The Jarque-Bera test quantifies departure from normality by combining skewness and excess kurtosis into a single chi-squared statistic. A rolling JB value near zero means the window looks Gaussian. Values exceeding 5.991 (5% significance) reject normality. Financial returns almost always fail this test, which is precisely why the test matters.
## Historical Context
Carlos Jarque and Anil Bera published the test in 1980, building on earlier work by Bowman and Shenton (1975). The insight was elegant: under normality, skewness is zero and kurtosis is three, so any deviation from these values indicates non-Gaussianity. The test statistic combines both deviations into a single number that follows a chi-squared distribution with two degrees of freedom.
Most implementations compute JB on static samples. This rolling implementation maintains O(1) updates by tracking running sums of powers (x, x², x³, x⁴), matching the approach used in the companion Skew indicator but extended to the fourth moment.
## Architecture
### 1. Running Power Sums
Four accumulators track $\sum x_i$, $\sum x_i^2$, $\sum x_i^3$, $\sum x_i^4$ over a sliding window of size $n$. When a new value enters and the oldest exits, each accumulator updates via simple addition/subtraction. This yields O(1) complexity per update.
### 2. Central Moments from Power Sums
Central moments are computed from raw power sums without explicitly centering each value:
Floating-point drift accumulates in running sums. Every 1000 ticks, the accumulator is rebuilt from the buffer contents. This bounds error growth without degrading amortized complexity.
QuanTAlib allows period >= 3 (minimum for meaningful moments), though periods below 10 produce unstable estimates.
## Performance Profile
### Operation Count (Scalar, per bar)
| Operation | Count | Cycle Cost |
|:----------|:------|:-----------|
| ADD/SUB | 20 | 1 |
| MUL | 16 | 3 |
| DIV | 5 | 15 |
| SQRT | 1 | 15 |
| FMA | 1 | 4 |
### Batch Mode (SIMD/AVX2)
Vectorized path processes 4 bars per iteration using prefix-sum accumulators for all four power sums. Available when `Avx2.IsSupported` and input contains no NaN values.
- Skewed data produces larger JB than symmetric data
- JB is always non-negative (sum of squares)
- Batch, streaming, span, and event modes produce identical results
## Common Pitfalls
1.**Small windows inflate JB.** With n < 10, moment estimates are noisy. The test's chi-squared approximation requires n >= 30 for reliable p-values. QuanTAlib allows n >= 3 for computation but interprets results cautiously below n = 20.
2.**JB tests population skewness, not sample.** This implementation uses population moments (dividing by n, not n-1), matching the original Jarque-Bera formulation and the PineScript reference. Sample-adjusted versions exist but produce different critical values.
3.**Zero variance data returns JB = 0.** When all values in the window are identical, m2 = 0 and the formula is undefined. The implementation returns 0, which correctly indicates no evidence against normality (a degenerate distribution is trivially "normal-shaped").
4.**Financial returns almost always reject normality.** Fat tails (positive excess kurtosis) are universal in financial data. A persistently high JB is normal for markets. The indicator is most useful for detecting *changes* in the degree of non-normality.
5.**FP drift in x⁴ accumulator.** The fourth power amplifies floating-point errors more than lower moments. The resync interval of 1000 ticks keeps drift bounded, but for very long-running streams (>100k ticks), consider shorter resync intervals.
6.**NaN handling substitutes last valid.** Non-finite inputs are replaced with the most recent finite value. This maintains continuity but can mask data quality issues. Monitor NaN frequency separately.
7.**Memory: 4 doubles of running state.** The O(1) update carries sum, sumSq, sumCu, sumQu plus previous-state copies for bar correction. Total state footprint is ~128 bytes excluding the RingBuffer.
## References
- Jarque, C. M.; Bera, A. K. (1980). "Efficient tests for normality, homoscedasticity and serial independence of regression residuals." *Economics Letters*, 6(3), 255-259.
- Bowman, K. O.; Shenton, L. R. (1975). "Omnibus test contours for departures from normality based on √b₁ and b₂." *Biometrika*, 62(2), 243-250.
