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

ABSTRACT

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;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 P_𝜃{X>Y}. All theoretical properties are adequately validated by large-scale simulations.

KEYWORDS:

• Bounded-accuracy
• confidence coefficient
• confidence intervals
• fixed-accuracy
• fixed-sample-size estimation
• fixed-width
• maximum likelihood estimator
• “Nile” example
• purely sequential sampling
• sample size determination

MATHEMATICS SUBJECT CLASSIFICATION:

• 62L12
• 62F25
• 62E17
• 62B10

Acknowledgments

Yan Zhuang gave her debut invited presentation based on a preliminary version of this paper at the 8th International Workshop in Applied Probability (IWAP), June 20–23, 2016, held in Toronto, Canada. We are grateful for the encouraging feedback received after that presentation during IWAP. We also thank an Associate Editor and the reviewers for their helpful suggestions.