Survival Analysis and Hazard of Log Logistic Distribution on Type I Censored Data Parametrically

Estimating the failure rate of the log-logistic distribution by smooth adaptive and bias-correction methods

The log-logistic distribution is more comprehensively applied in the area of survival and reliability engineering analysis for modeling the lifetime data practices of both human and electronic designs. The goal of this paper is to develop a generalization of the classical pattern log-logistic distribution, known as the neutrosophic log-logistic distribution (NLLD), to model various survival and reliability engineering data with indeterminacies. The developed distribution is especially useful for modeling indeterminate data that is roughly positively skewed. This paper discusses the developed NLLD’s main statistical properties such as neutrosophic survival function, neutrosophic hazard rate, neutrosophic moments, and neutrosophic mean time failure. Furthermore, the neutrosophic parameters are estimated using the well-known maximum likelihood (ML) estimation method in a neutrosophic environment. A simulation study is carried out to establish the achievement of the estimated neutrosophic parameters. As a final point, the proposed NLLD applications in the real world have been discussed with the help of real data. The real data illustrated that the efficiency of the proposed model as compared with the existing models.

The probabilistic models achieved considerable attention in the field of seismology due to their dynamic properties. In this study, an attempt has been made to quantify the uncertainty observed in the earthquake data of the Hindu Kush region by using probabilistic models. The conditional probabilities of major earthquake (Mw ≥ 5.5) occurrence in the Hindu Kush region are estimated by using the Weibull, lognormal, gamma, and log-logistic probability distributions. The maximum likelihood estimation method has been used to estimate the model parameters. Results based on the three model selection criteria suggest that the Weibull and log-logistic distributions provide the most appropriate fit, while the lognormal and gamma distributions appeared as the second and third appropriate fitted distributions to the elapsed time data of the Hindu Kush region. Using the most appropriate fitted Weibull and log-logistic distributions, it has been observed that the estimated earthquake occurrence probability of magnitude 5.5 or greater in the Hindu Kush region reaches 0.94–0.99 within next 3 to 6 (2021–2024) years since the last event occurred in 2018 in the study region. The conditional probability values are continuously increasing with increasing time interval and it approximately reaches to one in next 6 to 12 (2024–2030) years without any elapsed time, i.e., (T = 0) years. The estimated earthquake conditional probability curves for different elapsed time and time interval values are also presented to understand the earthquake genesis in the study region. The earthquake mean recurrence interval is 0.68, 0.81, 0.68, and 1.11 years by using the Weibull, lognormal, gamma, and log-logistic models respectively. According to the obtained results, the earthquake of magnitude Mw ≥ 5.5 occurrence probability in the study region is significantly high. The obtained results provide important information’s regarding the underlying physical mechanism of earthquake generation process in the Hindu Kush region.

The hazard function describes the unobserved rate at which failures occur. In this paper, we present a new inverted U-shape hazard function, developed for risk-events related to human activities. This hazard shape has received less attention in the literature relative to monotone hazard functions, likely due to the mathematical complexity of their regression models. We present a new survivor function, based on a logarithm transformation, and time displacement of Weibull's survivor function, that has an inverted U-shape hazard function similar to the log-logistic one. The advantage of this hazard function relative to the Weibull, and log-logistic hazard functions is in its unique curvature, which may fit better in some cases. We use simulation studies to present the new distribution characteristics in comparison to the Weibull, and the log-logistic distributions.

In this paper, we have derived and studied new probability distributions by extending the 2‐dimensional Rayleigh distribution (RD). First, we extend the RD to 3 dimensions and then generalize it to k dimensions for any positive integer k ≥ 3. The distributions are named the 3‐dimensional Rayleigh distribution (3‐DRD) and k‐dimensional Rayleigh distribution (k‐DRD), respectively. For both 3‐DRD and k‐DRD, detailed mathematical and statistical properties including derivations of the corresponding cumulative distribution, probability density, survival, and hazard functions, moments, moment generating functions, mode, skewness, kurtosis, and differential entropy are obtained in closed forms. Parameter estimation is done for both models using the maximum likelihood estimation method and some statistical properties of the estimator are discussed for each case. Interestingly, the commonly known Normal, Rayleigh, Maxwell–Boltzmann, chi‐square, gamma, and Erlang distributions are related to the newly developed extended RDs as special cases. For the 3‐DRD, plots of cumulative distribution, probability density, survival, and hazard functions are exhibited, a simulation study is carried out, and random samples are generated using the standard accept–reject (AR) algorithm to check the efficiency of the maximum likelihood estimates of the parameter. Moreover, the new 3‐DRD model is fitted to one simulated and three real datasets, revealing good performance compared to four existing Rayleigh‐based distributions. This study will contribute new knowledge to the field of applied statistics and probability, and the findings will be used as a basis for future research in the field.

It is known that various types of location privacy attacks can be carried out using a personalized transition matrix that is learned for each target user, or a population transition matrix that is common to all target users. However, since many users disclose only a small amount of location information in their daily lives, the training data can be extremely sparse. The aim of this paper is to clarify the risk of location privacy attacks in this realistic situation. To achieve this aim, we propose a learning method that uses tensor factorization (or matrix factorization) to accurately estimate personalized transition matrices (or a population transition matrix) from a small amount of training data. To avoid the difficulty in directly factorizing the personalized transition matrices (or population transition matrix), our learning method first factorizes a transition count tensor (or matrix), whose elements are the number of transition counts that the user has made, and then normalizes counts to probabilities. We focus on a localization attack, which derives an actual location of a user at a given time instant from an obfuscated trace, and compare our learning method with the maximum likelihood (ML) estimation method in both the personalized matrix mode and the population matrix mode. The experimental results using four real data sets show that the ML estimation method performs only as well as a random guess in many cases, while our learning method significantly outperforms the ML estimation method in all of the four data sets.

In this paper, we introduce the alternative methods to estimation for the new weibull-pareto distribution parameters. We discussed of point estimation and interval estimation for parameters of the new weibull-pareto distribution. We have also discussed the method of Maximum Likelihood estimation, the method of Least Squares estimation, the method of Weighted Least Squares estimation and the method of Maximum Product Spacing estimation. In addition, we discussed the raw moment of random variable X and the reliability functions (survival and hazard functions). Further, we compared between the results of the methods that have been discussed using Monte Carlo Simulation method and application study.

This study presents the Kumaraswamy Marshal–Olkin Flexible Weibull Extension (KMO-FWE) distribution, a newly developed probability model designed to extend existing lifetime distributions. This extension enhances the ability to model various hazard rate patterns with greater flexibility. The study explores the fundamental properties of this distribution, including the cumulative distribution function (CDF), probability density function (PDF), survival function, hazard function, moments, moment generating function, quantile function, and order statistics. To estimate the distribution’s parameters, several statistical methods are assessed, with a primary focus on Maximum Likelihood Estimation (MLE). Additionally, alternative estimation techniques such as Maximum Product Spacing (MPS), Least Squares (LS), Weighted Least Squares (WLS), and Percentile Estimation (PE) are considered. A Monte Carlo simulation study is conducted to evaluate the accuracy and efficiency of these estimation approaches under both complete data and Type-II censoring conditions. The results indicate that MLE is highly effective for large sample sizes but may introduce bias in smaller samples. In contrast, MPS provides stable parameter estimates under more complex conditions. While LS and WLS are computationally straightforward, their performance diminishes in the presence of censoring. Meanwhile, PE proves beneficial in reliability analysis based on percentile estimation. Overall, the findings highlight the KMO-FWE distribution as a versatile and reliable model, suitable for applications in reliability engineering, survival analysis, and biomedical research.

A new two-parameter family of discrete distributions.

Breast cancer is the most common cancer in women and the leading cause of cancer-related death in Indonesia. Analysis of survival data is important for improving the treatment and care of breast cancer patients. This study aims to estimate the parameters, find the survival function, and hazard function of breast cancer patients using a parametric method with an exponential distribution. Previous studies have shown that the Maximum Likelihood Estimation (MLE) method is suitable for estimating the survival function from exponential survival data by censoring. In this study, the exponential distribution was found to be the best for data on breast cancer patients from Surabaya Ontology Hospital. The estimated parameters are θ = 33.9157, and the survival function is calculated using The estimated hazard function for patient death or failure is 0.0295. The results of this study can contribute to the development of better treatment and care strategies for breast cancer patients. However, further research is needed because this study only used monthly time units.

In applications of survival analysis, the failure rate function may frequently present a unimodal shape. In such cases, the log-normal and log-logistic distributions are used. In this paper, we shall be concerned only with parametric forms, so a location-scale regression model based on the odd log-logistic Weibull distribution is proposed for modelling data with a decreasing, increasing, unimodal and bathtub failure rate function as an alternative to the log-Weibull regression model. For censored data, we consider a classic method to estimate the parameters of the proposed model. We derive the appropriate matrices for assessing local influences on the parameter estimates under different perturbation schemes and present some ways to assess global influences. Further, for different parameter settings, sample sizes and censoring percentages, various simulations are performed. In addition, the empirical distribution of some modified residuals is determined and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the new regression model applied to censored data. We analyse a real data set using the log-odd log-logistic Weibull regression model.

The study of failure times is essential in evaluating the reliability and longevity of components in engineering systems. Reliability analysis relies on probability models to predict the likelihood of failure within a given time frame. Length biased probability distributions arise in scenarios where the probability of sampling an object or event is directly proportional to its length, size or duration. These distributions are especially relevant in areas such as reliability theory, survival analysis, and epidemiology. The mathematical framework of length-biased distributions is based on transforming the probability density function of a baseline distribution by weighting it according to the length of the object. In this paper, we obtained a novel length biased probability distribution, the length biased quasi Garima distribution, characterized by two parameters. We explored and derived its statistical properties including moments, moment generating function, characteristic function, hazard rate function and survival function. Further we also studied about order statistics and entropy measures. We estimated distribution parameters using maximum likelihood estimation method. We presented the likelihood ratio test for comparing the goodness of fit of the length biased quasi Garima distribution over quasi Garima distribution. Additionally, we performed simulation study to evaluate the reliability of maximum likelihood estimators. Finally, length biased quasi Garima distribution is applied to a real-life data set related to failure time. the distribution proved to be a better fit than other established well-known distributions.

Objectives: To investigate the Modified Inverse Weibull Distribution (MIWD) to determine its effectiveness in modelling real customer transaction data, which is often skewed and exhibits heavy-tailed characteristics. Methods: Parameters of the model were estimated using the Maximum Likelihood Estimation (MLE) technique. The model's performance was evaluated using goodness-of-fit measures, such as the log-likelihood (ℓ), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Kolmogorov–Smirnov (K–S) statistic. The probability density function (PDF), cumulative distribution function (CDF), hazard function, and survival function were derived analytically and visualized with Python to assess model behavior and data alignment. Findings: The MIWD demonstrated a better fit for the customer transaction data than its sub-models and other competing distributions. The estimated probability density function (PDF), cumulative distribution function (CDF), hazard, and survival functions highlighted the model's adaptability in representing both skewness and heavy-tailed characteristics in actual transaction data. Novelty: The main contribution of this study is the empirical validation of the four-parameter MIWD using actual customer transaction data, demonstrating its superiority over established sub-models based on various quantitative criteria. The findings provide compelling evidence that the MIWD is a robust and adaptable alternative for modelling skewed, heavy-tailed transaction and lifetime data, with practical applications in customer behavior analysis, reliability engineering, and risk modelling. Keywords: Modified Inverse Weibull Distribution, Maximum Likelihood Estimation, Customer transaction data, Parameters, Rayleigh distribution

This paper discusses estimating the two scale parameters of Exponential-Rayleigh distribution for singly type one censored data which is one of the most important Rights censored data, using the maximum likelihood estimation method (MLEM) which is one of the most popular and widely used classic methods, based on an iterative procedure such as the Newton-Raphson to find estimated values for these two scale parameters by using real data for COVID-19 was taken from the Iraqi Ministry of Health and Environment, AL-Karkh General Hospital. The duration of the study was in the interval 4/5/2020 until 31/8/2020 equivalent to 120 days, where the number of patients who entered the (study) hospital with sample size is (n=785). The number of patients who died during the period of study was (m=88). And the number of patients who survived during the study period was (n-m=697), then utilized one of the most important non-parametric tests which is the Chi-square test to determine if the sample (data) corresponded with the Exponential-Rayleigh distribution (ER). then, after estimating the parameters of ER distribution for singly type-I censoring data, compute the survival function, hazard function, and probability density function.

Developing compound probability distributions is very important in the field of probability and statistics because there are different datasets from different fields with different features. These features range from high skewness, peakedness (kurtosis), bimodality, highly dispersed, and so on. Existing distributions might not easily fit well to these emerging data of interest. So, there is a need to develop more robust and flexible distributions that are positively skewed, negatively skewed, and bathup shape, to handle some of these features in the emerging data of interest. This paper, therefore, proposed a new four-parameter distribution called the Normal-Power{logistic} distribution. The proposed distribution was characterized by its density, distribution, survival, hazard, cumulative hazard, reversed hazard, and quantile functions. Properties such as the r-th moment, heavy tail property, stochastic ordering, mean inactive time were obtained. A useful transformation of the proposed distribution to normal distribution was shown to help generate its quantiles. The method of Maximum Likelihood Estimation (MLE) was used to estimate the model parameters. A simulation study was carried out to test the consistency of the maximum likelihood parameter estimates. The result of the simulation shows that the biases reduce as the sample size increases for different parameter values. The importance of the new distribution was proved empirically using a real-life dataset of gauge lengths of 10mm. The proposed distribution was compared with five other competing distributions, and the results show that the proposed Normal-Power{logistic} distribution (NPLD) performed favourably than the other five distributions using the AIC, CAIC, BIC, HQIC criteria.