On powerful distributional tests based on sample spacings (Q1083795)

Tests for uniformity based on sum functions of m-spacings are considered. The ratio of m to the sample size n plays a vital role in this context. If m increases at a rate slower than \(n^{1/2}\), then such tests are good for alternatives departing from uniformity by \(0((mn)^{-1/4})\), while for m increasing at a rate faster than \(n^{1/2}\), some other modifications are needed. In particular, when m/n\(\to p\), \(0<p<1\), then such a spacing test has adequate power against local or fixed alternatives if and only if p is irrational. Only in the case of fixed alternatives, the asymptotic distribution is normal.