## The Annals of Statistics

### Asymptotically Optimal Tests for Conditional Distributions

#### Abstract

Let $(X_1,Y_1),\cdots,(X_n,Y_n)$ be independent replicates of the random vector $(X,Y)\in \mathbb{R}^{d+m}$, where X is $\mathbb{R}^d$-valued and Y is $\mathbb{R}^m$-valued. We assume that the conditional distribution $P(Y\in\cdot|X=x)=Q_\vartheta(\cdot)$ of Y given X = x is a member of a parametric family, where the parameter space $\Theta$ is an open subset of $\mathbb{R}^k$ with $0\in\Theta$. Under suitable regularity conditions we establish upper bounds for the power functions of asymptotic level-$\infty$ tests for the problem $\vartheta=0$ against a sequence of contiguous alternatives, as well as asymptotically optimal tests which attain these bounds. Since the testing problem involves the joint density of (X,Y) as an infinite dimensional nuisance parameter, its solution is not standard. A Monte Carlo simulation exemplifies the influence of this nuisance parameter. As a main tool we establish local asymptotic normality (LAN) of certain Poisson point processes which approximately describe our initial sample.

#### Article information

Source
Ann. Statist., Volume 21, Number 1 (1993), 45-60.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349014

Digital Object Identifier
doi:10.1214/aos/1176349014

Mathematical Reviews number (MathSciNet)
MR1212165

Zentralblatt MATH identifier
0770.62014

JSTOR