Hazard Rate Function Research Articles

Alpha Logarithm Transformed Half Logistic Distribution: Properties and Applications

Abstract: In this paper, we introduce a new two parameter Alpha Logarithm Transformed Half Logistic (ALTHL) distribution. A new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility .The proposed family may be named as Alpha Logarithm Transformed Half Logistic. This can be obtained via reparameterizing the exponentiated Kumaraswamy G-logarithmic family and the alpha logarithmic family of distributions. The recommended distribution reveals increasing, decreasing and bath-shaped probability Density, Distribution and Hazard rate function and Survival function. Some distributional properties of new model are investigated which include the Density Function, Distribution Function (DF), Quantile Function(QF), Moments, Moment Generating Function (MGF), Cumulative Generating Function (CGF).

Exponentiated Inverse Chen distribution: Properties and applications

A new distribution having three parameters called Exponentiated Inverse Chen distribution is proposed in this study. Important statistical properties like Survival function, hazard rate function, skewness and kurtosis etc are studied. Some methods of estimation Least Square, Maximum likelihood and Cramer-Von Mises methods are used using R programming software. A data set is discussed and Validity of the model is tested by analyzing P-P and Q-Q plots. Different information criteria such as Akanke’s information criterion, Bayesian information criterion, Corrected Akanke’s information criterion, and Hannan – Quinn information criterion are applied for model comparisons. For testing the goodness of fit of the proposed model, Kolmogrov-Smirnov, Anderson darling and Cramer –Von Mises statistics are used. The proposed model called Exponentiated Inverse Chen distribution is more applicable as compared to some existing probability model. It is found that MLEs are better with respect to LSE and CVM. PDF curve of model has shown that it can have various shapes like increasing as well as decreasing, monotonically increasing, constant as well as inverted bathtub shaped based hazard function is seen. Applicability and suitability of model is evaluated. For this, we have considered a real-life dataset. We found distribution that EIC is much flexible.

A New Life-Time Model with Bathtub and Inverted Bathtub Failure Rate Function

In this paper, a three parameter model called Zubair- Kumaraswamy (Z-Kum) distribution is proposed. The extension was done using Zubair G-Family [1] of continuous probability distribution to extend the well knownKumaraswamy distribution to make it more flexible in modeling and predicting real world phenomenon. Some basic structural properties of the new distributions like cdf, pdf, quantile functions, moments, moment generating functions, characteristics functions and order statistics was obtained. Survival function, hazard function, reversed hazard rate function and a cumulative hazard rate function was also obtained. Behavior of the hazard rate plot exhibit increase, decrease, Bathtub and inverted Bathtub shape. Maximum likelihood estimate was used to estimate the Z-Kum distribution parameters. Monte Carlo simulation was carried out to evaluate the performance of MLE method adopted. Result of the simulation studies indicates that MLE is good for the estimation of our distribution parameters. To compare the proposed model with the other fitted existing models, analytical measure of goodness of fit of some information criterion was considered using three real life data sets. From the results obtained, it is evident that our proposed model gives better fit than the other competing models and therefore, our proposed model provide greater flexibility in modeling real world phenomenon.

Role of Radical Surgery Following Preoperative Chemoradiotherapy in Octogenarians With Rectal Cancer: A SEER Analysis.

There has been a growing trend toward a watch-and-wait strategy to avoid surgery. This study evaluated the prognostic outcomes of nonoperative management following chemoradiotherapy (CRT) in the very old patients with rectal cancer. Using the Surveillance, Epidemiology, and End Results database, we conducted a retrospective analysis of octogenarians (age ≥80 years) with stage II-III rectal cancer who received neoadjuvant CRT with or without radical surgery between 2005 and 2016. Long-term survival outcomes of the two treatment strategies were compared. Propensity-matched cohorts were identified and analyzed for CRT followed by radical surgery vs. CRT alone (n=782). The 7-year rates of overall survival (OS) and disease-specific survival (DSS) of the two groups were 43% vs. 11% and 57% vs. 26%, respectively (p<0.001 for both comparisons). Radical surgery after CRT was the strongest prognostic factor associated with improved OS and DSS [hazard ratio (HR)=2.66 and 95% confidence interval (CI)=2.15-3.28 for OS; HR=2.50 and 95%CI=1.94-3.24 for DSS]. Based on the time-course hazard rate function plots of disease-specific mortality, short-term and late risk increments were observed in patients who underwent nonoperative management. This study highlights the importance of an active treatment strategy with radical surgery even in the highest age population with rectal cancer. Omitting surgery may not generally be considered safe when it is considered solely on chronological age. Further research is needed to identify the appropriate indications for nonoperative management.

A decreasing failure rate model with a novel approach to enhance the artificial neural network's structure for engineering and disease data analysis

The study focuses on key metrics used to examine the characteristics of a lifetime random variable distribution in reliability and survival theory research. In this analysis, metrics including the probability density function time, mean residual lifespan, mean time between failures, hazard rate, and reliability function are essential. The focus of the inquiry is these important parameters in relation to the Burr-Hatke exponential model specifically. The study focuses on key metrics used to examine the characteristics of a lifetime random variable distribution in reliability and survival theory research. In this analysis, metrics including the probability density function time, mean residual lifespan, mean time between failures, hazard rate, and reliability function are essential The focus of the inquiry is these important parameters in relation to the Burr-Hatke exponential model specifically. A key component of the research is a comparison of the outcomes from the artificial intelligence approach and those from conventional literature-based methodologies. This comparison study sheds light on how well the artificial neural network framework performs while evaluating the Burr-Hatke exponential model's technical features. The study allows a comprehensive analysis of the training and prediction capabilities of the growing neural network by calculating multiple performance measures. This comprehensive strategy improves our comprehension of the model's survival traits and reliability, offering significant contributions to the larger field of study. The network structure's mean square error was estimated to be 5.19E-04, and its coefficient of determination value was 0.99987 for the first neural network model. For the second neural network model, the coefficient of determination value was 0.99999 and the mean square error value was 4.58E-06. The outcomes amply revealed the neural network structure's extraordinarily high prediction accuracy and the degree to which the prediction outputs agree with those of the Maximum Likelihood Estimation technique.

Risk evolution of crude oil pipeline under periodic maintenance based on dynamic bayesian network

Pipeline failure can result in a chain reaction of accidents, including oil and gas leaks. Traditional risk analysis methods fail to fully capture the degradation and imperfect maintenance processes of multi-state systems. The model utilizes a hybrid failure rate recursive rule to accurately depict the changes in system reliability under varying maintenance intervals. The robustness of the model was validated through a case study of crude oil pipelines and the probability of pipeline failure under different maintenance strategies was predicted. The results suggest that semi-annual maintenance can significantly reduce the probability of pipeline failure compared to annual maintenance. If the equipment has experienced multiple instances of imperfect maintenance, it will accumulate excessive fatigue damage, leading to a gradual increase in the growth rate of the hazard rate function. Therefore, after 36 months, the advantage of semi-annual incomplete maintenance gradually diminishes, indicating that there is an upper limit to the number of equipment maintenance actions. To maximize failure risk reduction and minimize losses, a combined maintenance strategy (semi-annual and annual maintenance) should be implemented. Finally, through sensitivity and diagnostic analysis, key factors leading to pipeline failure were identified and appropriate measures to prevent pipeline failure were proposed.

Modelling the progression of illicit substance use patterns from real-world evidence.

To investigate an innovative pharmacometrics approach that addresses the challenges of using real-world evidence to model the progression of illicit substance use. The modelling strategy analysed real-world data from the National Longitudinal Study of Adolescent to Adult Health (AddHealth) survey using survival analyses and differential equations. Respondents were categorized into drug-naïve, active users and nonusers. The transitions between categories were modelled using interval-censored parametric survival analysis. The resulting hazard rate functions were used as time-dependent rate constants in a differential equation system. Covariate models for sex and depression status were assessed. AddHealth enrolled 6504 American teenagers (median age 16 years, range 11-21 years); this cohort was followed with five interviews over a 22-year period; the median age at the last interview was 38 years (range 34-45 years). The percentages of illicit drug users at Interviews 1-5 were 7.7%, 5.9%, 15.8%, 21.4% and 0.98%, respectively. The generalized gamma distribution emerged as the preferred model for the survival functions for transitions between categories. Age-dependent prevalence was obtained from the differential equation system. Active drug use was more prevalent in males, increased in adolescence and college years, peaked at 24 years, and decreased to low levels by 35 years. Depression, which was more frequent in females, increased the drug-naïve-active user transition rates but not the active user-nonuser and nonuser-active user transition rates. The evidence did not support an interaction between sex and depression. The model provided a satisfactory approximation for the age-dependent progression of illicit substance use from preadolescence to early middle age.

A study of a survival data using kernel estimates of hazard rate and aging intensity functions

Analyzing survival (life-testing) data and drawing inferences about them is a part of engineering and health sciences. So far, various statistical tools, e.g., survival (reliability) function (s f ), probability density function (pd f ), and hazard rate function (HR) were available among decision-making scientists to handle time-to-event data (complete or censored). But because functions (pd f ) estimators were interval (window) based, they mostly gave qualitative ideas having pictorial representation resembling step functions, ordinate remain constant when abscissa vary over an interval, thereby giving incomplete information. However, it can be sorted out with the use of kernel estimates of the above mentioned functions, resulting into smooth estimators. Moreover, the metric based on aging intensity function (AI) gives an alternative way of studying lifetime or clinical datasets as it is a quantitative measure (not interval-based), thereby depicting a broader view of a given data. In our study, we primarily focus on AI and HR functions estimated using four different kernels. We apply them to a case study of patients with primary malignant tumors of sternum (cf. Daniel and Cross, 2014) with the right-censored data. Our result shows that kernel estimates of HR and AI functions for patients with high grade tumor (HGT) are higher than for patients with low grade tumor (LGT), as expected. Thus, the study opens up a new direction for applying AI and HR functions in health sciences and engineering studies.

NEW LIFE-TIME CONTINUOUS PROBABILITY DISTRIBUTION WITH FLEXIBLE FAILURE RATE FUNCTION

Recently, the area of distribution theory has been receiving increased interest in generating or defining new classes of continuous probability distributions by way of extending the existing distributions. The new generated distribution is expected to be more flexible and have wider acceptability in modeling and predicting real world data sets. In this research, we proposed and study an extension of Ikum distribution using Zubai G- Family (2018) of distribution called Z-Ikum distribution. Expression of some basic structural properties of the new distribution such as cdf, pdf, quantile functions, moments, moment generating function, characteristics function and order statistics was derived. Survival function, hazard rate function, commutative hazard rate function and reversed hazard rate function was also discussed. Plots of the hazard rate function showincrease, decrease and bathtub shapes.Estimation of the proposed distribution parameters was carried out using MLE method. Performance of the parameter estimation was also evaluated via simulation studies. Result of the simulation studies indicates that our estimator is consistent. Three life data sets were used to evaluate the performance of our proposed distribution over some existing distributions. Result of the empherical study revealed that our proposed distribution performwell in modeling real life data than the competing distributions.

A versatile model for lifetime of a component under stress

In this study, a versatile model, called alpha-monotone inverse Weibull distribution (alphaIW), for lifetime of a component under stress is introduced by using the alpha-monotone concept. The alphaIW distribution is also expressed as a scale-mixture between the inverse Weibull distribution and uniform distribution on (0, 1). The alphaIW distribution includes alpha-monotone inverse exponential and alpha-monotone inverse Rayleigh distributions as submodels and converenges to the inverse Weibull, inverse exponential, and inverse Rayleigh distributions as limiting cases. Also, slash Weibull, slash Rayleigh, and slash exponential distribuitons can be obtained under certain variable transformation and parameter settings. The alphaIW distribution is characterized by its hazard rate function and characterizing conditions are provided as well. Maximum likelihood, maximum product of spacing, and least squares methods are used to estimate distribution parameters. A Monte-Carlo simulation study is conducted to compare the efficiencies of the considered estimation methods. In the application part, two practical data sets, Kevlar 49/epoxy and Kevlar 373/epoxy, are modeled via the alphaIW distribution. Modeling performance of the alphaIW distribution is compared with its rivals by means of some well-known goodness-of-fit statistics and results show that alphaIW distribution performs better modeling than them. Results of comparison also indicate that obtaining the alphaIW distribution by using the alpha-monotone concept is cost effective since the new shape parameter added to the distribution by using the alpha-monotone concept significantly increases the modeling capability of the IW distribution. As a result of this study, it is shown that the alphaIW distribution can be an alternative to the well-known and recently-introduced distributions for modeling purposes.

Failure rate, vitality, and residual lifetime measures: Characterizations based on stress-strength bivariate model with application to an automated life test data

In this article, we introduce some reliability concepts for the bivariate Pareto Type II distribution including joint hazard rate function, CDF for parallel and series systems, joint mean residual lifetime, and joint vitality function. The maximum likelihood and Bayesian estimation methods are utilized to estimate the model parameters. Simulation is carried out to assess the performance of the maximum likelihood and Bayesian estimators, and it is found that the two approaches work quite well in estimation process. Finally, a real lifetime data is analyzed to show the flexibility and the importance of the introduced bivariate mode.

A weighted Gompertz-G family of distributions for reliability and lifetime data analysis

This article is set to push new boundaries with leading-edge innovations in statistical distribution for generating up-to-the-minute contemporary distributions by a mixture of the second record value of the Gompertz distribution and the classical Gompertz model (weighted Gompertz model) using T-X characterization, especially used for two-sided schemes that provide an accurate model. The quantile, ordinary, and complete moments, order statistics, probability, and moments generating functions, entropies, probability weighted moments, Lin’s condition random variable, reliability in multicomponent stress strength system, reversed, and moments of residuals life and other reliability characteristics in engineering, actuarial, economics, and environmental technology were derived in their closed form. To investigate and test the flexibility, viability, tractability, and performance of the proposed Weighted Gompertz-G (WGG) generated model, the shapes of some sub-models of the WGG model were examined. The shapes of the sub-models indicated J-shapes, increasing, decreasing, and bathtub hazard rate functions. The maximum likelihood estimation of the WGG-generated model parameters was examined. An illustration with simulation and real-life data analysis indicated that the WGG-generated model provides consistently better goodness-of-fit statistics than some competitive models in the literature.

The odd log- logistic Power Inverse Lindley distribution: Model, Properties and Applications

In this article, we introduce a new three-parameter odd log-logistic power inverse Lindley distribution and discuss some of its properties. These include the shapes of the density and hazard rate functions, mixture representation, the moments, the quantile function, and order statistics. Maximum likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Three algorithms are proposed for generating random data from the proposed distribution. A simulation study is carried out to examine the bias and root mean square error of the maximum likelihood estimators of the parameters. An application of the model to three real data sets is presented finally and compared with the fit attained by some other well-known two and three-parameter distributions for illustrative purposes. It is observed that the proposed model has some advantages in analyzing lifetime data as compared to other popular models in the sense that it exhibits varying shapes and shows more flexibility than many currently available distributions.

The Topp-Leone Odd Burr X-G Family of Distributions: Properties and Applications

This paper proposes a new generalized family of distributions called the Topp-Leone odd Burr X-G (TLOBX-G) distribution and its special model, Topp-Leone odd Burr X-Weibull (TLOBX-W) is studied in detail. Structural properties are derived, including the hazard rate function, quantile function, density expansion, moments, R'enyi entropy, and order statistics. The maximum likelihood technique is used to estimate the parameters of the new family of distributions and a simulation study was carried out to assess the accuracy and consistency of these estimators. Finally, the applicability, usefulness, and flexibility of TLOBX-W distribution are illustrated using two real-life datasets.

Model and Properties of Exponentiated Generalized Odd Lomax Exponential Distribution

This study is based on the formulation of new probability model called Exponentiated Generalized Odd Lomax Exponential Distribution using Odd Lomax Generator and the exponential distribution. Some statistical properties like survival function, hazard rate function, quantile function and random deviate generation of the model is studied. Parameters of the model are estimated using Maximum likelihood, least square and Cramer – von Mises method methods of estimation. Applicability of the model is studied taking a real COVID - 19 data set. For model comparison, different information criteria like Akanke, Bayesian, and corrected Akaike criteria are used and the goodness of fit of the proposed model is test using Kolmogrov - Smirnov, Cramer – von Mises and Anderson Darling method. To compare the suitability of the model, some already defined probability models are considered and compared on basis of different criteria. To study of the performance of MLEs, Monte - Carlo simulation is presented. All the calculations are performed using R programming language.

Unit upper truncated Weibull distribution with extension to 0 and 1 inflated model – Theory and applications

A two-parameter unit distribution and its regression model plus its extension to 0 and 1 inflation is introduced and studied. The distribution is called the unit upper truncated Weibull (UUTW) distribution, while the inflated variant is called the 0−1 inflated unit upper truncated Weibull (ZOIUUTW) distribution. The UUTW distribution has an increasing and a J-shaped hazard rate function. The parameters of the proposed models are estimated by the method of maximum likelihood estimation. For the UUTW distribution, two practical examples involving household expenditure and maximum flood level data are used to show its flexibility and the proposed distribution demonstrates better fit tendencies than some of the competing unit distributions. Application of the proposed regression model demonstrates adequate capability in describing the real data set with better modeling proficiency than the existing competing models. Then, for the ZOIUUTW distribution, the CD34+ data involving cancer patients are analyzed to show the flexibility of the model in characterizing inflation at both endpoints of the unit interval.

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.

A New Approach of Presenting Lindley Expontiated Gumbel Distribution with Application to Environmental Data

The Lindley exponentiated Gumbel (LEGu) Distribution, a new family of probability distribution developed for modelling environmental data is introduced in this research paper. The investigation defines the new distribution's basic statistical properties, such as its shape, density function, hazard rate function, moment generation function, and maximum likelihood estimates of its model parameters. The study shows that, in comparison to other baseline distributions and comparators, increasing the number of parameters in the distribution improves its robustness and adaptability, making it more flexible in accommodating different types of data. The model is now more adaptable and can handle a wider range of data sets, making it a useful and effective tool for environmental modelling. The creation of the Lindley exponentiated Gumbel Distribution and the statistical qualities that go along with it bring up new possibilities for the analysis of environmental data, making a significant contribution to the larger initiatives in climate research and decision-making. The results of this study have the potential to advance our comprehension of environmental mechanisms and raise the precision of climate assessments and projections.

A Better Alternative to the Generalized Bilal Distribution: A New Model and Applications

In 2017, A. M. Abd-Elrahman [A new two-parameter lifetime distribution with decreasing, increasing or upside-down bathtub-shaped failure rate, Commun. Stat. - Theory Methods 46 (2017) 8865–8880, doi:10.1080/03610926.2016.1193198] introduced a generalization of the Bilal distribution, where a new two-parameter distribution, generalized Bilal distribution (GBD), was presented. He showed that its failure rate function can be upside-down bathtub-shaped. The failure rate can either be decreasing or increasing due to some mathematical and statistical reasons, which will be given below. In this paper, we introduce a simple and better alternative to the GBD, which will be denoted by WMD. We show that the WMD is a two-parameter distribution which can fit five different types of data sets with respect to their empirical hazard rate functions. Most properties of the WMD are investigated. Point and interval estimation procedures for the two unknown parameters are presented. The existence and uniqueness of the maximum likelihood estimates are proved. The moment estimates are obtained and we showed that one of these estimates is the minimum variance unbiased estimate (MVUE) for its corresponding parameter. A simulation study is provided and the paper is motivated by applications to four different real data sets. A detailed analysis for the Meeker and Escobar data is provided by the book of Meeker and Escobar [Statistical Methods for Reliability Data, 2nd edn. (John Wiley, 1998)]. The results may show that the new distribution provides a better fit than some other most recent existing and already known distributions in the literature. Finally, some concluding remarks are presented.

Unit Exponential Probability Distribution: Characterization and Applications in Environmental and Engineering Data Modeling

Distributions with bounded support show considerable sparsity over those with unbounded support, despite the fact that there are a number of real-world contexts where observations take values from a bounded range (proportions, percentages, and fractions are typical examples). For proportion modeling, a flexible family of two-parameter distribution functions associated with the exponential distribution is proposed here. The mathematical and statistical properties of the novel distribution are examined, including the quantiles, mode, moments, hazard rate function, and its characterization. The parameter estimation procedure using the maximum likelihood method is carried out, and applications to environmental and engineering data are also considered. To this end, various statistical tests are used, along with some other information criterion indicators to determine how well the model fits the data. The proposed model is found to be the most efficient plan in most cases for the datasets considered.