Empty-boxes test

A statistical test for verifying the hypothesis $H_0$ that an independent sample belongs to a given distribution. More explicitly, let $X_1,\ldots,X_n$ be an independent sample taken from a continuous distribution $F(x)$. The points $z_0=-\infty<z_1<\ldots<z_{N-1}<z_n=\infty$ are chosen so that $F(z_k)-F(z_{k-1})=1/N$, $k=1,\ldots,N$. The test is constructed on the basis of the statistic $\mu_0$ equal to the number of half-intervals $(z_{k-1},z_k]$ in which there is not a single observation $x_i$. This test has the following form: If $\mu_0\leq C$, then the hypothesis $H_0$ is accepted; if $\mu_0>C$, then $H_0$ is rejected. The constant $C$ is chosen from the condition that the error of the first kind, that is, the probability that $H_0$ is rejected while it is true, is equal to a given value. One may calculate the constant $C$ and estimate the power of the empty-boxes test for large $n$ and $N$ by using limit theorems for random distributions.

References

[1]	V.F. [V.F. Kolchin] Kolčin, B.A. [B.A. Sevast'yanov] Sevast'janov, V.P. [V.P. Chistyakov] Čistyakov, "Random allocations" , Winston (1978) (Translated from Russian)