## An infinite-dimensional approximation for nearest neighbor goodness of fit tests.(English)Zbl 0532.62076

The author considers a goodness of fit test for multi-dimensional densities based on quadratic functionals of weighted empirical distribution functions of the variables $$W_ i=\exp \{-ng(X_ i)V(R_ i)\}.$$ Here g is the density under $$H_ 0$$ completely specified and $$V(R_ i)$$ is the volume of the m-sphere with radius $$R_ i$$, the Euclidean distance of $$X_ i$$ from its nearest neighbour. The problem of determining the distribution is intractable for $$m>1$$, even for simple weight functions such as w(x)$$\equiv 1.$$
The author proposes an unorthodox approximation to the limiting process (when sample size tending to infinity) by letting the dimension $$m\to \infty$$ and using $$m=1$$, $$m=\infty$$ to approximate the situation when $$1<m<\infty$$. Monte Carlo studies for $$m=3,5$$ are carried out and comparisons are made and it appears that for the weight function identically equal to one cut off points for $$m=1$$, $$n=\infty$$ are liberal but for $$m=\infty$$, $$n=\infty$$ are conservative.
Reviewer: B.K.Kale

### MSC:

 62M99 Inference from stochastic processes 62G10 Nonparametric hypothesis testing 62H15 Hypothesis testing in multivariate analysis 62E20 Asymptotic distribution theory in statistics
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