Hi all I have a question about a proof that I don't understand, enter image description here

My question is about the line after "We also have that....", I don't understand how $P(\hat{\theta_n} \geq \theta -\frac{x}{n})$ becomes $1-(1-\frac{x}{\theta*n})^n$

Could someone kindly explains?

  • 2
    $\begingroup$ Isn’t that simply $P(\hat{\theta}_n \geq ...) = 1 - P(\hat{\theta}_n \leq ...)$ and for the latter expression one used the equation above...? $\endgroup$ Dec 14, 2019 at 5:22
  • 1
    $\begingroup$ (and then pull out the $\theta$ in $\theta-x/n$...) $\endgroup$ Dec 14, 2019 at 5:24
  • 1
    $\begingroup$ please add the self-study tag and provide more details $\endgroup$
    – Xi'an
    Dec 14, 2019 at 9:33

1 Answer 1


The result follows since $$\begin{align*}P(\hat\theta_n\geq\theta-x/n)&=1-P(\hat\theta_n<\theta-x/n)\\&=1-P(\hat\theta_n\leq\theta-x/n)\\&=1-\Big(\frac{\theta-x/n}{\theta}\Big)^n\\&=1-(1-x/(\theta n))^n,\end{align*}$$ where the second equality follows by continuity of $\theta_n$ (it is continuous because its cdf is continuous), and where the third equality follows from the fact that since $x\in[0,\theta]$, $(\theta-x/n)/\theta\in[1-1/n,1]$, assuming $\theta\geq1$.


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