Cramér–von Mises criterion

Cramér–von Mises criterion is a type of statistical hypothesis testing method used in the field of statistics to determine the goodness of fit of a probability distribution. Named after the Swedish mathematician Harald Cramér and the Austrian mathematician Richard von Mises, this criterion is a measure of the discrepancy between the observed and the theoretical cumulative distribution functions (CDF) of a sample. It is widely used in various statistical applications, including econometrics, biostatistics, and psychometrics.

Overview

The Cramér–von Mises criterion is based on the squared differences between the empirical cumulative distribution function (ECDF) of the sample data and the CDF of a theoretical distribution. The test statistic, often denoted as \( \omega^2 \), is calculated by integrating the square of the difference between the ECDF and the theoretical CDF over the range of the data. The smaller the value of \( \omega^2 \), the better the fit of the theoretical distribution to the observed data.

Formulation

Given a sample \( X_1, X_2, \ldots, X_n \) of \( n \) independent and identically distributed observations, the empirical cumulative distribution function \( F_n(x) \) is defined as: \[ F_n(x) = \frac{1}{n} \sum_{i=1}^{n} I(X_i \le x) \] where \( I \) is the indicator function, equal to 1 if \( X_i \le x \) and 0 otherwise.

The Cramér–von Mises statistic is then given by: \[ \omega^2 = n \int_{-\infty}^{\infty} [F_n(x) - F(x)]^2 dF(x) \] where \( F(x) \) is the CDF of the theoretical distribution being tested.

Applications

The Cramér–von Mises criterion is applied in various fields to test the fit of a theoretical distribution to observed data. It is particularly useful when the form of the distribution is not known a priori, making it a non-parametric test. In econometrics, it helps in model selection and validation. In biostatistics, it is used to assess the fit of survival models. In psychometrics, it aids in evaluating the fit of item response theory models.

Comparison with Other Tests

The Cramér–von Mises criterion is often compared with other goodness-of-fit tests such as the Kolmogorov-Smirnov test and the Anderson-Darling test. While the Kolmogorov-Smirnov test focuses on the maximum difference between the ECDF and the CDF, the Cramér–von Mises criterion considers the overall difference. The Anderson-Darling test is a modification of the Cramér–von Mises criterion that gives more weight to the tails of the distribution.

Limitations

One limitation of the Cramér–von Mises criterion is its sensitivity to sample size. With a large sample size, even small deviations from the theoretical distribution can result in a significant \( \omega^2 \) value, leading to the rejection of the null hypothesis. Additionally, the test requires the selection of a specific theoretical distribution, which may not always be straightforward.

Conclusion

The Cramér–von Mises criterion is a powerful and versatile tool in statistical hypothesis testing for assessing the goodness of fit of a theoretical distribution to observed data. Its non-parametric nature makes it applicable across a wide range of disciplines and scenarios. However, like all statistical tests, it has its limitations and should be used judiciously, often in conjunction with other tests and methods.

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