Estimation of <i>R = P(X &gt; Y)</i> for the New Exponentiated Arctan‐X Family Under Doubly Censored Data: Sensitivity Analysis With a Cancer Study

ABSTRACT This paper investigates the estimation of the stress‐strength reliability parameter , which measures the probability that one random variable exceeds another, for the exponentiated arctan‐X (EAT‐X) family of distributions–a new, flexible class of exponentiated distributions recently introduced in the literature–under doubly censored data. Three common practical scenarios are examined: the case where all parameters of both and are unknown (Case 1); the case where and share equal but unknown baseline parameters (Case 2); and the case where their baseline parameters are known (Case 3). For each scenario, maximum likelihood estimators of and the corresponding asymptotic confidence intervals are derived. For Case 3, a modified maximum likelihood estimator for yielding an explicit, closed‐form expression is also proposed. The performance of the proposed estimators is evaluated through an extensive Monte Carlo simulation study based on the exponentiated arctan‐exponential distribution, a specific member of the EAT‐X family. A sensitivity analysis is conducted to examine the impact of varying censoring patterns on estimation efficiency. Finally, the practical utility of the methodology is demonstrated through two real‐data examples: survival times of head and neck cancer patients (comparing radiotherapy alone versus combined therapy) and interfailure times of air conditioning systems from two Boeing aircraft.