Cramér-von Mises test

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2020 Mathematics Subject Classification: Primary: 62G10 [MSN][ZBL]

A non-parametric test for testing a hypothesis $H _{0}$ which states that independent and identically-distributed random variables $X _{1} \dots X _{n}$ have a given continuous distribution function $F (x)$. The Cramér–von Mises test is based on a statistic of the type

$$\omega _ n^{2} [ \Psi (F (x))] \ = \ \int\limits _ {- \infty} ^ {+ \infty} \left [ \sqrt n (F _{n} (x) - F (x)) \right ]^{2} \Psi (F (x)) \ dF (x),$$

where $F _{n} (x)$ is the empirical distribution function constructed from the sample $X _{1} \dots X _{n}$ and $\Psi (t)$ is a certain non-negative function defined on the interval $[0,\ 1]$ such that $\Psi (t)$, $t \Psi (t)$ and $t^{2} \Psi (t)$ are integrable on $[0,\ 1]$. Tests of this type, based on the "square metric" , were first considered by H. Cramér [C] and R. von Mises [M]. N.V. Smirnov proposed putting $\Psi (t) \equiv 1$, and showed that in that case, if the hypothesis $H _{0}$ is valid and $n \rightarrow \infty$, the statistic $\omega^{2} = \omega _ n^{2}$ has in the limit an "omega-squared" distribution, independent of the hypothetical distribution function $F (x)$. A statistical test for testing $H _{0}$ based on the statistic $\omega _ n^{2}$, is called an $\omega^{2}$( Cramér–von Mises–Smirnov) test, and the numerical value of $\omega _ n^{2}$ is found using the following representation:

$$\omega _ n^{2} \ = \ { \frac{1}{12n} } + \sum _ {j = 1} ^ n \left [ F (X _{(j)} ) - \frac{2j - 1}{2n} \right ]^{2}$$

where $X _{(1)} \leq \dots \leq X _{(n)}$ is the variational series based on the sample $X _{1} \dots X _{n}$. According to the $\omega^{2}$ test with significance level $\alpha$, the hypothesis $H _{0}$ is rejected whenever $\omega _ n^{2} \geq \omega _ \alpha^{2}$, where $\omega _ \alpha^{2}$ is the upper $\alpha$- quantile of the distribution of $\omega^{2}$, i.e. ${\mathsf P} \{ \omega^{2} < \omega _ \alpha^{2} \} = 1 - \alpha$. T.W. Anderson and D.A. Darling proposed a similarly constructed test, based on the statistic $\omega _ n^{2} [(1 - F (x))/F(x)]$( see [AD]).

References

 [C] H. Cramér, "Sannolikhetskalkylen och nåcgra av dess användningar" , Stockholm (1926) [M] R. von Mises, "Mathematical theory of probability and statistics" (1964) (Translated from German) [S] N.V. Smirnov, "On the $\omega^2$-distribution of von Mises" Mat. Sb. , 2 : 5 (1937) pp. 973–993 (In Russian) (French abstract) [BS] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) [AD] T.W. Anderson, D.A. Darling, "Asymptotic theory of certain "goodness-of-fit" criteria based on stochastic processes" Ann. of Math. Stat. , 23 (1952) pp. 193–212

Usually, the choice $\Psi (t) \equiv 1$ is simply called the Cramér–von Mises test in Western literature. However, Smirnov first proposed making this choice and rewrote the statistic in the distribution-free form above. The limit distribution of $\omega _ n^{2}$ is independent of $F$ whatever the choice of $\Psi$. (The term "square metric" refers to the expression $[ \sqrt n (F _{n} (x) - F (x))]^{2}$, not to some choice of $\Psi$.) Cramér actually considered the test with $\Psi (F (x)) \ dF (x)$ replaced by $dx$, while von Mises used $\lambda (x) \ dx$.