The Generalised Problem of the Nile: Robust Confidence Sets for Parametric Functions
The Annals of Statistics · 1983 · Volume 11 · Issue 1
Abstract
The pivotal model is described and applied to the estimation of parametric functions $\phi(\theta)$. This leads to equations of the form $H(x; \theta) = G\{p(x, \theta)\}$. These can be solved directly or by the use of differential equations. Examples include various parametric functions $\phi(\theta, \sigma)$ in a general location-scale distribution $f(p), p = (x - \theta)/\sigma$ and in two location-scale distributions. The latter case includes the ratio of the two scale parameters $\sigma_1/\sigma_2$, the difference and ratio of the two location parameters $\theta_1 - \theta_2$ and the common location $\theta$ when $\theta_1 = \theta_2 = \theta$. The use of the resulting pivotals to make inferences is discussed along with their relation to examples of non-uniqueness occurring in the literature.
Keywords
62A99, 62F35, Ancillary statistics, conditional inferences, confidence intervals for parametric functions, pivotal quantities, robust
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