Cramér-von Mises and Watson Tests

You've fit a normal distribution to a return series and want to check whether the fit is any good. The Kolmogorov-Smirnov test answers this by finding the single largest gap between your data's empirical distribution and the fitted model — but a model can be systematically, mildly wrong everywhere without ever having one dramatic gap, and KS can miss exactly that kind of diffuse mismatch. The Cramér-von Mises test (and its close relative, the Watson test) fix this by summing up the squared gap across the entire distribution rather than reporting only its worst point, making them more sensitive to broad, evenly spread departures from the model.

An analogy: judging a tailor by one bad seam versus overall fit

Imagine grading two tailors on how well a suit fits. Tailor A's suit is perfect everywhere except one seam that gapes noticeably at the shoulder. Tailor B's suit has no single glaring flaw, but is a little loose in the waist, a little tight in the chest, and a touch long in the sleeve — every measurement slightly off, nowhere dramatically so. A judge who only checks "what's the single worst measurement" (like Kolmogorov-Smirnov) might flag Tailor A as the worse fit and pass Tailor B. A judge who adds up every measurement's deviation, squared, across the whole suit (like Cramér-von Mises) would correctly recognize that Tailor B's suit, while never egregiously wrong at any one point, is actually a worse overall fit once every small gap is counted.

The mechanics, one piece at a time

Let $F_n(x)$ be the sample's empirical distribution function (the fraction of observations $\le x$) and $F(x)$ be the fitted model's cumulative distribution function. The Cramér-von Mises statistic is

$$\omega^2 = n \int_{-\infty}^{\infty} \left[ F_n(x) - F(x) \right]^2 dF(x),$$

In plain English: at every point along the distribution, measure the squared vertical gap between the data's cumulative curve and the model's cumulative curve, then integrate (sum continuously) that squared gap across the whole range, weighting more heavily where the model expects more probability mass. Unlike KS, which reports only $\max_x |F_n(x) - F(x)|$ (the single worst gap), $\omega^2$ accumulates evidence from everywhere the two curves disagree, so it detects diffuse, whole-distribution mismatches that a single-point statistic can miss. The Watson test is a close variant, $U^2$, that additionally corrects for the curve's overall vertical shift before integrating — making it specifically useful when you suspect the model is right about spread and shape but the data might be centered slightly differently (relevant on a circular scale, like time-of-day effects, where "shift" doesn't have a fixed reference point).

Worked example 1: a diffusely miscalibrated volatility model

A risk model assumes standardized daily returns follow a standard normal. Comparing 250 days of actual standardized returns against the normal CDF: KS finds a maximum gap of 0.04 at the 90th percentile — not statistically alarming on its own (critical value around 0.086 for $n=250$ at 5% significance). But the Cramér-von Mises statistic, summing squared gaps across all 250 comparison points, comes out to $\omega^2 = 0.61$, comfortably above its 5% critical value of about 0.461 — flagging the fit as poor even though no single point looked dramatically off. The model isn't badly wrong anywhere in particular; it's mildly wrong almost everywhere, understating tail thickness a bit and overstating central density a bit, and only a whole-curve statistic like $\omega^2$ picks that up.

Worked example 2: comparing test power at a glance

Simulate 500 samples of size 100 from a Student-t distribution (5 degrees of freedom) and test each against a fitted normal. Kolmogorov-Smirnov correctly rejects normality (at 5% significance) in roughly 38% of samples; Cramér-von Mises rejects in roughly 55% of samples — because the t-distribution's departure from normal is spread broadly across the whole curve (heavier everywhere in the tails, not just at one location), which is precisely the kind of mismatch Cramér-von Mises is built to detect more reliably than a single-worst-gap statistic.

What this means in practice

Use Cramér-von Mises or Watson over Kolmogorov-Smirnov whenever you suspect a fitted distribution is subtly wrong across its whole shape rather than badly wrong in one spot — common when checking whether standardized returns really are normal, whether a copula fit captures joint tail behavior, or whether a simulated model's output distribution matches historical data. The Watson variant specifically suits circular or seasonal data (intraday time, day-of-week effects) where there's no natural fixed starting point for the comparison.