Arc-Tangent Exponential Distribution with Applications to Weather and Chemical Data under Classical and Bayesian Approach

Abstract

This paper introduces the Arctan exponential distribution, a novel two-parameter trigonometric distribution. Various statistical properties of the distribution are examined, including hazard rate functions, cumulative hazard rate functions, mean deviation, reliability function, moments, conditional moments, incomplete moments, quantile function, entropy, Lorenz and Bonferroni curves, order statistics, and symmetry measures such as skewness and kurtosis. The parameters of the proposed distribution are estimated using the maximum likelihood estimation method, and a simulation study is conducted to assess its performance. Two real datasets are utilized to demonstrate the significance of the proposed distribution, showing that it performs comparably or better than well-known distributions. Furthermore, the suggested Arctan exponential distribution is employed within the Bayesian framework. The model's parameters are estimated and predicted using posterior samples generated through the application of the Markov Chain Monte Carlo (MCMC) technique. The application of the suggested model involves employing the Stan software in conjunction with the Hamiltonian Monte Carlo (HMC) algorithm and its adaptive variant known as the No-U-turn sampler (NUTS). A real dataset is utilized to showcase the methodology, and both numerical and graphical Bayesian analyses are performed, employing weakly informative priors. A posterior predictive check is also conducted to evaluate the model's predictability. The tools and methods employed in this study adhere to the Bayesian approach and are implemented using the R statistical programming language.