Sharpe Ratio Distribution: When Normal’s a No-Go, Why Not SHASHo? (2/4)
In the first post (1/4), a GAMLSS fit picked SHASHo over Normal for the Sharpe ratio estimator (López de Prado et al., 2026) — all four parameters statistically significant, a better GAIC than Normal, and an exact transform ($R$ transform) to standard Normal already in hand.
The goal of this post is to reach the probability $P(\widehat{SR}\le\widehat{SR}^*)$ — built up across all three blocks below, and ultimately resolved in block 3). That result is the base for all four measures from Post 1/4 — not just testing the hypothesis ($H_0: SR\le SR_0$ vs. $H_1: SR>SR_0$) via p-value and critical value, but also characterizing the test itself via test power and Minimum Track Record Length — and everything else here builds toward it.
1️⃣ Block 1) shows why it's reachable. In SHASHo, $\mu$ is a location parameter, not the mean — the actual mean equals $\mu$ plus a shift term built from the already-estimated scale and shape parameters. Since the Sharpe ratio is identified with that mean, solving the identity backwards pins down $\mu$ directly, for any Sharpe ratio value, without ever estimating $\mu$ from the data itself.
2️⃣ Block 2) applies a location-scale adjustment to $\widehat{SR}$, $Z=(\widehat{SR}-\mu)/\hat\sigma$, using that anchored $\mu$ together with the same fitted scale and shape parameters; evaluated at the observed estimate, this gives $Z^*$, which passes through the $R$ transform into $R^*$.
3️⃣ Block 3) closes the loop: because $Z$ and $R$ are both strictly monotonic transforms of $\widehat{SR}$, the probability of $\widehat{SR}$ not exceeding any value equals the probability of $R$ not exceeding the corresponding $R^*$ — exactly, not approximately — which is just the value of the standard Normal CDF at $R^*$, $\Phi(R^*)$. That's what makes the next post's derivations so direct.
With this scaffolding fixed, the next post (3/4) sets out two results for each of the four measures — p-value, critical value, test power, and Minimum Track Record Length — López de Prado and colleagues' original, and mine under SHASHo.
Reference
López de Prado, M., Lipton, A., & Zoonekynd, V. (2026). How to use the Sharpe ratio. SSRN.
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5520741
← Start from Part 1 of this series
