On fixed-accuracy and bounded accuracy confidence interval estimation problems in Fisher's "Nile" example

• Published in: Sequential analysis Vol. 35; no. 4; pp. 516 - 535
• Main Authors: Mukhopadhyay, Nitis, Zhuang, Yan
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 01.10.2016; Taylor & Francis Ltd
• Subjects: "Nile" example; Accuracy; Approximation; Asymptotic properties; Bounded-accuracy; Confidence; confidence coefficient; Confidence intervals; Economic models; Estimating techniques; fixed-accuracy; fixed-sample-size estimation; fixed-width; Mathematical analysis; maximum likelihood estimator; Monte Carlo simulation; Probability density functions; purely sequential sampling; Sample size; sample size determination; Simulation; Strategy
• ISSN: 0747-4946; 1532-4176
• DOI: 10.1080/07474946.2016.1238264

Fisher'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; ) = exp(− x− −1 y), 0<x,y<∞, where >0 is a single unknown parameter. We develop (i) fixed-width and (ii) fixed-accuracy confidence intervals for 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 P {X>Y}. All theoretical properties are adequately validated by large-scale simulations.