Purely Sequential and Two-Stage Bounded-Length Confidence Interval Estimation Problems in Fisher’s “Nile” Example

Fisher’s “Nile” example is a classic which involves a bivariate random variable (X, Y) having a joint probability density function given by f(x, y; θ) = exp(−θx − θ−1y), 0 < x, y < ∞, where θ > 0 is a single unknown parameter. We develop bounded-length confidence interval estimations for Pθ(X > a) with a preassigned confidence coefficient using both purely sequential and two-stage methodologies. We show: (i) Both methodologies enjoy asymptotic first-order efficiency and asymptotic consistency properties; (ii) Both methodologies enjoy second-order efficiency properties. After presenting substantial theoretical investigations, we have also implemented extensive sets of computer simulations to empirically validate the theoretical properties.