Cramer-Von Mises goodness-of-fit testΒΆ
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.
API:
Examples: