Timeline for Convergence in Probability of the minimum
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Nov 12, 2013 at 19:15 | comment | added | whuber♦ | The right endpoint makes no difference to the limiting distribution of the minimum. If you prefer, use $f_\theta(x) = (x-\theta)\exp(\theta-x)$ when $x\ge \theta.$ | |
| Nov 12, 2013 at 18:30 | comment | added | bdeonovic | Hmm, is your example not excluded by the assumption that the support of $f$ is $[\theta,\infty)$? | |
| Nov 12, 2013 at 13:35 | comment | added | whuber♦ | The key ideas are all clearly displayed. There is a tiny (but interesting) error, though: your conclusion can be false when $x=\theta$. For instance, let $f_\theta(x)=(x-\theta)/2$ for $\theta\le x\le\theta+2$ and $f_\theta(x)=0$ otherwise. Then $F(\theta)=0,$ implying the limiting value of $F_{X_{[1]}}(x)$ is $0$, not $1$. This is interesting because it highlights the importance of invoking the definition of support somewhere in the proof and because it reveals the nature of the convergence is a tiny bit trickier than you might think. | |
| Nov 12, 2013 at 12:55 | history | asked | bdeonovic | CC BY-SA 3.0 |