We present a new two parameter Burr XII distribution. The new density can be right skewed with no peak, unimodal-right skewed, left skewed and symmetric. The new failure rate can be decreasing, unimodal and increasing. Properties related to the new model are derived. Simple type copula-based construction is presented for deriving some new bivariate and multivariate type distributions. The maximum likelihood estimation, Anderson Darling estimation, right tail Anderson Darling estimation and left tail Anderson Darling estimation methods are used to estimate the model parameters. A new data set is analyzed for comparing estimations methods and the competitive models.

After defining a new log-logistic model and studying its properties, some new bivariate type versions using “Farlie-Gumbel-Morgenstern Copula”, “modified Farlie-Gumbel-Morgenstern Copula”, “Clayton Copula”, and “Renyi’s entropy Copula” are derived. Then, using the Bagdonavicius-Nikulin goodness-of-fit (BN-GOF) test for validation, we proposed a goodness-of-fit test for a new log-logistic model. The modified test is applied for the “right censored” real dataset of survival times. All elements of the modified test are explicitly derived and given. Three real data applications are presented for measuring the flexibility and the importance of the new model under the uncensored scheme. Two other real datasets are analyzed for censored validation.

In this paper, we first study a new two parameter lifetime distribution. This distribution includes “monotone” and “non-monotone” hazard rate functions which are useful in lifetime data analysis and reliability. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Renyi entropy, δ-entropy, order statistics and probability weighted moments are derived. Non-Bayesian estimation methods such as the maximum likelihood, Cramer-Von-Mises, percentile estimation, and L-moments are used for estimating the model parameters. The importance and flexibility of the new distribution are illustrated by means of two applications to real data sets. Using the approach of the Bagdonavicius–Nikulin goodness-of-fit test for the right censored validation, we then propose and apply a modified chi-square goodness-of-fit test for the Burr X Weibull model. The modified goodness-of-fit statistics test is applied for the right censored real data set. Based on the censored maximum likelihood estimators on initial data, the modified goodness-of-fit test recovers the loss in information while the grouped data follows the chi-square distribution. The elements of the modified criteria tests are derived. A real data application is for validation under the uncensored scheme.

We introduce a new extension of the reciprocal Exponential distribution for modeling the extreme values. We used the Morgenstern family and the clayton copula for deriving many bivariate and multivariate extensions of the new model. Some of its properties are derived. We assessed the performance of the maximum likelihood estimators (MLEs) via a graphical simulation study. The assessment was based on the sample size. The new reciprocal model is employed for modeling the skewed and the symmetric real data sets. The new reciprocal model is better than some other important competitive models in statistical modeling.

In this paper, the authors introduce a new three-parameter lifetime probability distribution known as the Marshall-Olkin-generated log-logistic (LL) distribution. They thoroughly examine and describe this distribution, providing insights into its characteristics and its suitability for various applications. This newly constructed distribution's density function exhibits characteristics of both symmetry and right-skewness, providing modelling flexibility across a range of datasets. Because of its skewness coefficient, which can take both positive and negative values, a wide range of data asymmetries can be represented. The Marshall-Olkin generated LL distribution's corresponding hazard rate displays a variety of characteristics, including monotonic increase, increasing-constant, constant, upside-down, and monotonic drop. Because of its adaptability, the distribution can successfully capture various risk or failure rate patterns across time. Using a number of techniques, the researchers expand this distribution to the bivariate domain. Its utility in modelling multivariate lifetime data and inter-variable relationships is improved by these extensions. The researchers use the maximum likelihood method to estimate the parameters of the distribution, which ensures reliable and effective parameter estimation from observed data. They carry out an extensive simulation research to analyse biases and mean squared errors in a range of scenarios and sample sizes in order to evaluate the finite behaviour of the maximum likelihood estimators. In real-life and reliability applications, this meticulous methodology aids in evaluating the estimators' precision and dependability. Because it may offer a comprehensive and nuanced knowledge of high financial risks, the Peaks Over a Random Threshold Value-at-Risk (PORT-VaR) study is crucial for evaluating Norwegian fire insurance claims. This financial analysis is given extra consideration.

The paper introduces a new reliability Burr Pareto type-II model, showcasing its versatility and effectiveness in engineering applications, particularly in analyzing the failure and service times of aircraft windshields. The BUPII model's application in failure analysis offers insights into the probabilistic behavior of windshield failures, aiding in risk prediction and management. Similarly, its extension to service time analysis demonstrates its utility in optimizing maintenance schedules and operational efficiency. Moreover, the paper conducts a rigorous mean-of-order P analysis under both failure and service time datasets, validating the new model's reliability assessment capabilities. Furthermore, employing the peaks over random threshold value at risk analysis highlights the model's practical relevance in quantifying financial risks associated with extreme events. Overall, the novel probability distribution emerges as a valuable tool for engineers and researchers involved in reliability and risk analysis, promising advancements in understanding and managing the reliability of engineering systems. Future research could explore broader applications and refined methodologies to further enhance predictive capabilities and decision-making support.