Introduction to GOF Tests

All of these tests can only detect deviations from CDF $F$. This means when accepting the Null Hypothesis, there is quite a large Type 2 Error. The data could for example come from a distribution that is really similar to a normal distribution.

We can do tests for a particular distribution with the qq GOF Test for Particular Distribution. However sometimes we only want to know whether the datas random mechanism belongs to some class of distributions (for example whether it is normally distributed). In this case we can use the qq GOF Test for Class of Distributions.

In discrete cases we can use the Chi-Squared GOF Test for finitely many values.

This can also be applied when

• Class Condition is not met
• infinitely many values are present
• continous distribution is used

by grouping (Binning) the values: $l_1=\left(-\infty ,a_1\right],l_2=\left(a_1,a_2\right],\dots ,I_s=\left(a_{s-1},\infty \right)\left(a_1<a_2<\dots <a_{s-1}\right)$ For large $n$ one can calculate the relative interval frequency which will coincide with $p_j$. Then use GOF test above.