Cramér–von Mises criterion

In statistics the Cramér–von Mises criterion is a criterion used for judging the goodness of fit of a cumulative distribution function $F^*$ compared to a given empirical distribution function $F_n$, or for comparing two empirical distributions. It is also used as a part of other algorithms, such as minimum distance estimation. It is defined as

$\omega^2 = \int_{-\infty}^{\infty} [F_n(x)-F^*(x)]^2\,\mathrm{d}F^*(x)$

In one-sample applications $F^*$ is the theoretical distribution and $F_n$ is the empirically observed distribution. Alternatively the two distributions can both be empirically estimated ones; this is called the two-sample case.

The criterion is named after Harald Cramér and Richard Edler von Mises who first proposed it in 1928–1930.[1] [2] The generalization to two samples is due to Anderson.[3]

The Cramér–von Mises test is an alternative to the Kolmogorov–Smirnov test.

Cramér–von Mises test (one sample)

Let $x_1,x_2,\cdots,x_n$ be the observed values, in increasing order. Then the statistic is[3]:1153[4]

$T = n \omega^2 = \frac{1}{12n} + \sum_{i=1}^n \left[ \frac{2i-1}{2n}-F(x_i) \right]^2.$

If this value is larger than the tabulated value, then the hypothesis that the data come from the distribution $F$ can be rejected.

Watson test

A modified version of the Cramér–von Mises test is the Watson test[5] which uses the statistic U2, where[4]

$U^2= T-n( \bar{F}-\tfrac{1}{2} )^2,$

where

$\bar{F}=\frac{1}{n} \sum F(x_i).$

Cramér–von Mises test (two samples)

Let $x_1,x_2,\cdots,x_N$ and $y_1,y_2,\cdots,y_M$ be the observed values in the first and second sample respectively, in increasing order. Let $r_1,r_2,\cdots,r_N$ be the ranks of the x's in the combined sample, and let $s_1,s_2,\cdots,s_M$ be the ranks of the y's in the combined sample. Anderson[3]:1149 shows that

$T = N \omega^2 = \frac{U}{N M (N+M)}-\frac{4 M N - 1}{6(M+N)}$

where U is defined as

$U = N \sum_{i=1}^N (r_i-i)^2 + M \sum_{j=1}^M (s_j-j)^2$

If the value of T is larger than the tabulated values,[3]:1154–1159 the hypothesis that the two samples come from the same distribution can be rejected. (Some books[specify] give critical values for U, which is more convenient, as it avoids the need to compute T via the expression above. The conclusion will be the same).

The above assumes there are no duplicates in the $x$, $y$, and $r$ sequences. So $x_i$ is unique, and its rank is $i$ in the sorted list $x_1,...x_N$. If there are duplicates, and $x_i$ through $x_j$ are a run of identical values in the sorted list, then one common approach is the midrank[6] method: assign each duplicate a "rank" of $(i+j)/2$. In the above equations, in the expressions $(r_i-i)^2$ and $(s_j-j)^2$, duplicates can modify all four variables $r_i$, $i$, $s_j$, and $j$.

References

1. ^ H. Cramér, On the composition of elementary errors, Scandinavian Actuarial Journal 1928
2. ^ R. E. von Mises, Wahrscheinlichkeit, Statistik und Wahrheit, Julius Springer 1928
3. ^ a b c d Anderson, TW (1962). "On the Distribution of the Two-Sample Cramer–von Mises Criterion" (PDF). The Annals of Mathematical Statistics (Institute of Mathematical Statistics) 33 (3): 1148–1159. doi:10.1214/aoms/1177704477. ISSN 0003-4851. Retrieved June 12, 2009.
4. ^ a b Pearson, E.S., Hartley, H.O. (1972) Biometrika Tables for Statisticians, Volume 2, CUP. ISBN 0-521-06937-8 (page 118 and Table 54)
5. ^ Watson, G.S. (1961) "Goodness-Of-Fit Tests on a Circle", Biometrika, 48 (1/2), 109-114 JSTOR 2333135
6. ^ Ruymgaart, F. H., (1980) "A unified approach to the asymptotic distribution theory of certain midrank statistics". In: Statistique non Parametrique Asymptotique, 1±18, J. P. Raoult (Ed.), Lecture Notes on Mathematics, No. 821, Springer, Berlin.
• M. A. Stephens (1986). "Tests Based on EDF Statistics". In D'Agostino, R.B. and Stephens, M.A. Goodness-of-Fit Techniques. New York: Marcel Dekker. ISBN 0-8247-7487-6.