## The Annals of Statistics

- Ann. Statist.
- Volume 11, Number 4 (1983), 1175-1189.

### Two Sample Rank Estimators of Optimal Nonparametric Score-Functions and Corresponding Adaptive Rank Statistics

Konrad Behnen, Georg Neuhaus, and Frits Ruymgaart

#### Abstract

In the general two-sample testing problem, $X_1, \cdots, X_m$ i.i.d. with continuous c.d.f. $F, Y_1, \cdots, Y_n$ i.i.d. with continuous c.d.f. $G$, and null hypothesis $H_0: F = G$ versus alternative $H_1: F \leq G, F \neq G$, we construct uniformly consistent and tractable rank estimators of the underlying optimal nonparametric score-function for a large subclass of (fixed) alternatives. Moreover, we prove asymptotic normality of the corresponding adaptive rank statistics under any fixed alternative $(F, G)$ from the same subclass, and compare the results with the corresponding results for the (local) asymptotically optimum linear rank statistic for $H_0$ versus $(F, G)$. In addition we prove some results on the estimation of a density and its derivative in the i.i.d. case if the support is [0,1], which are needed for a comparison argument in the case of rank estimators, but which may be of interest in other situations, too.

#### Article information

**Source**

Ann. Statist., Volume 11, Number 4 (1983), 1175-1189.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346330

**Digital Object Identifier**

doi:10.1214/aos/1176346330

**Mathematical Reviews number (MathSciNet)**

MR720262

**Zentralblatt MATH identifier**

0548.62029

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G10: Hypothesis testing

Secondary: 62E20: Asymptotic distribution theory 62G20: Asymptotic properties

**Keywords**

Adaptive rank tests two-sample problem stochastically-larger-alternatives rank estimators of score-functions Chernoff-Savage theorems with estimated score-functions kernel estimators density estimation

#### Citation

Behnen, Konrad; Neuhaus, Georg; Ruymgaart, Frits. Two Sample Rank Estimators of Optimal Nonparametric Score-Functions and Corresponding Adaptive Rank Statistics. Ann. Statist. 11 (1983), no. 4, 1175--1189. doi:10.1214/aos/1176346330. https://projecteuclid.org/euclid.aos/1176346330