A random variable is said to have probability density function

abitinomaq1
2022-03-25
Answered

Confidence Interval for Pareto Distribution

A random variable is said to have probability density function

${f}_{X}\left(x\right)=\frac{\alpha {k}^{\alpha}}{{x}^{\alpha +1}},{\textstyle \phantom{\rule{1em}{0ex}}}\alpha ,k>0;\text{}\text{and}\text{};xk.$

A random variable is said to have probability density function

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Tristatex9tw

Answered 2022-03-26
Author has **18** answers

Step 1

$\hat{k}>x$ if and only if $[{X}_{1}>x\cdots {X}_{n}x]$

and the probability of that is $\left(\mathrm{Pr}\right({X}_{1}>x){)}^{n}$

$Pr\left({X}_{1}>x\right)={\int}_{x}^{\mathrm{\infty}}\frac{\alpha {k}^{\alpha}}{{u}^{\alpha +1}},du={\left(\frac{k}{x}\right)}^{\alpha},$

so

$Pr\left(\hat{k}>x\right)={\left(\frac{k}{x}\right)}^{n\alpha}.$

Thus,

$Pr\left({x}_{1}<\hat{k}<{x}_{2}\right)={\left(\frac{k}{{x}_{1}}\right)}^{n\alpha}-{\left(\frac{k}{{x}_{2}}\right)}^{n\alpha}.$

$Pr\left(Ak<\hat{k}<Bk\right)={A}^{-n\alpha}-{B}^{-n\alpha}.$

$Pr\left(A<\frac{\hat{k}}{k}<B\right)=\cdots$

$Pr\left(\frac{1}{B}<\frac{k}{\hat{k}}<\frac{1}{A}\right)=\cdots$

$Pr\left(\frac{\hat{k}}{B}<k<\frac{\hat{k}}{A}\right)=\cdots$

This gives you a confidence interval for k if $\alpha$ is known. Since $\alpha$ 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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