# Cramer-Von Mises goodness-of-fit test¶

This method deals with the modelling of a probability distribution of a random vector . It seeks to verify the compatibility between a sample of data and a candidate probability distribution previous chosen. The Cramer-von-Mises Goodness-of-Fit test allows to answer this question in the one dimensional case , and with a continuous distribution. The current version is limited to the case of the Normal distribution.

Let us limit the case to . Thus we denote . This goodness-of-fit test is based on the distance between the cumulative distribution function of the sample (see ) and that of the candidate distribution, denoted . This distance is no longer the maximum deviation as in the Kolmogorov-Smirnov test but the distance squared and integrated over the entire variation domain of the distribution:

With a sample , the distance is estimated by:

The probability distribution of the distance is asymptotically known (i.e. as the size of the sample tends to infinity). If is sufficiently large, this means that for a probability and a candidate distribution type, one can calculate the threshold / critical value such that:

• if , we reject the candidate distribution with a risk of error ,

• if , the candidate distribution is considered acceptable.

Note that depends on the candidate distribution being tested; it is currently is limited to the case of the Normal distribution.

An important notion is the so-called -value of the test. This quantity is equal to the limit error probability under which the candidate distribution is rejected. Thus, the candidate distribution will be accepted if and only if is greater than the value desired by the user. Note that the higher , the more robust the decision.

Examples: