In this study, we introduce a new class of length–biased distribution, the length–biased Weibull–Rayleigh (LBWR) distribution, and provide its properties such as the limit behavior, survival function, hazard rate function, \(r\)th moment, and moment generating function. Moreover, maximum likelihood estimation is used in the parameter estimation. The LBWR distribution was fitted to two hydrological datasets and its efficacy compared with Rayleigh, Weibull, Pareto, and Weibull–Rayleigh distributions, the results of which show that the novel distribution provides a better fit than the others.

This study proposes a new five-parameter distribution called the gamma-exponentiated Weibull–Poisson (GEWP) distribution. As an extension of the exponentiated Weibull–Poisson family, the GEWP distribution offers a more flexible tool for analyzing a wider variety of data due to its theoretically and practically advantageous properties. It encompasses established distributions like the exponential, Weibull, and exponentiated Weibull. The development of the GEWP distribution proposed in this paper is obtained by combining the gamma–exponentiated Weibull (GEW) and the exponentiated Weibull–Poisson (EWP) distributions. Therefore, it serves as an extension of both the GEW and EWP distributions. This makes the GEWP a viable alternative for describing the variability of occurrences, enabling analysis in situations where GEW and EWP may be limited. This paper analyzes the probability distribution functions and provides the survival and hazard rate functions, the sub-models, the moments, the quantiles, and the maximum likelihood estimation of the GEWP distribution. Then, the numerical experiments for the parameter estimation of GEWP distribution for some finite sample sizes are presented. Finally, the comparative study of GEWP distribution and its sub-models is investigated via the goodness of fit test with real datasets to illustrate its potentiality.

Weather derivatives are financial instruments influenced by temperature fluctuations, impacting industries such as agriculture, tourism, and energy. Accurate temperature modeling is essential for improving risk assessment and hedging strategies. This study evaluates the effectiveness of two forecasting hybrid approaches: the Fourier Ornstein-Uhlenbeck (OU) process, a widely used stochastic model, and the Fourier-Elman Recurrent Neural Network (ERNN), a hybrid neural network-based model. Daily temperature data from Chiang Mai, Thailand, spanning January 2005 to December 2021, were analyzed. The predictive performance of each model was assessed using root mean square error (RMSE). The results indicate the Fourier ERNN model (RMSE = 0.106) significantly outperforms the Fourier OU process (RMSE = 2.299), demonstrating superior accuracy in capturing both seasonal and stochastic variations in temperature dynamics. Thus, deep learning-based hybrid models provide a more effective framework for temperature forecasting. The proposed approach has potential applications in climate risk management, weather derivative pricing, and decision-making in climate-sensitive sectors.

In this study, the application of the Expectation Maximization (EM) algorithm for parameters estimation of a Zero – Inflated Gamma Distribution (ZIG) is of interest in which we were used several experiments in the simulation. Maximum likelihood estimation (MLE) and moment methods were estimated parameters to determine the best parameter estimation method for a ZIG distribution. By examining minimal mean square error (MSE) and average relative bias (AvRB) values of the test, it was found that the EM algorithm and MLE were the best parameter estimation method for ZIG distribution when the sample size was small, but, the EM algorithm was the best parameter estimation method when the sample size was large. The moment method was suitable for some parameter since its MSE and AvRB value was very high. In order to test for weather, the rainfall data in the summer season from Yom River in Northern Thailand follows a ZIG distribution using the three estimation methods, it was found that the estimated parameter values using MLE and EM algorithm were similar. Furthermore, we analyzed drought trends of all. We found that no tendency towards drought was detected.