Theory and Methods of Statistics

7.4 Frequency ${\chi }^2$

An important class of hypothesis testing problems arises in the context of the multinomial distribution Multi(n;π₁,…,π_m), where π_j ≥ 0 with $\sum {\pi }_j=1$ are the probabilities of a random sample belonging to the classes (1,…,m). The data consist of a random sample (X₁,…,X_n) where the X_i’s are m-dim iid rv’s with

$\begin{array}{l}\hfill f({\mathit{e}}_j,\mathit{\pi })=P({\mathit{X}}_i={\mathit{e}}_j)={\pi }_j,j=1,\dots ,m,\end{array}$

where e_j is the m-dim vector with 1 at the jth coordinate and 0 for the rest of the coordinates, ...