Consider the multinomial version of the PCGF test given in (8 23) with (p j ) replaced by ( widehat p j N 1 sum i F j left( mathbf x i , widehat boldsymbol theta ight) ) Show that PCGF can be expressed as CGF in (8 27) with ( widehat mathbf V operatorname Diag left N hat p i ight ) Conclude that in...

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Consider the multinomial version of the PCGF test given in (8.23) with (p_{j}) replaced by (widehat{p}_{j}=N^{-1} sum_{i}

Question:

Consider the multinomial version of the PCGF test given in (8.23) with $p_{j}$ replaced by $\widehat{p}_{j}=N^{-1} \sum_{i} F_{j}\left(\mathbf{x}_{i}, \widehat{\boldsymbol{\theta}}\right)$. Show that PCGF can be expressed as CGF in (8.27) with $\widehat{\mathbf{V}}=\operatorname{Diag}\left[N \hat{p}_{i}\right]$. [Conclude that in the multinomial case Andrew's test statistic simplifies to Pearson's statistic.] image text in transcribed