On fixed-accuracy and bounded accuracy confidence interval estimation problems in Fisher’s “Nile” example

Abstract

ABSTRACTFisher’s “Nile” example is a classic that 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 (i) fixed-width and (ii) fixed-accuracy confidence intervals for 𝜃 with a preassigned confidence coefficient. In problem (i), we develop a purely sequential estimation strategy along with its asymptotic properties. In problem (ii), we determine that a fixed-sample-size estimation strategy will suffice and yet the requisite sample size would have to be found. We have done that both exactly as well as approximately and we report that for all practical purposes the approximations nearly provide the exact sample size whether it is small, moderate, or large. The last problem we address is bounded-accuracy fixed-sample-size estimation of P𝜃{X>Y}. All theoretical properties are adequately validated by large-scale simulations.