Ordering properties of the smallest order statistics from generalized Birnbaum–Saunders models with associated random shocks

Let $$X_{1},\ldots , X_{n}$$ X 1 , … , X n be lifetimes of components with independent non-negative generalized Birnbaum–Saunders random variables with shape parameters $$\alpha _{i}$$ α i and scale parameters $$\beta _{i},~ i=1,\ldots ,n$$ β i , i = 1 , … , n , and $$I_{p_{1}},\ldots , I_{p_{n}}$$ I p 1 , … , I p n be independent Bernoulli random variables, independent of $$X_{i}$$ X i ’s, with $$E(I_{p_{i}})=p_{i},~i=1,\ldots ,n$$ E ( I p i ) = p i , i = 1 , … , n . These are associated with random shocks on $$X_{i}$$ X i ’s. Then, $$Y_{i}=I_{p_{i}}X_{i}, ~i=1,\ldots ,n,$$ Y i = I p i X i , i = 1 , … , n , correspond to the lifetimes when the random shock does not impact the components and zero when it does. In this paper, we discuss stochastic comparisons of the smallest order statistic arising from such random variables $$Y_{i},~i=1,\ldots ,n$$ Y i , i = 1 , … , n . When the matrix of parameters $$(h({\varvec{p}}), {\varvec{\beta }}^{\frac{1}{\nu }})$$ ( h ( p ) , β 1 ν ) or $$(h({\varvec{p}}), {\varvec{\frac{1}{\alpha }}})$$ ( h ( p ) , 1 α ) changes to another matrix of parameters in a certain mathematical sense, we study the usual stochastic order of the smallest order statistic in such a setup. Finally, we apply the established results to two special cases: classical Birnbaum–Saunders and logistic Birnbaum–Saunders distributions.