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).


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

  • $\begingroup$ 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 '15 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 '15 at 11:22
  • $\begingroup$ Got it cheers August! $\endgroup$ – Bobby May 26 '15 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 '15 at 11:40
  • $\begingroup$ What's your result ? $\endgroup$ – Augustin May 26 '15 at 11:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.