What is the Anderson-Darling test, and why is it preferred over Kolmogorov-Smirnov for testing normality in risk management applications?

Question

AcadiFi · Accepted Answer

The **Anderson-Darling (AD) test** is a goodness-of-fit test that measures how well a sample matches a specified theoretical distribution. It is particularly valuable in risk management because it places **extra weight on the tails** of the distribution — exactly where risk managers care most.

**How It Differs from Kolmogorov-Smirnov (KS)**

The KS test measures the maximum absolute distance between the empirical CDF and theoretical CDF. It treats all parts of the distribution equally. The AD test, by contrast, uses a weighted integral that amplifies discrepancies in the tails.

| Feature | Kolmogorov-Smirnov | Anderson-Darling |
|---|---|---|
| Statistic | Max |F_n(x) - F(x)| | Weighted integral of squared CDF differences |
| Tail sensitivity | Low — focuses on center | **High — upweights tail deviations** |
| Power for fat tails | Moderate | **Superior** |
| Best use case | General distribution testing | Financial returns with potential tail risk |
| Null hypothesis | Data follows specified distribution | Same |

**The AD Statistic**

**A^2 = -n - (1/n) SUM [((2i-1) x (ln F(Y_i) + ln(1 - F(Y_{n+1-i}))))]**

where Y_i are the ordered observations and F is the theoretical CDF. Higher values indicate greater departure from the theoretical distribution.

**Worked Example**

Calloway Risk Analytics tests whether 500 daily returns from its multi-strategy hedge fund are normally distributed.

| Test | Statistic | Critical Value (5%) | Result |
|---|---|---|---|
| KS test | 0.031 | 0.061 | **Fail to reject** — looks normal |
| AD test | 2.847 | 0.752 | **Reject** — significant tail departure |

The KS test is fooled because the center of the distribution matches the normal well, but the AD test detects that the 1st and 99th percentile returns are far more extreme than a normal distribution predicts.

```mermaid
flowchart LR
    A["500 daily returns"] --> B["KS Test: p = 0.14"]
    A --> C["AD Test: p < 0.001"]
    B --> D["Fail to reject normality"]
    C --> E["Reject normality — tail divergence"]
    E --> F["Switch to t-distribution or EVT for VaR"]
```

**Implications for Risk Models**

If a risk manager relies on KS alone, they might validate a normal VaR model that severely underestimates tail losses. The AD test's tail sensitivity makes it the preferred diagnostic for:
- Validating distributional assumptions in parametric VaR
- Back-testing P&L distributions against model predictions
- Regulatory model validation (Basel requires evidence of distributional fit)

**Exam Tip:** If asked which test is most appropriate for detecting departures in the tails of a distribution, Anderson-Darling is the answer.

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What is the Anderson-Darling test, and why is it preferred over Kolmogorov-Smirnov for testing normality in risk management applications?

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