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

Schilling, Mark F.

Ann. Stat. 11, 13-24 (1983).

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

Keywords:

nearest neighbor goodness of fit tests; infinite-dimensional approximation; Fredholm equation; quadratic functionals of weighted empirical distribution functions