Question

Recall the statistical set-up above. Recall that X_1, \ldots , X_ n \stackrel{iid}{\sim } \mathbf{P} are iid from an unknown distribution \mathbf{P}. For all 1 \leq i \leq n, Y_ i is a discrete random variable supported on \{ 1, \ldots , 5\} that denotes which bin contains the realization of X_ i.

Let \mathbf{P}_{\mu , \sigma ^2} = \mathcal{N}(\mu , \sigma ^2) and let (\widehat{\mu }, \widehat{\sigma }^2) denote the MLE for the statistical model (\mathbb {R}, \{ P_{\mu , \sigma ^2} \} _{\mu \in \mathbb {R}, \sigma ^2\in (0,\infty )}), i.e. Gaussian with unknown mean and unknown variance. For 1 \leq j \leq 5, let N_ j denote the frequency of j (i.e. number of times that j appears) in the data set Y_1, \ldots , Y_ n.

Define the \chi ^2 test statistic

T_ n = n \sum _{j = 1}^5 \frac{\left(\frac{N_ j}{n} - P_{\widehat{\mu }, \widehat{\sigma }^2}(Z \in A_ j)\right)^2}{P_{\widehat{\mu }, \widehat{\sigma }^2}(Z \in A_ j)}.

where Z \sim \mathcal{N}( \widehat{\mu }, \widehat{\sigma }^2 ). Then it holds that

T_ n \xrightarrow [n \to \infty ]{(d)} \chi ^2_\ell

for some constant \ell > 0.

What is \ell?

Hint: Use the result on the very last page of Lecture 15.

l=\quad