Resolved: "Confidence Interval for Pareto DistributionA random variable is..."

abitinomaq1

Answered question

2022-03-25

Confidence Interval for Pareto Distribution
A random variable is said to have probability density function

f_{X} (x) = \frac{α k^{α}}{x^{α + 1}}, α, k > 0; and; x > k .

Answer & Explanation

Tristatex9tw

Beginner2022-03-26Added 18 answers

Step 1
$\hat{k} > x$ if and only if $[X_{1} > x & \dots & X_{n} > x]$
and the probability of that is $(\Pr (X_{1} > x))^{n}$
$P r (X_{1} > x) = \int_{x}^{\infty} \frac{α k^{α}}{u^{α + 1}}, d u = {(\frac{k}{x})}^{α},$
so
$P r (\hat{k} > x) = {(\frac{k}{x})}^{n α} .$
Thus,
$P r (x_{1} < \hat{k} < x_{2}) = {(\frac{k}{x_{1}})}^{n α} - {(\frac{k}{x_{2}})}^{n α} .$
$P r (A k < \hat{k} < B k) = A^{- n α} - B^{- n α} .$
$P r (A < \frac{\hat{k}}{k} < B) = \dots$
$P r (\frac{1}{B} < \frac{k}{\hat{k}} < \frac{1}{A}) = \dots$
$P r (\frac{\hat{k}}{B} < k < \frac{\hat{k}}{A}) = \dots$
This gives you a confidence interval for k if $α$ is known. Since $α$ is not known, there is more work to do. (You have to choose A and B to get you the probablities that you want.)

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