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Title Two-Stage Sequential Interval Estimation of Functions of the Exponential Scale Parameters
Posted by Bernadette Tubo
Authors Tubo, Bernadette F. ; Polestico, Daisy Lou L. and Serquiña, Ruth P.
Publication date 2020
Journal Asia Pacific Journal of Science, Mathematics and Engineering
Volume 6
Issue 2
Pages 24-27
Publisher ISSN:2244-5471
Abstract Let X_1,X_2,…,X_n and Y_1,Y_2,…,Y_n be random samples from two exponential populations, with scale parameters, σ_1 and σ_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 θ=h(σ_1,σ_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 θ through the stopping rules m_d and N_d defined in equations (3) and (4), respectively. Under the assumption that σ_1>σ_L and σ_2>σ_G, where σ_L,σ_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<∞)=1. Moreover, we also provide the coverage probability of the interval estimates I_n guaranteeing asymptotic consistency for the parameter θ. Performances of the proposed two-stage methodology is illustrated via simulation using the R programming language on a parameter of the forms θ=(σ_1⁄σ_2 )^r and θ=|σ_1-σ_2 |^r for r>0. Simulation results show that the proposed two-stage procedure is asymptotically consistent.
Index terms / Keywords two-stage sequential procedure, confidence interval, stopping rule, exponential distribution