Hazard Rate Function Research Articles

Medical and Engineering Applications for Estimation and Prediction of a New Competing Risks Model: A Bayesian Approach

The Bayesian approach offers a flexible, interpretable and powerful framework for statistical analysis, making it a valuable tool to help in making optimal decisions under uncertainty. It incorporates prior knowledge or beliefs about the parameters, which can lead to more accurate and informative results. Also, it offers credible intervals as a measure of uncertainty, which are often more interpretable than confidence intervals. Hence, the Bayesian approach is utilized to estimate the parameters, reliability function, hazard rate function and reversed hazard rate function of a new competing risks model. A squared error loss function as a symmetric loss function and a linear exponential loss function as an asymmetric loss function are employed to derive the Bayesian estimators. Credible intervals of the parameters, reliability function, hazard rate function and reversed hazard rate function are obtained. Predicting future observations is important in many fields, from finance and weather forecasting to healthcare and engineering. Thus, two-sample prediction (as a special case of the multi-sample prediction) for future observation is considered. An adaptive Metropolis algorithm is applied to conduct a simulation study to evaluate the performance of the Bayes estimates and predictors. Moreover, two applications of medical and engineering data sets are used to test and validate the theoretical results, ensuring that they are accurate, applicable to real-world scenarios and contribute to the understanding of the world and inform decision-making.

A New Bivariate Survival Model: The Marshall-Olkin Bivariate Exponentiated Lomax Distribution with Modeling Bivariate Football Scoring Data

This paper focuses on applying the Marshall-Olkin approach to generate a new bivariate distribution. The distribution is called the bivariate exponentiated Lomax distribution, and its marginal distribution is the exponentiated Lomax distribution. Numerous attributes are examined, including the joint reliability and hazard functions, the bivariate probability density function, and its marginals. The joint probability density function and joint cumulative distribution function can be stated analytically. Different contour plots of the joint probability density function and joint reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are given. The unknown parameters and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution are estimated using the maximum likelihood method. Also, the Bayesian technique is applied to derive the Bayes estimators and reliability and hazard rate functions of the bivariate exponentiated Lomax distribution. In addition, maximum likelihood and Bayesian two-sample prediction are considered to predict a future observation from a future sample of the bivariate exponentiated Lomax distribution. A simulation study is presented to investigate the theoretical findings derived in this paper and to evaluate the performance of the maximum likelihood and Bayes estimates and predictors. Furthermore, the real data set used in this paper comprises the scoring times from 42 American Football League matches that took place over three consecutive independent weekends in 1986. The results of utilizing the real data set approve the practicality and flexibility of the bivariate exponentiated Lomax distribution in real-world situations, and the bivariate exponentiated Lomax distribution is suitable for modeling this bivariate data set.

A new Marshall-Olkin generalized-k family of distributions with applications

Abstract In this article, we describe a new Marshall-Olkin family of distributions based on the Marshall-Olkin family of distributions by employing a different generator. We have derived several statistical properties of the new Marshall-Olkin family, including survival function, hazard rate function, quantile function, moments, and linear representation. We have taken the Weibull distribution as a baseline distribution and found that the new Marshall-Olkin Weibull distribution exhibits a variety of shapes for its hazard rates and densities. We have tested the performance of various estimation methods for the new Marshall-Olkin family of distributions, including eight different estimation methods. Additionally, we conducted extensive simulations to assess the robustness of the estimation methods across various scenarios and determine which methods are most effective for varying sample sizes. To further validate the new Marshall-Olkin Weibull distribution, we have used two different data sets to compare the fitted new Marshall-Olkin Weibull distribution with the well-known comparative models. By testing the performance of different estimation methods and using multiple data sets, we can demonstrate the versatility and usefulness of the new Marshall-Olkin family of distributions for modeling a wide range of phenomena.

A New Method of Generating Truncated Bivariate Families of Distributions

The modeling of complex data has attracted several researchers for the quest of generating new probability distributions. The joint modeling of two variables asks for some additional complexities as a bivariate distribution is needed. The field of research in developing bivariate families of distributions is somewhat new. In certain situations, the domain of data is restricted and some truncated distribution is required. Several univariate truncated families of distributions are available for modeling of a single variable but the bivariate truncated families of distributions has not been studied and in this paper, we have proposed a new bivariate truncated families of distributions. A specific sub-family has been proposed by using the bivariate Burr as a base-line distribution, resulting in a bivariate truncated Burr family of distributions. Some important statistical properties of the proposed family has been studied, which include the marginal and conditional distributions, bivariate reliability, and bivariate hazard rate functions. The maximum likelihood estimation for the parameters of the family is also carried out. The proposed bivariate truncated Burr family of distributions is studied for the Burr baseline distributions, giving rise to the bivariate truncated Burr-Burr distribution. The new bivariate truncated Burr-Burr distribution is explored in detail and several statistical properties of the new distribution are studied, which include the marginal and conditional distributions, product, ratio, and conditional moments. The maximum likelihood estimation for the parameters of the proposed distribution is done. The proposed bivariate truncated Burr-Burr distribution is used to model some real data sets. It is found that the proposed distribution performs better than the other distributions considered in this study.

Discrete Joint Random Variables in Fréchet-Weibull Distribution: A Comprehensive Mathematical Framework with Simulations, Goodness-of-Fit Analysis, and Informed Decision-Making

This paper introduces a novel four-parameter discrete bivariate distribution, termed the bivariate discretized Fréchet–Weibull distribution (BDFWD), with marginals derived from the discretized Fréchet–Weibull distribution. Several statistical and reliability properties are thoroughly examined, including the joint cumulative distribution function, joint probability mass function, joint survival function, bivariate hazard rate function, and bivariate reversed hazard rate function, all presented in straightforward forms. Additionally, properties such as moments and their related concepts, the stress–strength model, total positivity of order 2, positive quadrant dependence, and the median are examined. The BDFWD is capable of modeling asymmetric dispersion data across various forms of hazard rate shapes and kurtosis. Following the introduction of the mathematical and statistical frameworks of the BDFWD, the maximum likelihood estimation approach is employed to estimate the model parameters. A simulation study is also conducted to investigate the behavior of the generated estimators. To demonstrate the capability and flexibility of the BDFWD, three distinct datasets are analyzed from various fields, including football score records, recurrence times to infection for kidney dialysis patients, and student marks from two internal examination statistical papers. The study confirms that the BDFWD outperforms competitive distributions in terms of efficiency across various discrete data applications.

A New Generalized Family of Weibull-Exponentiated Half Logistic-G Distribution with Applications

We propose a new extension of the Weibull-exponentiated half logistic-G (WEHL-G) distribution, called the Marshall-Olkin-Weibull-exponentiated half logistic-G (MO-WEHL-G) family of distributions. The properties of this family of distributions including quantile function, distribution of the order statistics, hazard rate function, Rényi entropy and moments of residual life are presented. To estimate the parameters of the MO-WEHL-G family of distributions, six different estimation approaches are used, namely, maximum likelihood, Anderson-Darling, Ordinary Least Squares, Weighted Least Squares, Cramér-von Mises and Maximum Product of Spacing. The consistency properties of the six estimation methods were assessed using Monte Carlo simulations for a special case of the MO-WEHL-G family of distributions. The flexibility and importance of the proposed model was assessed using numerous goodness-of-fit statistics on three different data sets.

The entropy-transformed Gompertz distribution: Distributional insights and cross-disciplinary utilizations

A novel two-parameter continuous model titled the entropy-transformed Gompertz (ETGPZ) distribution has been developed via the entropy transform. A new framework has been investigated and found to meet the criteria of the probability function. By significantly improving the functional shape and having the ability to model the most likely form of the hazard rate function, this new modification has increased the adaptability of the typical distribution. Some of its core characteristics, such as its statistical and computational features, are clearly presented. A thorough simulation analysis has been done to examine the final behavior of maximum likelihood estimators while estimating model parameters. We assess the performance and practical applicability of the ETGPZ distribution using eight real datasets from engineering and biomedical fields. The results demonstrate that the ETGPZ outperforms the baseline Gompertz (GPZ) distribution, highlighting its superiority and broader potential for various applications.

A New Lifetime Distribution and its Application to Cancer Data

Introduction: Recently, researchers have introduced new generated families of univariate lifetime distributions. These new generators are obtained by adding one or more extra shape parameters to the underlying distribution or compounding two distributions to get more flexibility in fitting data in different areas such as medical sciences, environmental sciences, and engineering. The addition of parameter(s) has been proven useful in exploring tail properties and for improving the goodness-of-fit of the family of the proposed distributions. Methods:A new Three-Parameter Weibull-Generalized Gamma distribution which provides more flexibility in modeling lifetime data is developed using a two-component mixture of Weibull distribution (with parameters θ and λ) and Generalised Gamma distribution (with parameters α=4,θ and λ). Some of its mathematical properties such as the density function, cumulative distribution function, survival function, hazard rate function, moment generating function, Renyi entropy and order statistics are obtained. The maximum likelihood estimation method was used in estimating the parameters of the proposed distribution and a simulation study is performed to examine the performance of the maximum likelihood estimators of the parameters. Results: Real life applications of the proposed distribution to two cancer datasets are presented and its fit was compared with the fit attained by some existing lifetime distributions to show how the Three-Parameter Weibull-Generalized Gamma distribution works in practice Conclusion: The results suggest that the proposed model performed better than its competitors and it’s a useful alternative to the existing models.

Physical aspects of quantile residual lifetime sequence

Abstract The modeling of count data is found in many fields, such as statistical physics, public health, medicine, epidemiology, applied science, sociology, and agriculture. In many physical situations, it has been observed that many times in the real world, the original variables may be continuous in nature, but discrete by observation. In this study, the α {\rm{\alpha }} -quantile residual life function for discrete lifetime models is defined and some attributes are investigated. The relation between this measure and the hazard rate function is studied. We discuss how this measure could be useful for finding the burn-in time of a lifetime dataset. Then, a new stochastic order based on the α {\rm{\alpha }} -quantile residual life is proposed and studied.

The Hybrid Odd Exponential-Exponential Distribution: Statistical Properties and Applications

In this paper, we study a new statistical distribution called the hybrid odd exponential-exponential distribution (HOE-E distribution for short) we the aim of applying it to real data sets. This new distribution appears as a sub-model of the larger HOEE-Φ family in (Mahdi et al., 2024). We explore many mathematical and statistical properties of this distribution such as the series expansion of the PDF, the quantile function, moments and their generating function, incomplete moments, the shape of the PDF, and the shape of the hazard rate function, order statistics, and parameter estimation using the maximum likelihood estimation. We use this distribution to model a plasma concentration data set. Moreover, this we show that the suggested HOE-E distribution outperformance many other distributions based on various statistical criteria like AIC, CAIC, BIC, HQIC, W, and A.

Unit extended exponential distribution with applications

In this paper, the unit extended exponential distribution is presented. The extended exponential distribution is a two-parameter lifespan distribution that has been demonstrated to be very adaptable. Our goal is to convert this flexibility between the unit interval and the extended exponential distribution. Several different types of probability density functions, including decreasing, left-skewed, right-skewed, and unimodal, have been shown. However, the hazard rate function may take on various forms, including J-shaped, U-shaped, and bathtub shapes. A few statistical properties, such as moments, mean, variance, skewness, kurtosis, expectation-weighted moments, incomplete moments, conditional moments, mean deviation, Lorenz and Bonferroni curves, mean residual life, mean inactivity time, and various entropy measures, round out the theoretical section. The maximum likelihood technique is utilized to estimate model parameters using unit data. This study provides simulation data to evaluate the proposed strategy. Two applications utilizing actual data sets emphasize the significance of the new model compared to existing unit models.

Sine Type II Topp-Leone Burr XII Distribution: Mathematical Properties and Application to Biomedical and Environmental Data

This paper introduces a novel three-parameter lifetime model known as the Sine Type II Top-Leone Burr XII distribution. We derive various mathematical properties of this distribution, including its survival function, hazard rate function, quantile and generating functions, moments, Rényi entropy, moment generating function, and order statistics. The parameters of the proposed model were estimated using the maximum likelihood method. A simulation study examines the consistency of these estimates across both small and large sample sizes. To demonstrate the flexibility and significance of the distribution, we apply it to model Biomedical and Environmental data sets.

Constant Stress-Partially Accelerated Life Tests of Vtub-Shaped Lifetime Distribution under Progressive Type II Censoring

In lifetime tests, the waiting time for items to fail may be long under usual use conditions, particularly when the products have high reliability. To reduce the cost of testing without sacrificing the quality of the data obtained, the products are exposed to higher stress levels than normal, which quickly causes early failures. Therefore, accelerated life testing is essential since it saves costs and time. This paper considers constant stress-partially accelerated life tests under progressive Type II censored samples. This is realized under the claim that the lifetime of products under usual use conditions follows Vtub-shaped lifetime distribution, which is also known as log-log distribution. The log–log distribution is highly significant and has several real-world applications since it has distinct shapes of its probability density function and hazard rate function. A graphical description of the log–log distribution is exhibited, including plots of the probability density function and hazard rate. The log–log density has different shapes, such as decreasing, unimodal, and approximately symmetric. Several mathematical properties, such as quantiles, probability weighted moments, incomplete moments, moments of residual life, and reversed residual life functions, and entropy of the log–log distribution, are discussed. In addition, the maximum likelihood and maximum product spacing methods are used to obtain the interval and point estimators of the acceleration factor, as well as the model parameters. A simulation study is employed to assess the implementation of the estimation approaches under censoring schemes and different sample sizes. Finally, to demonstrate the viability of the various approaches, two real data sets are investigated.

A novel asymmetric extension of power XLindley distribution: properties, inference and applications to engineering data

It is impossible to overstate the importance of using trigonometric functions appropriately in distribution theory. The main contribution of the research is to construct a flexible trigonometric extension of the power XLindley distribution. More specifically, we build an innovative two-parameter lifetime distribution known as the sine power XLindley distribution (SPXLD) using characteristics from the sine-generated family of distributions. As the main motivational fact, it provides an attractive alternative to the power Lindley, power XLindley, weighted Lindley, and extended power Lindley distributions; it may be better able to model lifetime phenomena presenting data of leptokurtic and platkurtic nature. In contrast to the increasing, decreasing, and reversed-j-shaped hazard rate function, the density exhibits asymmetric shapes with varying peakedness levels. Several significant characteristics are illustrated, including moments, the quantile function, the probability density function in series representation, the stress-strength reliability, and incomplete moments. To analyze the behavior of the suggested distribution, sixteen estimation techniques are applied, such as the maximum likelihood, percentiles, some methods of minimum distances, some methods based on minimum and maximum spacing distances, and the Kolmogorov method. After that, an extensive simulation study and the examination of two survival real datasets are used to show the viability, usefulness, and adaptability of the SPXLD. Relevant goodness of fit criteria demonstrates that the SPXLD fits several current distributions.

On Positive Inflated Geometric Distribution: Properties and Applications

One-inflation in zero-truncated count data has recently found considerable attention. In this regard, zero-truncated Geometric distribution and distribution to a point mass at one are used to create a one-inflated model, namely, one-inflated zero-truncated Geometric distribution. Its reliability characteristics, generating functions, and distributional properties are investigated in detail, which includes survival function, hazard rate function, reverse hazard rate function, probability generating function, characteristic function, variance, skewness, and kurtosis. Monte Carlo simulation have been undertaken to evaluate the effectiveness of the maximum likelihood estimators. To test the compatibility of our proposed model, the baseline model and the proposed model are distinguished by using the two different test procedures. The adaptability of the suggested model is demonstrated using two real-life datasets from separate domains by taking various performance measures into consideration.

Likelihood and pivotal-based reliability inference for inverse exponentiated Rayleigh distribution under block progressive censoring

ABSTRACT In this paper, focus is on the estimation of survival characteristics, specifically the reliability function, hazard rate function, median time to failure, and differences in different test facilities, using block progressive censored data. Estimations are carried out through both maximum likelihood and pivotal methods, assuming the lifetime distribution of test units follows an inverse exponentiated Rayleigh distribution. The paper derives maximum likelihood estimators for unknown parameters, exploring their existence and uniqueness properties. Approximate confidence intervals for survival characteristics are constructed using the delta method and likelihood theory. Moreover, point and generalized confidence interval estimators are developed through a pivotal quantity-based method. A simulation study is conducted to compare the performance of the proposed approaches, revealing that the pivotal quantity-based approach yields superior estimation results. Finally, the proposed estimates are applied to analyze two real datasets, illustrating their practical applicability.

Modified Ramos-Louzada-G family with baseline Weibull distribution: Properties, characterizations, regression, and applications

The paper introduced a novel family of distributions, called the Kumaraswamy Ramos-Louzada-G (KumRL-G) class, focusing on the five-parameter Kumaraswamy Ramos-Louzada Weibull (KumRLW) distribution. This new family of distributions, which includes existing and numerous new sub-models, offers improved flexibility and accuracy in modeling and analyzing survival data. Key statistical properties, including quantile function, moments, and entropy measures underlying the distribution have been derived, and characterizations have also been provided based on the ratio of two truncated moments and the hazard rate function. The maximum likelihood estimation (MLE) is employed to estimate the parameters of the proposed probability distribution, and Monte Carlo simulation analysis is performed to demonstrate the effectiveness of this method. The significance and adaptability of the new family of distributions are revealed through applications to COVID-19 and survival rate to age 65 of male cohort datasets from Ghana, Nigeria, and Canada. A new location-scale regression model was subsequently formulated from the new KumRLW distribution. Its practicality was demonstrated using survival data on hypertension from Ghana with gender as a covariate. The regression analysis showed that gender is a significant factor in the length of time before hypertension develops. The new KumRL-G family with baseline Weibull distributions provides more flexibility and improved fit in modeling various shapes and behaviors in the survival datasets surpassing its existing sub-models and other notable distributions.

Binomial Poisson Ailamujia model with statistical properties and application

In this study, an improved extension of the Poisson-Ailamujia distribution is introduced. The new distribution was derived using a binomial mixing approach, and the new model is named the “Binomial Poisson-Ailamujia (Bin-PA)” distribution. Some important statistical properties are derived, including mode, quantile function, moments and their associated measures, actuarial (risk) measures, and reliability features such as survival, hazard (failure) rate, and mean residual life function. The parameters of the proposed distribution are estimated using the maximum likelihood estimation method. A comprehensive simulation study is also carried out to access the behavior-derived maximum likelihood estimators. Furthermore, a new count-regression model was also introduced. Two datasets are utilized to demonstrate the applicability and usefulness of the new model. It is concluded that the Binomial Poisson-Ailamujia distribution is more flexible and efficiently analyzed both datasets as compared to competitive discrete distributions.

A new family of generalized-G distributions with properties and applications

We introduce and study a new generalized family of distributions herein referred to as the Marshall–Olkin Exponentiated Half Logistic-Generalized-G (MO-EHL-GG). A generalized distribution is a broader class of probability distributions that includes various specific distributions. It has parameters that allow for flexibility in modeling different types of data. By combining the Marshall–Olkin generator, the exponentiated half logistic generator and the generalized generator, the MO-EHL-GG family of distributions is developed. The primary objective behind introducing this new distribution is its enhanced flexibility and the ability of its hazard rate function to exhibit diverse shapes, making it valuable for statistical analysis and modeling purposes. Special cases of the new model are presented. Mathematical and statistical properties of the distribution are investigated. Estimates of the parameters are provided and simulation studies are conducted to examine the consistency of the model’s estimates. The significance of the new model is finally investigated through applications to real-life data sets. Three datasets were analyzed, demonstrating superior performance of our proposed distribution compared to competing models with the same number of parameters.

Epidemiological modeling of COVID-19 data with Advanced statistical inference based on Type-II progressive censoring

This research proposes the Kavya-Manoharan Unit Exponentiated Half Logistic (KM-UEHL) distribution as a novel tool for epidemiological modeling of COVID-19 data. Specifically designed to analyze data constrained to the unit interval, the KM-UEHL distribution builds upon the unit exponentiated half logistic model, making it suitable for various data from COVID-19. The paper emphasizes the KM-UEHL distribution's adaptability by examining its density and hazard rate functions. Its effectiveness is demonstrated in handling the diverse nature of COVID-19 data through these functions. Key characteristics like moments, quantile functions, stress-strength reliability, and entropy measures are also comprehensively investigated. Furthermore, the KM-UEHL distribution is employed for forecasting future COVID-19 data under a progressive Type-II censoring scheme, which acknowledges the time-dependent nature of data collection during outbreaks. The paper presents various methods for constructing prediction intervals for future-order statistics, including maximum likelihood estimation, Bayesian inference (both point and interval estimates), and upper-order statistics approaches. The Metropolis-Hastings and Gibbs sampling procedures are combined to create the Markov chain Monte Carlo simulations because it is mathematically difficult to acquire closed-form solutions for the posterior density function in the Bayesian framework. The theoretical developments are validated with numerical simulations, and the practical applicability of the KM-UEHL distribution is showcased using real-world COVID-19 datasets.