Cramer-von Mises Test

A Cramer-von Mises Test is a hypothesis test that compares the goodness-of-fit of a cumulative distribution function to an empirical distribution function.

• AKA: Cramer-Von Mises Criterion.
• See: Hypothesis Test, Kolmogorov-Smirnov test, Anderson–Darling test, Shapiro–Wilk test.

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References

2016

• (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/Cram%C3%A9r%E2%80%93von_Mises_criterion Retrieved 2016-07-30
  • In statistics the Cramér–von Mises criterion is a criterion used for judging the goodness of fit of a cumulative distribution function [math]\displaystyle{ F^* }[/math] compared to a given empirical distribution function [math]\displaystyle{ F_n }[/math], 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

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

In one-sample applications [math]\displaystyle{ F^* }[/math] is the theoretical distribution and [math]\displaystyle{ F_n }[/math] 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.

1962

• (Anderson, 1962) ⇒ Anderson, T. W. (1962). On the distribution of the two-sample Cramer-von Mises criterion. The Annals of Mathematical Statistics, 1148-1159. http://projecteuclid.org/euclid.aoms/1177704477
  • The Cramer-von Mises [math]\displaystyle{ \omega^2 }[/math] criterion for testing that a sample, [math]\displaystyle{ x_1, \cdots,x_N }[/math], has been drawn from a specified continuous distribution [math]\displaystyle{ F(x) }[/math] is

[math]\displaystyle{ \omega^2=\in_{-\infty}^{\infty}[F_N(x)−F(x)]^2\,dF(x) \quad(1) }[/math]

where [math]\displaystyle{ F_N(x) }[/math] is the empirical distribution function of the sample; that is, [math]\displaystyle{ FN(x)=k/N }[/math] if exactly [math]\displaystyle{ k }[/math] observations are less than or equal to [math]\displaystyle{ x(k=0,1,⋯,N) }[/math]. If there is a second sample, [math]\displaystyle{ y_1,\cdots,y_M }[/math] a test of the hypothesis that the two samples come from the same (unspecified) continuous distribution can be based on the analogue of [math]\displaystyle{ N\omega^2 }[/math], namely

[math]\displaystyle{ T=[NM/(N+M)]\int^\infty_{−\infty}[F_N(x)−G_M(x)]^2dH_{N+M}(x), \quad (2) }[/math]

where [math]\displaystyle{ GM(x) }[/math] is the empirical distribution function of the second sample and [math]\displaystyle{ H_{N+M(x)} }[/math] is the empirical distribution function of the two samples together [that is, [math]\displaystyle{ (N+M)H_{N+M}(x)=NF_N(x)+MG_M(x)] }[/math]. The limiting distribution of [math]\displaystyle{ N\omega^2 }[/math] as [math]\displaystyle{ N\rightarrow \infty }[/math] has been tabulated [2], and it has been shown ([3], [4a], and [7]) that TT has the same limiting distribution as [math]\displaystyle{ N\rightarrow \infty }[/math], [math]\displaystyle{ M\rightarrow \infty }[/math], and [math]\displaystyle{ N/M\rightarrow \lambda }[/math], where [math]\displaystyle{ \lambda }[/math] is any finite positive constant. In this note we consider the distribution of [math]\displaystyle{ T }[/math] for small values of [math]\displaystyle{ N }[/math] and [math]\displaystyle{ M }[/math] and present tables to permit use of the criterion at some conventional significance levels for small values of [math]\displaystyle{ N }[/math] and [math]\displaystyle{ M }[/math] . The limiting distribution seems a surprisingly good approximation to the exact distribution for moderate sample sizes (corresponding to the same feature for [math]\displaystyle{ N\omega^2 }[/math] [6]). The accuracy of approximation is better than in the case of the two-sample Kolmogorov-Smirnov statistic studied by Hodges [4].

1928

• (Cramer, 1928) ⇒ Cramér, H. (1928). On the composition of elementary errors. Almqvist & Wiksells. DOI: http://dx.doi.org/10.1080/03461238.1928.10416862

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