Parameter Estimation of Generalized Modified Weibull Using the Maximum Likelihood on Simulation and Real-World Data

Almost all survival data is censored, and censor imputation is necessary. This study aimed to investigate the performance of the Bayesian Approach (BA) in the imputation of censored records in simulated and Breast Cancer (BC) data. Due to the difference in the distribution of time to event in survival analysis, two well-known the Weibull and Birnbaum-Saunders (BS) distributions have been used to test the performance of the BA. For each of the censored, 10,000 times were simulated using the BA in R and BUGS software, and their median or mean was imputed instead of each censor. The eligibility of both imputation methods was investigated using different curves, different censoring percentages, and sample sizes, as well as the Deviance Information Criteria (DIC), Effective Sample Size, and the Geweke diagnostic in simulated and especially real BC data. The BC data, which contains 220 patients who were identified and followed up between 2015 and 2023, was made accessible on February 1, 2023. The Kaplan-Meier, the BA, and other survival curves were drawn for the observed times. Findings indicated that the performance of the BA under the Weibull and BS distributions in simulated data is similar. The DIC index in the BC data under the BS distribution (1510) is less than the Weibull distribution (1698). Therefore, the BS distribution is preferred over the Weibull for imputation of censoring times in real BC data.

In this article, an attempt is made to propose a novel method of lifetime distributions with maximum flexibility using a popular T–X approach together with an exponential distribution, which is known as the New Generalized Logarithmic-X Family (NGLog–X for short) of distributions. Additionally, the generalized form of the Weibull distribution was derived by using the NGLog–X family, known as the New Generalized Logarithmic Weibull (NGLog–Weib) distribution. For the proposed method, some statistical properties, including the moments, moment generating function (MGF), residual and reverse residual life, identifiability, order statistics, and quantile functions, were derived. The estimation of the model parameters was derived by using the well-known method of maximum likelihood estimation (MLE). A comprehensive Monte Carlo simulation study (MCSS) was carried out to evaluate the performance of these estimators by computing the biases and mean square errors. Finally, the NGLog–Weib distribution was implemented on four real biomedical datasets and compared with some other distributions, such as the Alpha Power Transformed Weibull distribution, Marshal Olkin Weibull distribution, New Exponent Power Weibull distribution, Flexible Reduced Logarithmic Weibull distribution, and Kumaraswamy Weibull distribution. The analysis results demonstrate that the new proposed model performs as a better fit than the other competitive distributions.

Research on survival data is quite important. Many modeling problems can be made related to survival data. One of the survival data methods is the Type III Censored method, which is a type of censoring that is limited by time, with individuals entering the study at different times during a certain period of time. In this study, the research focus is to obtain point estimates of the Weibull distribution parameters using the Maximum Likelihood Estimator method. The method is used to examine the survival data of Weibull distribution censored type III on secondary data of patients with lung cancer at Dr. Kariadi Hospital Semarang. Based on the results of the analysis, the Weibull distribution with a point estimator for the scale parameter is α̂ = 94.875997 and the shape parameter is β̂ = 1.638958.

Mixture of Weibull distributions has wide application in modeling of heterogeneous data sets. The parameter estimation is one of the most important problems related to mixture of Weibull distributions. In this paper, we propose a L-moment estimation method for mixture of two Weibull distributions. The proposed method is compared with maximum likelihood estimation (MLE) method according to the bias, the mean absolute error, the mean total error and completion time of the algorithm (time) by simulation study. Also, applications to real data sets are given to show the flexibility and potentiality of the proposed estimation method. The comparison shows that, the proposed method is better than MLE method.

This study introduces the Odd Rayleigh-G (OR-G) family of distribution and explores its mathematical properties, applications, and performance comparisons. The Odd Rayleigh-Weibull distribution (ORWD) is developed by incorporating the "Odd" transformation into the Rayleigh and Weibull distribution, resulting in a flexible model suitable for various real-life and survival data applications. The probability density function (PDF), cumulative distribution function (CDF), hazard function, and survival function of the ORWD are derived and analyzed. Parameter estimation is performed using the Maximum Likelihood Estimation (MLE) method, and the performance of the ORWD is assessed through simulation studies. The simulations for parameter estimates at 100 sample sizes were conducted and the plot of the simulated data on the PDF, CDF, survival and hazard function demonstrate a comprehensive view of the characteristics of the Odd Rayleigh Weibull distribution. This information is useful for understanding the behaviour of the distribution and for applications in reliability analysis and survival studies. The results demonstrate the consistency and efficiency of the MLE method for the ORWD. The ORWD is compared with other distributions, including the Weibull, Power Rayleigh, and Rayleigh distributions, using goodness-of-fit measures such as the Akaike Information Criterion (AIC = 111.0238 and 87.4294), Bayesian Information Criterion (BIC = 117.2564 and 96.0320), and Kolmogorov-Smirnov (KS = 0.9559 and 0.9889) test with p-values (p-val = 7.772e-16 and 2.2e-16). The ORWD shows superior performance in fitting the mortality dataset and the Reddit advertisement dataset, highlighting its potential for modelling complex data structures. Overall, this study provides a comprehensive framework for the...

Univariate Weibull distribution is a well known lifetime distribution and has been widely used in reliability and survival analysis. In this paper, we introduce a new family of bivariate generalized Weibull (BGW) distributions, whose univariate marginals are exponentiated Weibull distribution. Different statistical quantiles like marginals, conditional distribution, conditional expectation, product moments, correlation and a measure component reliability are derived. Various measures of dependence and statistical properties along with aging properties are examined. Further, the copula associated with BGW distribution and its various important properties are also considered. The methods of maximum likelihood and Bayesian estimation are employed to estimate unknown parameters of the model. A Monte Carlo simulation and real data study are carried out to demonstrate the performance of the estimators and results have proven the effectiveness of the distribution in real-life situations.

The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues. In this article, we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences. A specific sub-model form of our suggested family, named as a new extended heavy-tailed Weibull distribution is examined in detail. Some basic characterizations, including quantile function and raw moments have been derived. The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method. To judge the performance of the maximum likelihood estimators, a simulation analysis is performed in detail. Furthermore, some important actuarial measures such as value at risk and tail value at risk are also computed. A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed. The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set. The practical application shows that the proposed model is more flexible and efficient than the other six competing models including (i) the two-parameter models Weibull, Lomax and Burr-XII distributions (ii) the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions, and (iii) a well-known four-parameter Kumaraswamy Weibull distribution.

Life is filled with uncertainty and risk. The analysis of lifetime is needed to be a tool that can manage uncertainty. Lifetime is defined as data that contains the time until the occurrence of an event. Based on its definition, lifetime data is like Hazard rate data or mortality data because mortality data can be defined as data that contains the probability of an object surviving until that moment per unit time interval. The analysis of mortality data aims to model the distribution of time to event and/or the determinants of time to event. One of the distribution models that can be used to analyze mortality data is Weibull distribution. However, the Weibull distribution is not very suitable for modeling the more complex versions of data. Therefore, an extension of the Weibull distribution that is more flexible in modeling data is used, namely the Extended Exponential Weibull (ExEW) distribution. The ExEW distribution has four parameters whose estimation can be calculated using the maximum likelihood estimation (MLE) method. However, parameters estimated with MLE are often too difficult to calculate analytically, hence the use of optimization methods. One of the optimization methods that can be used to determine the estimated parameters of the ExEW distribution is the conjugate gradient method. To date, many conjugate gradient methods have been developed, including the Liu-Feng-Zou (LFZ) spectral conjugate gradient method and the Jian-Yang-Jiang-Liu-Liu (JYJLL) spectral conjugate gradient method. Previous research suggests that the JYJLL spectral conjugate gradient method has more efficient computational performance than the LFZ spectral conjugate gradient method. Through data simulation, this study provides results that the JYJLL spectral conjugate gradient conjugate method has better accuracy than the LFZ spectral conjugate gradient method in parameter estimation of the ExEW distribution. In addition, the ExEW distribution is the most suitable distribution in modeling various forms of Hazard rate data compared to the Weibull and exponential distributions.

Estimating the parameters of 3-p Weibull distribution through differential evolution

The probability distributions of wind speeds and the availability of wind turbines were investigated by considering the vertical wind shear. Based on the wind speed data at the standard height observed at a wind farm, the power-law process was used to simulate the wind speeds at a hub height of 60 m. The Weibull and Rayleigh distributions were chosen to express the wind speeds at two different heights. The parameters in the model were estimated via the least square (LS) method and the maximum likelihood estimation (MLE) method, respectively. An adjusted MLE approach was also presented for parameter estimation. The main indices of wind energy characteristics were calculated based on observational wind speed data. A case study based on the data of Hexi area, Gansu Province of China was given. The results show that MLE method generally outperforms LS method for parameter estimation, and Weibull distribution is more appropriate to describe the wind speed at the hub height.

For lifetime data analysis, failure time analysis, or survival analysis, the Weibull distribution is commonly used due to its various hazard functions, which can be increasing, decreasing, or constant. We have extended the traditional twoparameter Weibull distribution to accommodate hazard functions that are increasing, decreasing, constant and bathtubshaped. Utilizing a competing risks approach, we applied this modified Weibull distribution to both simulated data and an observed mice dataset. Our findings indicate that the modified Weibull distribution provides a better fit to the mice dataset compared to the traditional Weibull distribution. We estimated the parameters of the modified Weibull distribution using Maximum likelihood estimation (MLE) and Bayesian methods. For MLE, we employed the Newton-Raphson numerical method, while for the Bayesian approach, we used the Metropolis-Hastings algorithm, an MCMC method. Additionally, we plotted hazard curves for both the simulated and mice datasets. The Kaplan-Meier survival curves were plotted along with the survival curve of the modified Weibull distribution.. KEYWORDS :Modified weibull distribution, Competing risks, MCMC, Information criterion, MLE.

In the fields of reliability engineering, survival analysis, and lifetime data modeling, accurately representing the failure times and life durations of systems, components, and organisms is a central concern. Traditional lifetime distributions—such as the exponential, Weibull, and gamma distributions—have been widely used due to their mathematical tractability and interpretability. However, these classical models often struggle to capture the complexity of real-world data, particularly when hazard rate behaviors vary, including increasing, decreasing, bathtub-shaped, or unimodal patterns. Although the Weibull distribution is popular for its flexibility, it may not sufficiently model datasets where the hazard function deviates from its typical monotonic form. Similarly, while the gamma distribution is effective in many stochastic and queuing contexts, it lacks the versatility to represent certain tail behaviors and multimodal characteristics observed in practice. To address these limitations, statisticians have developed hybrid and compound distributions that merge features from multiple distributions, enhancing both flexibility and applicability. One such development is the Weibull-Gamma distribution, derived by mixing the Weibull and Gamma distributions. The first four moments about the origin, as well as the mean, were calculated for this new distribution. Derived expressions also include the coefficient of variation, skewness, kurtosis, and index of dispersion. In addition, the moment-generating function, characteristic function, and Laplace transform were established. Key reliability functions—such as the survival function, hazard rate function, and mean residual life—were also derived. Parameter estimation was carried out using the Maximum Likelihood Estimation (MLE) method. The goodness-of-fit of the proposed distribution was evaluated against several existing related models using criteria such as the Akaike Information Criterion (AIC), Corrected Akaike Information Criterion (AICC), and Bayesian Information Criterion (BIC). These comparisons were based on real-world datasets. The results demonstrated that the Weibull-Gamma distribution outperformed the competing models, making it a promising alternative for modeling real-life lifetime data.

PCN28 METHOD TO ESTIMATE IMPACT OF PROGNOSTIC FACTORS ON SURVIVAL OUTCOMES USING MAXIMUM LIKELIHOOD ESTIMATION

Bayesian and frequentist approach for the generalized log-logistic accelerated failure time model with applications to larynx-cancer patients

This paper develops mathematical and computational methods for fitting, by the method of maximum likelihood (ML), the two‐parameter, right‐truncated Weibull distribution (RTWD) to life‐test or survival data. Some important statistical properties of the RTWD are derived and ML estimating equations for the scale and shape parameters of the RTWD are developed. The ML equations are used to express the scale parameter as an analytic function of the shape parameter and to establish a computationally useful lower bound on the ML estimate of the shape parameter. This bound is a function only of the sample observations and the (known) truncation point T. The ML equations are reducible to a single nonlinear, transcendental equation in the shape parameter, and a computationally efficient algorithm is described for solving this equation. The practical use of the methods is illustrated in two numerical examples.