The method of moments estimator of median

Question

In some textbook, it is written that the method of moment estimator can be applied even in estimating median. However, I can't come up with the idea.

In many cases the median is estimated by $x_{n/2}$ or $(x_{(n-1)/2} + x_{(n+1)/2})$ depending on whether n is even or odd. I think unfortunately it is not method of moment estimator.

Does someone know the way to get the estimator? To simplify the discussion, it is ok to think cumulative distribution function is continuous and strictly increasing function.

Answer 1

This isn't exactly the method of moments, but you can think of the median as the $n/2$ order statistic. Then you can use convergence of sample quantiles by way of order statistics. In particular, for the $p$-th quantile with a sample size of $n$, a standard Delta method/CLT result is $$ \sqrt{n} \{X_{([np])}-F^{-1}(p) \} \rightarrow_d N\left( 0, \dfrac{p(1-p)}{[f(F^{-1}(p))]^2} \right). $$ Taking $p=1/2$, this says the $n/2$ order stat converges in distribution to the median.

Just googling around, for example: http://www.math.ntu.edu.tw/~hchen/teaching/LargeSample/notes/noteorder.pdf

Answer 2

You can use the median $\tilde{x}$ in place of the mean: $\bar{x}$ for an estimation of the calculation of any moment $k$ >1:

$$ m_{(k)} = \frac 1 n \sum_{i=0}^n (x_{(i)} - \tilde{x})^k$$

The advantage is the increased robustness when outliers are present.

As with any estimation method, use common sense and visualizations to confirm the median is the better centroid value than the mean.

Any value between the median and mean can be used as $\tilde{x}$ with bounded error.

Approximations of the non-linear value of $\tilde{x}$ can be refined via iterations or with differential equations, if the distribution is known.

The following paper describes the error and asymptotically bounded correctness of the replacement of mean with median:

John T. Chu. Harold Hotelling. "The Moments of the Sample Median." Ann. Math. Statist. 26 (4) 593 - 606, December, 1955.

https://doi.org/10.1214/aoms/1177728419