Inverse Chen reliability function of P(X < Y < Z)

In models of reliability of stress-strength, the stress can fall under the influence of two strengths, and to varying degrees, the reliability of this factor can be expressed by the formula P(X<Y<Z). The paper aims to find the formula of reliability for two models: the first is the cascade model (1+1), where it consists of a main component and the second is a standby component, assuming that the component has two strength variables X and Z and is subjected to stress Y, the second is the cascade model (2+1), which consists of two main components and a third is standby component, assuming that both components have two strength variables X and Z, and the first component is subjected to stress Y1 and the second component to stress Y2, in both cases, the stress and robustness random variables will follow the inverse Chen distribution. The reliability function is estimated in two different methods maximum likelihood and percentile. Monte Carlo simulation is performed to compare the estimation results using the statistical criteria: meansquare error and mean absolute percentage error the results showed that the best estimator is ML.