Mean and variance of the median estimator

Asked Dec 16 '18 at 06:14

Active Dec 16 '18 at 08:01

Viewed 52 times

We are given an i.i.d sample of size n, $X_{1},...,X_{2}$. We also know that for all i, $EX_{i}=\mu,Var{X_{i}}=\sigma^{2}$.

Let us define $\hat{X}_{median}$ as the median of these samples: $\hat{X}_{median}(X_{1},...,X_{n})=median(X_{1},...,X_{n})$ .

Is there any known way to explicitly express the mean and variance of the median estimator ($E(\hat{X}_{median}), Var(\hat{X}_{median})$) In terms of $\mu,\sigma$?

Thank you!

edited Dec 16 '18 at 08:01

asked Dec 16 '18 at 06:14

Or Raveh

What distribution does $X_i$ follow? – user158565 Dec 16 '18 at 06:18
In the first line, I think you forgot the name of the variable in Var. It should be $Var(X_{i}) = \sigma^2$ – canovasjm Dec 16 '18 at 07:54
@user158565 The question is whether we can say something without knowing the distribution. – Or Raveh Dec 16 '18 at 08:02
This question has been asked before, and the answer is "no": https://stats.stackexchange.com/questions/41557/ – Gordon Smyth Dec 16 '18 at 09:06
You might find https://stats.stackexchange.com/questions/45124/central-limit-theorem-for-sample-medians/86804#86804 provides constructive approaches to the underlying problem. – whuber Dec 16 '18 at 21:26

Mean and variance of the median estimator

0 Answers0