New Extension Odd Generalized Rayleigh-Nadarajah Haghighi Distribution with Application and Simulated Data

Generating distributions from families is a classification system in the world of probability. Instead of having hundreds of isolated distributions, we groupd them into families based on their shared mathematical properties. This not only organizes our knowledge but also provides powerful tools for analyzing data and building effective statistical models.

The extended generalized Rayleigh-Nadarajah Haghighi (EOGRNH) distribution was introduced, and its basic statistical features were thoroughly studied. These features include important functions such as the cumulative function (cdf ), probability density function (pdf ), surviva, l and hazard functions. We obtain statistical features including moments, skewness, kurtosis, incomplete moments, order statistics, moment-generating, Rényi entropy and quantile functions. Maximum-Likelihood Estimation (MLE) and Ordinary Least Squares Estimation (OLS) are two common methods for estimating model parameters.

Monte Carlo simulations with different sample sizes (N = 50, 100, 200, and 5000 replications) evaluated the estimator performance using absolute bias and man square error (MSE). When applied to real reliability data, failures of 50 mechanical components per 1000 h, the EOGRNHD outperformed both Gompertz Nadarajah Haghighi (GoNH) and Nadarajah Haghighi (NH) distributions in terms of flexibility and accuracy. The Bayesian information criterion, Anderson-Darling, Hannan information, modified Akaike criterion, Akaike information criterion, Kolmogorov-Smirnov, and Cramer-von-Mises statistics all show that this is especially true under the MLE. The results show that the EOGRNH distribution is useful for reliability analysis and fault modeling.

The EOGRNH was built according to th T-X family of distributions by combining a generator function of extension odd Nadarajah Haghighi with a baseline generalized Rayleigh distribution to ensure the preservation of the properties of the probability. The proposed model can accommodate many known distributions as special cases, thus providing a general mathematical framework for unifying different families of distributions.