Kassandra Ross

## Answered question

2022-06-24

I`m trying to find an answer, but i have some problems, help.
Let ${\mathbb{P}}_{\theta }=U\left[0,\theta \right]$.
For $h,{\theta }_{0}>0$ and $Z\sim \mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{1}{{\theta }_{0}}\right)$ I have to show that:
$\frac{\mathrm{d}{\mathbb{P}}_{{\theta }_{0}-h/n}^{n}}{\mathrm{d}{\mathbb{P}}_{{\theta }_{0}}^{n}}\stackrel{d,{\mathbb{P}}_{{\theta }_{0}}^{n}}{⟶}{e}^{\frac{h}{{\theta }_{0}}}{\mathbb{1}}_{\left\{Z\ge h\right\}}$
I already proved that for ${Z}_{n}=n\left({\theta }_{0}-max\left\{{X}_{1},\dots ,{X}_{n}\right\}\right)$ with ${X}_{1},\dots {X}_{n}\sim {\mathbb{P}}_{{\theta }_{0}}$ holds ${Z}_{n}\stackrel{D}{\to }Z$ and this task seems like I have to prove that the pdf is converging too. I'm not sure which technical steps I need to show this and I'm not sure which kind of convergence is meant by $d,{\mathbb{P}}_{{\theta }_{0}}^{n}$.

### Answer & Explanation

Govorei9b

Beginner2022-06-25Added 21 answers

The convergence follows with Radon-Nikodym, Slutzky and continuous mapping theorem.

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