Bootstrap standard error of a median from four replicates

Question

You bootstrap a dataset of holding-period returns to get a standard error for the median, since the median has no tidy closed-form standard error. For a compact illustration you keep only four bootstrap replicates of the median:

$20, \quad 22, \quad 24, \quad 26.$

Estimate the bootstrap standard error of the median.

Accepted Answer

The bootstrap standard error is the standard deviation of the replicate values, using the sample formula that divides by B - 1 (here B = 4). Step 1, mean of the replicates. \bar{	heta}^ = \frac{20 + 22 + 24 + 26}{4} = \frac{92}{4} = 23. Step 2, squared deviations. (-3)^2 + (-1)^2 + (1)^2 + (3)^2 = 9 + 1 + 1 + 9 = 20. Step 3, divide by B - 1 = 3 and take the square root. 	ext{SE}{	ext{boot}} = \sqrt{\frac{20}{3}} = \sqrt{6.667} \approx 2.58. The median famously lacks a simple standard-error formula, which is exactly why resampling is so handy here: the spread of the resampled medians gives …

Bootstrap standard error of a median from four replicates

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