Confidence Intervals for Independent Exponential Series Systems

Abstract

Suppose X₁, X₂, ···, X_n are independent identically distributed exponential random variables with parameter λ₁. Let Y₁, Y₂, ···, Y_m also be independent identically distributed exponential random variables with parameter λ₂, and assume that X's and Y's are independent. The problem is to estimate R(t) = e^{−(λ₁+λ₂)t}. A procedure for determining on exact (1 − α) level lower confidence bound for R(t) is presented. let U = min(Σⁿ _{t = 1}X_i, Σ^m _{i = 1}Y_i), and K = {largest i ≤ n: Σ^j _{i − 1}X_i ≤ U} + {Largest i ≤ m: Σ^j _i = 1 Y_i ≤ U}. Then given K = k, it is shown that U has a gamma distribution with parameters k and λ₁ + λ₂. Hence, a lower confidence bound for R(t) can be obtained. The suggested procedure is then compared with others presented in the literature.