~56~ International Journal of Statistics and Applied Mathematics 2024; 9(2): 56-65 ISSN: 2456-1452 Maths 2024; 9(2): 56-65 © 2024 Stats & Maths www.mathsjournal.com Received: 18-01-2024 Accepted: 21-02-2024 Arun Kumar Chaudhary Department of Management Science, Nepal Commerce Campus, Tribhuvan University, Nepal Lal Babu Sah Telee Department of Management Science, Nepal Commerce Campus, Tribhuvan University, Nepal Vijay Kumar Department of Mathematics and statistics, DDU Gorakhpur University Gorakhpur, India Corresponding Author: Arun Kumar Chaudhary Department of Management Science, Nepal Commerce Campus, Tribhuvan University, Nepal Cauchy modified generalized exponential distribution: Estimation and Applications Arun Kumar Chaudhary, Lal Babu Sah Telee and Vijay Kumar DOI: https://doi.org/10.22271/maths.2024.v9.i2a.1682 Abstract We present a unique probability model in this study called the Cauchy Modified Generalized Exponential Distribution. This model is formulated by combining the Cauchy family of distributions with the Modified Generalized Exponential Distribution as the baseline distribution. Our objective is to employ this model in the analysis of lifetime data. We've crafted formulas for various statistical functions, like skewness, kurtosis, survival function, quantile function, hazard rate function, distribution function, and probability density function. Additionally, we've integrated visual depictions of the probability density and hazard rate curves. We gathered a dataset that included notable earthquakes (magnitude 7.0 and higher) that the USGS had documented between 1990 and 2018. Our proposed model's effectiveness was evaluated by applying it to a global dataset covering significant earthquakes of the same magnitude range. The model parameters were estimated using maximum likelihood estimation. Several statistical measures were applied in order to confirm the validity of the model, including the Bayesian Information Criterion, Corrected Akaike's Information Criterion, the Hannan-Quinn Information Criterion, and Akaike's Information Criterion. Additionally, Q-Q and P-P plots were employed for validation. We used the Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises tests to evaluate how well our model fit the data. These tests were conducted to determine the suitability of our model for analyzing the provided earthquake data. Our empirical results indicate that, compared to alternative lifetime distributions, our suggested distribution not only exhibits a better fit but also provides increased flexibility for analyzing lifetime data. This study advances our understanding of earthquake patterns and contributes to the ongoing efforts in seismic risk assessment and mitigation strategies. All numerical calculations were performed using the R programming language. Keywords: Cauchy family of distribution, earthquakes, failure rate function, maximum likelihood estimation, modified generalized exponential distribution 1. Introduction In recent decades, the exponential distribution has become a common baseline distribution for establishing new probability models. Numerous modifications of exponential distributions can be found in the literature. The Generalized Exponential Distribution (GED) created by (Gupta & Kundu, 2007) [17] is a statistical probability distribution that extends the traditional exponential distribution by incorporating an additional parameter in base line distribution to better capture the characteristics of real-world data. The inclusion of additional parameters results in the creation of new probability models. Typically, these adjusted models offer a more accurate representation of the data compared to conventional models. The GED introduces a shape parameter that changes the hazard function, allowing for a more flexible modeling approach than the usual exponential distribution, which assumes a constant hazard rate. This modification enables the distribution to better accommodate scenarios where the hazard rate varies over time, providing a more accurate representation of diverse phenomena in fields such as reliability engineering, biology, finance, life testing and survival analysis. While GED distribution is effective for analyzing datasets with a monotone (increasing/ decreasing) hazard function (HF), it cannot be applied to datasets with a unimodal or bathtub- shaped HF and upside-down bathtub shapes, such as those resembling the Weibull or gamma distributions. Several innovative probability models have been created through the