Confidence Interval Pareto Dist Let $X_1,...,X_n$ be iid random variables from Pareto distribution with the following distribution $\theta a^{\theta} x^{-(\theta+1)}$, $x>a, \theta >1, a>0$
I have to find a $100(1-a)\%$ CI for $\theta$ based on chi square critical values. Is someone willing to walk me throug this problem?
 A: To find a confidence interval, one could use a suitable pivotal quantity. A pivotal quantity $T=T(\boldsymbol X,\theta)$ for $\theta$ is a function of the sample $\boldsymbol X$ and of $\theta$ such that the distribution of $T$ is independent of $\theta$.

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*If $a$ is known, then $T=2\theta\sum\limits_{j=1}^n \ln\left(\frac{X_j}{a}\right)\sim \chi^2_{2n}$ is a pivot for $\theta$.


*If $a$ is not known, then a pivot is $T=2\theta\sum\limits_{j=1}^n \ln\left(\frac{X_j}{X_{(1)}}\right)\sim \chi^2_{2(n-1)}$.
For details, see Distribution of $\sum_{j=1}^n\ln\left(\frac{X_{(j)}}{X_{(1)}}\right)$ when $X_i$'s are i.i.d Pareto variables.
These pivots are based on sufficient statistics, so confidence intervals obtained from these pivots are expected to be 'good' estimators of $\theta$.
For a $100(1-\alpha)\%$ confidence interval, one starts from the probability statement
$$P_{\theta}\left\{c\le T\le d \right\}=1-\alpha \quad\forall\,\theta>0\,,$$ where $c$ and $d$ are appropriate quantiles of a $\chi^2$ distribution.
This is rearranged to get a confidence interval $I$ for $\theta$:
$$P_{\theta}\left\{\theta\in I \right\}=1-\alpha \quad\forall\,\theta$$
