Suppose $z_n=\min\{x_1,\dots,x_n\}$ where $x_i$'s are i.i.d. according to CDF $F$ over $[0,1]$.
Is it true that $Var(z_n)>Var(z_{n+1})$? What conditions would I need to ensure this monotonic relationship?
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$\begingroup$ Obviously the minimum is monotonically non-increasing as n increases. But that does not indicate that its variance will be non-increasing, A large drop in $x_n$ relative the minimum of the first n-1 $x_i$s could cause the sample variance to increase. $\endgroup$– Michael R. ChernickCommented May 24, 2017 at 3:28
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1$\begingroup$ Hmm I am not talking about sample variance though. I'm defining $z_n$ as a random variable, and I'm wondering whether the population variance of $z_n$ could be monotone in $n$. $\endgroup$– user341296Commented May 24, 2017 at 4:00
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$\begingroup$ Yes I know that. I am trying to find a way to use the fact the $G_n(x)$= 1-(1-F(x))$^n$ where F(x) is the distribution for each X and $G_n(x)$ is the minimum of the Xs. $\endgroup$– Michael R. ChernickCommented May 24, 2017 at 4:05
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$\begingroup$ Thank you! Yes, I've been thinking about that too. Another fact to note is that the $E[z_n]=\frac{n}{n+1}E[z_{n+1}]+\frac{1}{n+1}E[x_{2:n+1}]$, where $x_{2:n+1}$ is the second order statistic out of $n+1$ draws. The same formula would apply to $E[z_n^2]$, and I was hoping to use this fact to show the result... $\endgroup$– user341296Commented May 24, 2017 at 4:16
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$\begingroup$ The same question was asked on the mathematics site but without an answer thus far. $\endgroup$– Michael R. ChernickCommented May 24, 2017 at 4:17
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