Shapiro-Wilk and Anderson-Darling Tests

Before assuming returns are normal — to justify a t-test, a Sharpe ratio confidence interval, or a Black-Scholes assumption — it's worth formally testing that assumption rather than eyeballing a histogram. Jarque-Bera does this cheaply by checking whether sample skewness and kurtosis match the normal distribution's values of 0 and 3, but it can miss departures that don't show up strongly in those two summary numbers. The Shapiro-Wilk test instead compares the entire sorted sample against the pattern of order statistics you'd expect from a true normal sample, producing a single statistic $W$ between 0 and 1 (values close to 1 favor normality); it's widely regarded as one of the most powerful normality tests for small-to-moderate samples, meaning it's good at catching real non-normality without needing a huge sample size.

The Anderson-Darling test takes a different angle: it compares the empirical CDF of your data against the normal CDF, but weights discrepancies in the tails more heavily than discrepancies near the center. This makes it especially good at catching the fat-tailed departures from normality that are endemic in financial returns — exactly the kind of failure that matters most for risk management, where getting the tails wrong is far more costly than getting the middle wrong. Both tests return a p-value; a small p-value (conventionally below 0.05) rejects the null hypothesis that the data is normal.

In practice, a desk checking daily P&L for normality would run both: Shapiro-Wilk as a general-purpose check, and Anderson-Darling specifically to flag tail behavior that a risk model calibrated to a normal assumption would badly underestimate.