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So I'm given a Pareto distribution with parameters $\alpha >0$ and $k>0$ which is the form of \begin{equation*} f(x) = \frac{\alpha k^\alpha}{x^{\alpha+1}},~x > k. \end{equation*} I found the maximum likelihood estimator of $k$,

$\widehat{k}=$min$(x_{1}, x_{2},..., x_{n})$ and it's distribution which was, $f_{\widehat{k}}(x) = \frac{ank^{\alpha n}}{x^{\alpha n + 1}}$

1) How would I go about finding the distribution of $\frac{k}{\widehat{k}}$

2) Find a 95% confidence interval using that? I know I need a function of $k$ and the estimator $\widehat{k}$ and that's why I am asking for it in 1).

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I suggest you calculate the cumulative distribution function of $\frac{k}{\hat{k}}$.

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  • $\begingroup$ Yup then differentiate it to find the pdf right? But how would I set it up since $\frac{k}{\widehat{k}} = \frac{k}{min(x_{1}, ..., x_{n})}$ $\endgroup$
    – Bobby
    May 26, 2015 at 11:18
  • $\begingroup$ For $x\geq k>0$, $P\left(\frac{k}{\hat{k}}\leq x\right) = P\left( \hat{k}\geq \frac{k}{x}\right) = P\left(\min (x_1,\dots,x_n)\geq \frac{k}{x}\right) = \dots $ $\endgroup$
    – Augustin
    May 26, 2015 at 11:22
  • $\begingroup$ Got it cheers August! $\endgroup$
    – Bobby
    May 26, 2015 at 11:23
  • $\begingroup$ Just wondering what type of distribution for am I suppose to obtain, I get an unrecognisable distribution? $\endgroup$
    – Bobby
    May 26, 2015 at 11:40
  • $\begingroup$ What's your result ? $\endgroup$
    – Augustin
    May 26, 2015 at 11:44

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