This is a linkpost for http://putanumonit.com/2017/01/29/significantly-invested/
Glad to be of service :-)
My goal was to give an intuition about multiplicity
In which case you don't need the digression into Sharpe ratios at all. It just distracts from the main point.
the first method that people are taught is to: 1. Take the average daily return over a number of days, and multiply that by 252
Err... If I may offer more advice, don't breezily barge into subjects which are more complicated than they look.
The "average daily return" for people who are taught their first method usually means the arithmetic return (P1/P0 - 1). If so, you do NOT multiply that number by 252 because arithmetic returns are not additive across time. Log returns (log(P1/P0)) are, but people who are using log returns are usually already aware of how Sharpe ratios work.
the basic way that people are taught to test for statistical significance
This is testing the significance of the mean. I would probably argue that the most common context where people encounter statistical significance is a regression and the statistical significance in question is that of the regression coefficients. And for these, of course, it's a bit more complicated.
Still, both measurements are equally affected by testing multiple hypotheses
I don't understand what this means. If you do multiple tests and pick the best, any measurement is affected.
The "average daily return" for people who are taught their first method usually means the arithmetic return (P1/P0 - 1). If so, you do NOT multiply that number by 252 because arithmetic returns are not additive across time. Log returns (log(P1/P0)) are, but people who are using log returns are usually already aware of how Sharpe ratios work.
If your daily returns are so big that ln(P1/P0) is non-negligibly different from P1/P0 - 1, I'm interested in knowing what your investment strategy is. ;-)