Pearson fitting criterion or the χ2(chi-squared) criterion is used to test the hypothesis stating that the empirical distribution is consistent with the expected F(x) theoretical distribution with a large sample size (n ≥ 100). The criterion can be applied to any types of the F(x) function even with unknown values of function parameters on analyzing the mechanical test results. This shows its universality.
Using the χ2 criterion enables the user to break the sample range into intervals and define the number of observations (frequency) for each of the intervals. The intervals are selected of the equal length to facilitate estimation of the distribution parameters. The number of intervals depends on the sample size.
The disadvantage of the Pearson criterion is the loss of part of the initial information associated with grouping the observation results into intervals and combining separate intervals with a small number of observations. Thus, check the distributions consistency by the χ2 criterion using other criteria. This is especially the case when the sample size is small (n ≈ 100).
The following statistics is introduced to check the criterion:
Where:
. Estimated probability of falling within the i-th interval.
. Corresponding empirical value.
ni. The number of sample elements from the i-th interval.
This value is random in its turn (because X is random) and should follow the χ2 distribution.
If the obtained statistics exceeds quantile of the distribution rule χ2 of the specified significance level α with (k - 1) or (k - p - 1) degrees of freedom, where k is the number of observation or the number of intervals (for an interval variation series), and p — is the number of estimated parameters of the distribution rule, the hypothesis H0 is rejected. Otherwise the hypothesis is accepted for the specified significance level α.
See also: