Abstract |
Let X_1,X_2,â¦,X_n and Y_1,Y_2,â¦,Y_n be random samples from two exponential populations, with scale parameters, Ï_1 and Ï_2, respectively. This paper considers a two-stage sequential procedure to construct fixed-width confidence intervals I_n for functions of the exponential scale parameters of the form Î¸=h(Ï_1,Ï_2 ), where h is a real-valued, three-times continuously differential function defined on R_+^2. A two-stage sequential procedure is proposed for the estimation of Î¸ through the stopping rules m_d and N_d defined in equations (3) and (4), respectively. Under the assumption that Ï_1>Ï_L and Ï_2>Ï_G, where Ï_L,Ï_G>0 are lower bounds known to the experimenter from past experiences, we have shown that the stopping rule N_d is a good estimate of the optimal sample size n^*defined in (2). We have shown that the proposed two-stage sequential procedure will eventually stop with probability 1, that is, P(N_d<â)=1. Moreover, we also provide the coverage probability of the interval estimates I_n guaranteeing asymptotic consistency for the parameter Î¸. Performances of the proposed two-stage methodology is illustrated via simulation using the R programming language on a parameter of the forms Î¸=(Ï_1âÏ_2 )^r and Î¸=|Ï_1-Ï_2 |^r for r>0. Simulation results show that the proposed two-stage procedure is asymptotically consistent. |