Hazard Rate Function Research Articles

Modeling insurance loss data using novel approach of moment exponential model: Inference, actuarial measures and application

Asymmetrical probability models are helpful for analyzing skewed data sets since they allow you to describe the form of the distribution and anticipate the chance of extreme events. This article defines a novel approach to continuous moment exponential distribution called the exponentiated generalized moment exponential model. The extension has two additional parameters accounting for the distribution’s shape. We extend this distribution probability density, cumulative distribution, hazard rate, and survival functions and establish different key statistical properties. Parameter estimation is obtained using different procedures, notably maximum likelihood estimation, least square, and Bayesian methods. A Monte Carlo simulation experiment is conducted to assess parameter performance and indicator risk measures. This article examines two distinct actual data sets in order to highlight the significance of the proposed model as well as its application in a variety of settings. The new model is compared to a large number of well-known extensions that were developed by other businesses.

Fuzzy system reliability analysis based on new aging indexes

The concept of aging plays an important role in reliability analysis. The aim of this paper is providing the novel method to definition aging indexes based on fuzzy random variables and using the concept of α-pessimistic in both parametric and non-parametric approaches. In parametric approach, the concepts of survival function, hazard rate function and mean residual function were extended for fuzzy random variables. After that, we obtain and analyse these indexes for fuzzy exponential and fuzzy two-parameter Weibull distributions as two frequently used lifetime distribution in reliability theory. Moreover, with the assumption that the distribution of fuzzy observations is unknown (non-parametric approach), first an estimate for the fuzzy distribution function is presented, and then an estimate for the aging indices of the system is investigated. To do this, the concept of fuzzy empirical distribution function is investigated as an estimator of cumulative distribution function. Moreover, some properties of investigated estimator have been established.

The power new XLindley distribution: Statistical inference, fuzzy reliability, and applications

This paper introduces the power new XLindley (PNXL) distribution, a novel two-parameter distribution derived using the power transformation method applied to the XLindley distribution. We thoroughly explore the structural properties of the PNXL distribution, including the rth moment about the origin, moment generating function, survival rate function, distribution function, hazard rate function, skewness, kurtosis, and coefficient of variation. Additionally, we derive the quantile function, fuzzy reliability, reliability measures, stochastic ordering, and actuarial measures for this new distribution. To estimate the parameters of the PNXL distribution, we propose several estimators and evaluate their performance through extensive simulation studies. To demonstrate the applicability and superiority of the PNXL distribution over existing distributions, we fit it to two real datasets and compare its performance with potential competing models. The results highlight the PNXL distribution's effectiveness and potential as a robust tool for modeling and analyzing real-world data.

A New Family of Distributions Based on Amalgamation of Two Methods with an Application to the Rayleigh Model

In this paper, we present the Alpha power generalized odd generalized exponential-G (APGOGE-G) family of distributions and provides the most common shapes of the hazard rate function: increasing, decreasing, bathtub, and inverted bathtub. We provide some of its structural properties. We estimate the parameters by maximum likelihood estimation method, and perform a simulation study to verify the asymptotic properties of the estimator for the inverse Weibull baseline. The practicality of the new APGOGE-Rayleigh model is shown through application to uncensored real dataset.

Bias Reduction of Maximum Likelihood Estimation in the Inverse Xgamma Distribution

A distribution for modeling survival data that accounts for flexibility in modeling data with upside-down bathtub-shaped hazard rate functions is the inverse Xgamma distribution (IXG). The maximum likelihood method (MLE) is the most often used technique for parameter estimation of the IXG distribution. Conversely, the MLE is infamously biased for small sample sizes. This motivates us to produce almost unbiased estimators for IXG parameter. More precisely, we minimize MLE biases to the second degree of bias using two techniques for bias correction: bootstrap and analytical approaches. Two actual data applications and Monte Carlo simulations are used to compare the performances of these methods.

A new extension of the Gumbel distribution with biomedical data analysis

In the field of biomedical research, data characteristics often exhibit significant variability, challenging the applicability of classical Gumbel distribution for biomedical data modeling. To address this, this paper introduces a novel extension of the Gumbel model known as the odd beta prime Gumbel (OBP-Gum) model. Derived from the odd beta prime family, the new distribution exhibits greater kurtosis compared to the traditional Gumbel distribution. Importantly, the proposed distribution is designed to capture right-skewed, left-skewed, and nearly symmetric density functions, as well as increasing, decreasing, constant, and upside-down bathtub shapes for its hazard rate function, providing excellent curvature features for creating flexible statistical models for biomedical research. We derive the fundamental features of the OBP-Gum model, such as the quantile function, linear representations, moment generating function, moments, skewness, kurtosis, incomplete moments, and Rényi and Tsallis entropies. Parameter estimation for this new model is conducted using the maximum likelihood estimation method. A simulation study demonstrates the performance of the model parameters. The empirical findings, based on applications to two biomedical datasets, suggest that the OBP-Gum distribution outperforms existing models, particularly in handling extreme observations. Instead of relying on conventional models for decision-making, this research provides relevant stakeholders with an improved statistical distribution for more accurate biomedical data modeling.

A New Generalization of the Inverse Generalized Weibull Distribution with Different Methods of Estimation and Applications in Medicine and Engineering

Limitations inherent to existing statistical distributions in capturing the complexities of real-world data often necessitate the development of novel models. This paper introduces the new exponential generalized inverse generalized Weibull (NEGIGW) distribution. The NEGIGW distribution boasts significant flexibility with symmetrical and asymmetrical shapes, allowing its hazard rate function to be adapted to many failure patterns observed in various fields such as medicine, biology, and engineering. Some statistical properties of the NEGIGW distribution, such as moments, quantile function, and Renyi entropy, are studied. Three methods are used for parameter estimation, including maximum likelihood, maximum product of spacing, and percentile methods. The performance of the estimation methods is evaluated via Monte Carlo simulations. The NEGIGW distribution excels in its ability to fit real-world data accurately. Five medical and engineering datasets are applied to demonstrate the superior fit of NEGIGW distribution compared to competing models. This compelling evidence suggests that the NEGIGW distribution is promising for lifetime data analysis and reliability assessments across different disciplines.

Chen-Burr XII Model as a Competing Risks Model with Applications to Real-Life Data Sets

In this paper Chen-Burr XII distribution is constructed and graphical description of the probability density function, hazard rate and reversed hazard rate functions of the proposed model is obtained. Also, some statistical characteristics of the Chen-Burr XII distribution are discussed and some new models as sub-models from the Chen-Burr XII distribution are introduced. Moreover, maximum likelihood estimation of the parameters, reliability, hazard rate and reversed hazard rate functions of the Chen-Burr XII distribution are considered. Also, the asymptotic confidence intervals of the distribution parameters, reliability, hazard rate and reversed hazard rate functions are presented. Finally, three real life data sets are applied to prove how the Chen-Burr XII distribution can be applied in real life and to confirm its superiority over some existing distributions.

A new extended Fréchet model with different estimation methods and applications

In this study, we introduce a new extension of the Fréchet distribution known as the new extended Fréchet (NE_Fr) model. The NE_Fr is created by combining the new extended X family of distributions and the Fréchet distribution. The NE_Fr has more flexibility than the classical Fréchet distribution and some generalizations of the Fréchet distribution. The probability density function of the NE_Fr can be decreasing, unimodal and right skewed shape but it's hazard rate function can be decreasing or up-side-down shape. Several mathematical properties of the new model were derived by calculating the quantile function, the ordinary moments, the incomplete moments, the moment generating function, the conditional moment, the Bonferroni curve and the Lorenz curve. Several entropy measures were developed for this purpose. The inferences from the NE_Fr distribution were investigated using several established methods such as maximum likelihood estimation, least squares and weighted least squares estimation, and Anderson-Darling estimation. The simulation results demonstrated the computational efficiency of these techniques. The proposed NE_Fr distribution was demonstrated to be useful by examining three actual data sets.

A Generalized Area-Biased Power Ishita Distribution - Properties and Applications

Objectives: Generalized Area-Biased Power Ishita Distribution (GAPID) is a brand-new distribution that was suggested in this study. Its characterization is also mentioned in detail. Methods: The idea of weighted distributions is incorporated. This helps to derive new distributions that will be a best fit for many real-life data from many domains like agriculture, biomedical, financial, etc., where they may not match with the conventional distributions. Thus, it becomes necessary to create new distributions. The parameters have been calculated by maximum likelihood estimation. Findings: Its hazard rate function, survival function, and Moments, among other statistical features, were explored. Novelty: As results from classical distributions are insufficient for many Biomedical datasets, this novel distribution was fitted to a real data set of lung cancer patients' survival periods in months, allowing for a discussion of the data set's use. The superiority of the distribution is tested by comparing the same with known distributions and finding this one is better. Thereby the importance of the said distribution is established. Keywords: Estimate, Weighted distributions, Parameters, Reliability, Ishita distribution

Properties and applications of the Tan Weibull loss distribution

Probability models play crucial role in modeling loss in the finance and actuarial sciences. In this article, a new family of loss distributions known as the Tan F-Loss family of distributions is proposed with the Tan Weibull Loss distribution as a special case. The density exhibits decreasing, right skewed, symmetric, and approximately symmetric shapes. The hazard rate function shows decreasing and modified bathtub shapes. The statistical properties of the Tan Weibull Loss distribution including the quantile function, moments, expansion of the general rate, moment generating function, incomplete moment, and order statistics are studied. The maximum likelihood estimators of the distribution are also studied. Simulations are carried out to examine the behavior of the estimators. The results show that the estimators are consistent. The usefulness of the proposed distribution is demonstrated with two insurance loss datasets. The results show that the proposed distribution gives a better parametric to the two datasets compared with the competing distributions.

Marshall–Olkin Bivariate Weibull Model with Modified Singularity (MOBW-μ): A Study of Its Properties and Correlation Structure

We propose the “Marshall–Olkin Bivariate Weibull Model with Modified Singularity MOBW-μ”, which focuses on bivariate distributions essential for reliability and survival analyses. Distributions such as the Marshall–Olkin bivariate exponential (MOBE) and the Marshall–Olkin bivariate Weibull (MOBW) are discussed. The MOBW-μ model is introduced, which incorporates a lag parameter μ in the singular part, and probabilistic properties such as the joint survival function, marginal density functions, and the bivariate hazard rate function are explored. In addition, aspects such as the correlation structure and survival copulation are addressed and we show that the correlation of the MOBW-μ is always lower than that of its copula, regardless of the parameters. The latter result implies that the MOBW-μ does not have the Lancaster’s phenomenon that explains that any nonlinear transformation of variables decreases the correlation in absolute value. The article concludes by presenting a robust theoretical framework applicable to various disciplines.

Different Statistical Inference Algorithms for the New Pareto Distribution Based on Type-II Progressively Censored Competing Risk Data with Applications

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

Data analysis with applications in physics and engineering using XLindley model with improved adaptive Type-II progressively censored samples

Recently, a novel improved adaptive Type-II progressive censoring strategy has been suggested in order to obtain adequate data from trials that require a lengthy amount of time. Considering this scheme, this paper focuses on various classical and Bayesian estimation challenges for parameter and some reliability metrics for the XLindley distribution. Two classical estimation methods are considered from the classical perspective to get the point and interval estimations of the model parameter as well as reliability and hazard rate functions. In addition to the usual approaches, the Bayesian methodology is looked at by leveraging the squared error loss function and the Markov chain Monte Carlo technique. The Bayes point and credible intervals are obtained based on two forms of the posterior distribution. A simulation examination is implemented adopting multiple circumstances to distinguish between the conventional and Bayesian estimations. The simulation results demonstrate that the Bayesian approach using the likelihood function is superior for estimating the model parameter when compared with the other methods. In contrast, when estimating reliability metrics, it is advisable to utilize the Bayesian method with the spacings function. Two real-world data sets are analyzed to integrate the proposed approaches into practice, and the ideal progressive censoring strategy is chosen using some optimality criteria.

A New Extended Exponentiated Lomax (EEtLx) Distribution for Life Data

This paper proposes a new life distribution with both unbounded and bounded support. The proposed Extended Exponentiated Lomax (EEtLx) distribution is derived from the New Extended Exponentiated-G (NEET-G) family of Elgarhy, Haq, Gozel, and Nasir (2017). The maximum likelihood estimation method is used to estimate parameters of the proposed distribution and derive the relevant properties. The model is applied to patient relief times (minutes) after receiving a particular analgesic using different selection criteria (Log likelihood, Akaike Information Criteria (AIC), and Bayesian Information Criteria (BIC)). Resulting distributions are compared to well-established lifetime distributions in literature. The total time on test (TTT) plot is also used to determine whether the hazard rate function for this data increases or decreases, which subsequently reveals a decreasing trend. The proposed distribution remains the best fit compared to competing distributions. It is recommended that this distribution can be applied, not only in medical science, but also in reliability science, engineering, and economics fields.

Estimating the Parameters of the Generalized Inverted Kumaraswamy Distribution through the Utilization of a First Failure-Censored Sampling Plan

ThispaperfocusesonthedevelopmentofapproximateBayesestimatorsfortheshapeparametersofthegeneralizedinverted Kumaraswamy (GIKum) distribution. The estimators are based on a progressive first-failure censored plan. The study considers both maximum likelihood and Bayesian estimations using a gamma-informative prior distribution for the parameters, as well as the reliability function, hazard rate, and reversed hazard rate functions. To obtain the estimators, the paper employs Lindley’s approximation and utilizes Markov Chain Monte Carlo (MCMC) methods. The Bayes estimators are derived with respect to both symmetric (squared error) and asymmetric (linex and general entropy) loss functions. In order to assess the performance of the proposed estimators, the paper presents numerical results obtained through a simulation study involving different sample sizes.

A NEW INVERSE TWO-PARAMETER LINDLEY DISTRIBUTION AND ITS APPLICATION

This paper proposes an inverse two-parameter Lindley distribution by utilizing the one and two parameter Lindley distributions. The key properties of the novel distribution like, its survival function, shape characteristics of the density, entropy measure, hazard rate function, stochastic ordering and stress-strength reliability were examined. Two data sets were employed in the empirical studies. Method of maximum likelihood was used to estimate the parameters. The goodness- of -fit was accessed using the HQIC, BIC, CAIC, and AIC. The proposed distribution was compared with the inverse Lindley and the inverse Akash distributions in order to access its superiority over the two distributions. Ultimately, the new inverse two parameter Lindley distribution was found to be superior by providing a better fit.

On a New Transmuted Three-Parameter Lindley Distribution and Its Applications

In this paper, a new transmuted three-parameter Lindley distribution (TTHPLD) is established using the transmutation map method, which includes the Lindley distribution, two-parameter Lindley distribution, transmuted two-parameter Lindley distribution and three-parameter Lindley distribution as special cases. The statistical properties of the TTHPLD model, which are based on moments, order statistics, hazard rate functions, reliability functions, and Renyi entropy, have been studied. Moreover, the maximum likelihood estimators (MLEs) of the TTHPLD are obtained via differential evolution algorithms, and a simulation study is conducted to evaluate the consistency of the MLEs. Finally, the proposed distribution is applied to a real dataset and compared with other well-known models.

A Novel Three-Parameter Nadarajah Haghighi Model: Entropy Measures, Inference, and Applications

The fitting and modeling of skewed, complex, symmetric, and asymmetric datasets is an exciting research topic in many fields of applied sciences: notably, lifetime, medical, and financial sciences. This paper introduces a heavy-tailed Nadarajah Haghighi model by compounding the heavy-tailed family and Nadarajah Haghighi distribution. The model obtained has three parameters that account for the scale and shape of the distribution. The proposed distribution’s fundamental characteristics, such as the probability density, cumulative distribution, hazard rate, and survival functions, are provided, several key statistical properties are established, and several entropy information measures are proposed. Estimation of model parameters is performed via a maximum likelihood estimator procedure. Further, different simulation experiments are conducted to demonstrate the proposed estimator’s performance using measures like the average estimate, the average bias, and the associated mean square error. Finally, we apply our proposed model to analyze three different real datasets. In our illustration, we compare the practicality of the recommended model with several well-known competing models.

Enhanced Real-Life Data Modeling with the Modified Burr III Odds Ratio–G Distribution

In this study, we introduce the modified Burr III Odds Ratio–G distribution, a novel statistical model that integrates the odds ratio concept with the foundational Burr III distribution. The spotlight of our investigation is cast on a key subclass within this innovative framework, designated as the Burr III Scaled Inverse Odds Ratio–G (B-SIOR-G) distribution. By effectively integrating the odds ratio with the Burr III distribution, this model enhances both flexibility and predictive accuracy. We delve into a thorough exploration of this distribution family’s mathematical and statistical properties, spanning hazard rate functions, quantile functions, moments, and additional features. Through rigorous simulation, we affirm the robustness of the B-SIOR-G model. The flexibility and practicality of the B-SIOR-G model are demonstrated through its application to four datasets, highlighting its enhanced efficacy over several well-established distributions.