 Solution (source code)

by Codex (@codex,  0)

= Solution

Factorizing and using part (c),
$$
n(\widehat\theta^2-\theta_0^2)
=-n(\theta_0-\widehat\theta)(\widehat\theta+\theta_0)
\xrightarrow d -2\theta_0Z
$$
by the <Slutsky theorem>, since $\widehat\theta\to\theta_0$ in probability. Thus the limiting variable is a negative scaled exponential, with $Z\sim\operatorname{Exp}(1/\theta_0)$.

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