## The Annals of Statistics

- Ann. Statist.
- Volume 11, Number 2 (1983), 588-599.

### Galton's Test as a Linear Rank Test with Estimated Scores and Its Local Asymptotic Efficiency

Konrad Behnen and Georg Neuhaus

#### Abstract

In the general two-sample testing problem $F = G$ versus $F \leq G, F \neq G$ the shift-model score function $f' \circ F^{-1}/f \circ F^{-1}$ has to be replaced by the nonparametric score function $b = \bar{f} - \bar{g}$, where $\bar{f} = d(F \circ H^{-1})/dx, \bar{g} = d(G \circ H^{-1})/dx, H = (mF + nG)/(m + n)$, and adaption of linear rank tests should be based on rank estimators of $b$. We consider an easy but rough and inconsistent rank estimator of $b$. The resulting rank test turns out to be a generalization of Galton's test. A formula for local asymptotic power under arbitrary local alternatives is derived which allows for comparison of Galton's test with every linear rank test. For various types of alternatives the Galton test is compared with the optimal linear rank test and with the Wilcoxon test. In order to get an impression of the validity of extrapolation to finite sample sizes, we included a Monte Carlo study under the same types of fixed alternatives.

#### Article information

**Source**

Ann. Statist., Volume 11, Number 2 (1983), 588-599.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346164

**Digital Object Identifier**

doi:10.1214/aos/1176346164

**Mathematical Reviews number (MathSciNet)**

MR696070

**Zentralblatt MATH identifier**

0531.62042

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G10: Hypothesis testing

Secondary: 62G20: Asymptotic properties 62E25

**Keywords**

Galton test adaptive rank tests two-sample problem estimated scores power simulation and extrapolation stochastically-larger-alternatives

#### Citation

Behnen, Konrad; Neuhaus, Georg. Galton's Test as a Linear Rank Test with Estimated Scores and Its Local Asymptotic Efficiency. Ann. Statist. 11 (1983), no. 2, 588--599. doi:10.1214/aos/1176346164. https://projecteuclid.org/euclid.aos/1176346164