The Hypothesis Testing Framework

A backtest reports an annualised Sharpe ratio of $0.6$ on three years of daily returns. Is that skill, or noise dressed up as skill? Point estimation hands back a single number — $\widehat{\text{SR}}=0.6$ — and stops there; interval estimation wraps that number in a range. Hypothesis testing asks a sharper, binary question instead: is $0.6$ consistent with a genuine Sharpe ratio of zero, or is it surprising enough that "zero skill" should be rejected as an explanation? This note builds the formal vocabulary — null and alternative hypotheses, the two ways a test can be wrong, the power function that tracks both simultaneously, and the rejection region a decision rule ultimately reduces to. It is deliberately a vocabulary lesson: no test built here is yet claimed to be good. That question — what is the best possible test for a given testing problem — is exactly what the Neyman-Pearson lemma answers next, and this note's duality with confidence intervals — already stated from the interval-estimation side — is the same machinery worn inside out.

Learning objectives. By the end of this note you should be able to: