Quadratic Forms in Normal Vectors

The chi-square distribution with k degrees of freedom is (by definition) the distribution of for i.i.d. N(O, 1)-distributed variables, The sum of squares is the squared norm of the standard normal vector. The following lemma gives a characterization of the distribution of the norm of a general zero-mean normal vector.

17.1 Lemma. If the vector X is Nk-distributed, then is distributed as, the eigenvalues.

Proof. There exists an orthogonal matrix such that. Then the vector is diag -distributed, which is the same as the distribution of the vector has the same distribution.

The distribution of a quadratic form of the type is complicated in general. However, in the case that every is either, it reduces to a chi-square distribution. If this is not naturally the case in an application, then a statistic is often transformed to achieve this desirable situation. The definition of the Pearson statistic illustrates this.

Pearson Statistic

Suppose that we observe a vector with the multinomial distribution corresponding to trials and classes having probabilities. The Pearson statistic for testing the null hypothesis is given by

We shall show that the sequence converges in distribution to a chi-square distribution if the null hypothesis is true. The practical relevance is that we can use the chi-square table to find critical values for the test.