A novel discrete Poisson mixing probability distribution with two parameters has been developed by combining the Poisson distribution with the transmuted moment exponential distribution. It is possible to deduce several mathematical properties, such as the moment-generating function, ordinary moments, moments about the mean, skewness, kurtosis, and the dispersion index. The maximum likelihood estimation method is utilized to estimate the model’s parameters. A thorough simulation study is utilized to determine the behavior of the generated estimators. Estimating model parameters using a Bayesian methodology is another primary topic of this research. The behavior of Bayesian estimates is evaluated by first charting the trace, then generating 1,005,000 iterations of the Markov chain Monte Carlo method. In addition to this, we suggest a new count regression model that uses Poisson and negative binomial models in an alternating fashion. In conclusion, asymmetric datasets derived from various research areas are utilized for practical applications.

In this paper, the main aim is to define a statistical distribution that can be used to model COVID‐19 data in Mexico and Canada. Using the method of exponentiation on the gull alpha exponential distribution introduces a new distribution with three parameters called the exponentiated gull alpha power exponential (EGAPE) distribution. The distribution has the benefit of being able to represent monotonic and nonmonotonic failure rates, both of which are often seen in dependability issues. It is possible to determine the quantile function as well as the skewness, kurtosis, and order statistics of the suggested distribution. The approach of maximum likelihood is used in order to calculate the parameters of the model, and the RMSE and average bias are utilised in order to evaluate how successful the strategy is. In conclusion, the flexibility of the new distribution is demonstrated by modeling COVID‐19 data. From the practical application, we can conclude that the proposed model outperformed the competing models and therefore can be used as a better option for modeling COVID‐19 and other related datasets.

We propose a trigonometric generalizer/generator of distributions utilizing the quantile function of modified standard Cauchy distribution and construct a logistic‐based new G‐class disbursing cotangent function. Significant mathematical characteristics and special models are derived. New mathematical transformations and extended models are also proposed. A two‐parameter model logistic cotangent Weibull (LCW) is developed and discussed in detail. The beauty and importance of the proposed model are that its hazard rate exhibits all monotone and non‐monotone shapes while the density exhibits unimodal and bimodal (symmetrical, right‐skewed, and decreasing) shapes. For parametric estimation, the maximum likelihood approach is used, and simulation analysis is performed to ensure that the estimates are asymptotic. The importance of the proposed trigonometric generalizer, G class, and model is proved via two applications focused on survival and failure datasets whose results attested the distinct better fit, wider flexibility, and greater capability than existing and well‐known competing models. The authors thought that the suggested class and models would appeal to a broader audience of professionals working in reliability analysis, actuarial and financial sciences, and lifetime data and analysis.

Modeling real-world data is important in engineering, finance, environmental science, and medical science. Probability distributions are widely used to model such data. However, no single distribution can adequately fit all types of data, which creates a need for flexible distributions. In this study, a new extension of the Chris Jerry distribution (CJD), called the inverse Chris Jerry distribution (ICJD), is introduced using the inverse transformation technique. The probability density function of the ICJD can take right-skewed, reverse J, and inverted bathtub shapes. Its hazard rate function can also take upside-down and decreasing shapes. Several properties of the ICJD are derived, including the mode, harmonic mean, moments, inverse moments, conditional moments, incomplete moments, and order statistics. The model parameter is estimated using 15 estimation methods, and a simulation study is carried out to compare their performance. The simulation results show that the estimators perform consistently as the sample size increases, with the maximum likelihood method performing best. The modeling ability of the ICJD is demonstrated using two real datasets from environmental science and reliability engineering. The results show that the ICJD provides a better fit than the existing inverted distributions considered for both datasets, demonstrating its usefulness in modeling annual flood discharge rates and repair times of an airborne communication transceiver.

This paper introduces the Slash Q-Lindley (SQL) distribution, a new heavy-tailed model for positive data obtained as the ratio of a Q-Lindley variable to an independent power-distributed variable. The SQL inherits the flexibility of the baseline Q-Lindley while adding a tail parameter that controls polynomial decay. Its hazard rate tends to zero and, numerically, is decreasing or unimodal. Explicit expressions for the PDF, CDF, and hazard functions are derived using incomplete gamma and confluent hypergeometric functions. Key properties (moments, order statistics, stochastic ordering, entropy, and stress–strength reliability) are investigated. Parameter estimation is performed via maximum likelihood, Anderson–Darling, Cramér–von Mises, maximum product of spacings, ordinary and weighted least squares, and Bayesian MCMC methods. A simulation study evaluates finite-sample behavior, showing that the Bayesian estimator has the smallest overall rank, followed closely by the maximum product of spacings estimator, with maximum likelihood third. The practical advantage of the SQL is demonstrated on two real heavy-tailed datasets (flood discharges and coastal island areas), where it provides superior fits compared to several competing models based on multiple criteria (Akaike Information Criterion, Bayesian Information Criterion, Kolmogorov–Smirnov statistic, Anderson–Darling statistic, Cramér–von Mises statistic). The SQL offers a tractable and flexible alternative for modeling positive data with extreme values.

Bounded response variables, such as rates, proportions, and indices, frequently arise in the applied sciences and require regression models that respect the unit interval support, rather than relying on Gaussian regression. In this work, we propose a new regression model for continuous bounded data based on the unit Gumbel–logistic distribution as a flexible alternative to the existing models available in the literature. By reparameterizing the underlying distribution in terms of a location-type parameter directly linked to the covariate structure, our proposal enables the effect of explanatory variables on the response to be modeled in a natural and interpretable way. The proposed regression is derived from the reparameterization of the unit Gumbel–logistic distribution, which possesses a simple probability density function and a closed-form cumulative distribution function, which facilitates the derivation of its main properties. Parameter estimation is carried out using the maximum likelihood estimation method, and the finite-sample performance of the estimators is evaluated through an extensive Monte Carlo simulation study, comprising simulations for residual analysis and for the model. The proposed regression model is further compared with nine established competitors, including the beta, unit Kumaraswamy, unit Lindley, and unit log-log regression models, using two real-data applications. In both datasets, the unit Gumbel–logistic regression model achieved the highest log-likelihood and the lowest AIC, AICc, and BIC, outperforming its closest competitor, the beta regression model. Residual diagnostics and quantile-process analysis confirmed stable, well-behaved parameter estimates in both applications, supporting the unit Gumbel–logistic regression model as a valuable addition to the toolkit of regression models for bounded data.