress-strength parameter. To estimate the parameters of the proposed distribution, the maximum likelihood method is employed. It has been proved theoretically that the proposed distribution provides a better fit to the data with monotonic as well as non-monotonic hazard rate shapes. Moreover, two real data sets are used to evaluate the significance and flexibility of the proposed distribution as compared to other probability distributions.

Some new results on bathtub-shaped hazard rate models.

The most common non-monotonic hazard rate situations in life sciences and engineering involves bathtub shapes. This paper focuses on the quantile residual life function in the class of lifetime distributions that have bathtub-shaped hazard rate functions. For this class of distributions, the shape of the α-quantile residual lifetime function was studied. Then, the change points of the α-quantile residual life function of a general weighted hazard rate model were compared with the corresponding change points of the basic model in terms of their location. As a special weighted model, the order statistics were considered and the change points related to the order statistics were compared with the change points of the baseline distribution. Moreover, some comparisons of the change points of two different order statistics were presented.

A Gull Alpha Power Weibull distribution with applications to real and simulated data

In this paper, we produced a new family of distribution called Gull Alpha Power Family of distributions (GAPF). A Special case of GAPF is derived by considering the Weibull distribution as a baseline distribution called Gull Alpha Power Weibull distribution (GAPW). The suitability of the proposed distribution derives from its ability to model both the monotonic and non-monotonic hazard rate functions which are a common practice in survival analysis and reliability engineering. Various statistical properties were derived in addition to their special cases. The unknown parameters of the model are estimated using the maximum likelihood method. Moreover, the usefulness of the proposed distribution is supported by using two real lifetime data sets as well as simulated data.

A novel flexible additive Weibull distribution with real-life applications

This paper introduces a novel model with six parameters called “flexible additive Weibull distribution (FAWD).” The suggested model is capable to model the life time data with a non-monotonic hazard rate. The statistical properties of the new distribution including Renyi entropy, quantile function, maximum likelihood estimation, order statistics, moment generating function, probability generating function, factorial and characteristic function are discussed in details. The flexibility of the proposed distribution is judged based on AIC, CAIC, BIC and HQIC, the smaller is the value of these statistics, and the better is the results. The usefulness of the proposed distribution is illustrated by using two real data sets. The proposed distribution has shown better performance and fits the used data better than some other well-known distributions.

Reliability data analysis of systems in the wear-out phase using a (corrected) q-Exponential likelihood

Maintenance-related decisions are often based on the expected number of interventions during a specified period of time. The proper estimation of this quantity relies on the choice of the probabilistic model that best fits reliability-related data. In this context, the q-Exponential probability distribution has emerged as a promising alternative. It can model each of the three phases of the bathtub curve; however, for the wear-out phase, its usage may become difficult due to the “monotone likelihood problem”. Two correction methods (Firth's and resample-based) are considered and have their performances evaluated through numerical experiments. To aid the reliability analyst in applying the q-Exponential model, we devise a methodology involving original and corrected functions for point and interval estimates for the q-Exponential parameters and validation of the estimated models using the expected number of failures via Monte Carlo simulation and the bootstrapped Kolmogorov-Smirnov test. Two examples with failure data presenting increasing hazard rates are provided. The performances of the estimated q-Exponential, Weibull, q-Weibull and modified extended Weibull (MEW) models are compared. In both examples, the q-Exponential presented superior results, despite the increased flexibility of the q-Weibull and MEW distributions in modeling non-monotone hazard rates (e.g., bathtub-shaped).

A stochastic process for modeling failures of a system having a non-monotonic hazard rate function

Reliability literature on modeling failures of repairable systems mostly deals with systems having monotonically increasing hazard/failure rates. When the hazard rate of a system is non-monotonic, models developed for monotonically increasing failure rates cannot be effectively applied without making assumptions on the types of repair performed following system failures. For instance, for systems having bathtub-shaped hazard rates, it is assumed that during the initial, decreasing hazard rate phase, all repairs are minimal. These assumptions on the type of general repair can be restrictive. In order to relax these assumptions, it has been suggested that general repairs in the initially decreasing phase can be modeled as “aging” the system. This approach however does not preserve the order of effectiveness of the types of general repair as defined in the literature. In this article, we develop a set of models to address these shortcomings. We propose a new stochastic process to model consecutive failures of repairable systems having non-monotonic, specifically bathtub-shaped, hazard rates, where the types of general repair are not restricted and the order of the effectiveness of the types of repair is preserved. The proposed models guarantee that a repaired system is at least as reliable as one that has not failed (or equivalently one that has been minimally repaired). To illustrate the models, we present multiple examples and simulate the failure-repair process and estimate the quantities of interest.

Temporal Effects of Financial Globalization on Income Inequality

Through parametric accelerated failure time survival analysis on a panel of 120 countries for 1970-2018, the study finds empirical evidence that time to upsurge in income inequality declines after a country liberalizes investment across borders. This occurs with a significant non-monotonic hazard rate of upsurge, arguably via the channels of capital-skill complementarity and wage inequality. This effect is robust to distributional assumptions, alternative measurements of financial globalization and income inequality, sample restriction to pre-crisis period and residual heteroskedasticity. Moreover, it is reasonably generalizable across different economic arrangements due to employing diverse panel and handling issues of imputation and cross-country standardization in contemporary income inequality data.

Modified beta modified-Weibull distribution

We introduce a flexible modified beta modified-Weibull model, which can accommodate both monotonic and non-monotonic hazard rates such as a useful long bathtub shaped hazard rate in the middle. Several distributions can be obtained as special cases of the new model. We demonstrate that the new density function is a linear combination of modified-Weibull densities. We obtain the ordinary and central moments, generating function, conditional moments and mean deviations, residual life functions, reliability measures and mean and variance (reversed) residual life. The method of maximum likelihood and a Bayesian procedure are used for estimating the model parameters. We compare the fits of the new distribution and other competitive models to two real data sets. We prove empirically that the new distribution gives the best fit among these distributions based on several goodness-of-fit statistics.