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

Abstract

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 − 1 y ), 0 < x, y < ∞ , where θ > 0 is a single unknown parameter. We develop bounded-length confidence interval estimations for P θ ( 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 imple-mented extensive sets of computer simulations to empirically validate the theoretical properties.