## The Annals of Statistics

- Ann. Statist.
- Volume 11, Number 1 (1983), 104-113.

### The Generalised Problem of the Nile: Robust Confidence Sets for Parametric Functions

G. A. Barnard and D. A. Sprott

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 11, Number 1 (1983), 104-113.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346061

**Digital Object Identifier**

doi:10.1214/aos/1176346061

**Mathematical Reviews number (MathSciNet)**

MR684868

**Zentralblatt MATH identifier**

0514.62004

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62A99: None of the above, but in this section

Secondary: 62F35: Robustness and adaptive procedures

**Keywords**

Ancillary statistics conditional inferences confidence intervals for parametric functions pivotal quantities robust

#### Citation

Barnard, G. A.; Sprott, D. A. The Generalised Problem of the Nile: Robust Confidence Sets for Parametric Functions. Ann. Statist. 11 (1983), no. 1, 104--113. doi:10.1214/aos/1176346061. https://projecteuclid.org/euclid.aos/1176346061