Timeline for Is Var(sample min) decreasing in sample size?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 1, 2017 at 15:02 | comment | added | Yves | I think that the answer is no. The min can be replaced by max in the question; taking $F(x) := x^\alpha$ for $\alpha > 0$ we get $\text{Var}(x) = \alpha / (\alpha + 1)^2 / (\alpha+2)$. The cdf of $z_n$ is $G(z) := z^{n \alpha}$, with the same form. When $\alpha$ is close to $0$, the variance of $z_n$ does not decrease with $n$. As a guess, log-concavity of the distribution of $x_1$ could be a sufficient condition. | |
| May 24, 2017 at 23:31 | comment | added | user341296 | Could you provide a link to the math site? | |
| May 24, 2017 at 4:59 | history | edited | wolfies | CC BY-SA 3.0 |
Modified question to better reflect question
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| May 24, 2017 at 4:17 | comment | added | Michael R. Chernick | The same question was asked on the mathematics site but without an answer thus far. | |
| May 24, 2017 at 4:16 | comment | added | user341296 | 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... | |
| May 24, 2017 at 4:05 | comment | added | Michael R. Chernick | 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. | |
| May 24, 2017 at 4:00 | comment | added | user341296 | 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 $n$. | |
| May 24, 2017 at 3:28 | comment | added | Michael R. Chernick | 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. | |
| May 24, 2017 at 2:27 | history | asked | user341296 | CC BY-SA 3.0 |