# JB: Jarque-Bera Test

> *The assumption of normality is the most dangerous assumption in all of statistics.*

| Property         | Value                            |
| ---------------- | -------------------------------- |
| **Category**     | Statistic                        |
| **Inputs**       | Source (close)                          |
| **Parameters**   | `period`                      |
| **Outputs**      | Single series (Jb)                       |
| **Output range** | Varies (see docs)                     |
| **Warmup**       | `period` bars                          |
| **PineScript**   | [jb.pine](jb.pine)                       |

- The Jarque-Bera test quantifies departure from normality by combining skewness and excess kurtosis into a single chi-squared statistic.
- **Similar:** [Kurtosis](../kurtosis/Kurtosis.md), [Skew](../skew/Skew.md) | **Trading note:** Jarque-Bera test; tests if returns are normally distributed. Significant = fat tails present.
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.

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:

$$m_2 = \frac{\sum x_i^2 - \frac{(\sum x_i)^2}{n}}{n}$$

$$m_3 = \frac{\sum x_i^3 - 3\bar{x}\sum x_i^2 + 2n\bar{x}^3}{n}$$

$$m_4 = \frac{\sum x_i^4 - 4\bar{x}\sum x_i^3 + 6\bar{x}^2\sum x_i^2 - 3n\bar{x}^4}{n}$$

### 3. Periodic Resync

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.

## Mathematical Foundation

### Skewness

$$S = \frac{m_3}{m_2^{3/2}}$$

### Excess Kurtosis

$$K = \frac{m_4}{m_2^2} - 3$$

### Jarque-Bera Statistic

$$JB = \frac{n}{6}\left(S^2 + \frac{K^2}{4}\right)$$

Under $H_0$ (normality), $JB \sim \chi^2(2)$.

### Critical Values

| Significance | Critical Value |
|:-------------|:---------------|
| 10% (0.10)   | 4.605          |
| 5% (0.05)    | 5.991          |
| 1% (0.01)    | 9.210          |

### Parameter Mapping

| Parameter | PineScript | QuanTAlib |
|:----------|:-----------|:----------|
| Window    | `length`   | `period`  |
| Min Value | 10         | 3         |

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.

| Metric      | Scalar | AVX2   |
|:------------|:-------|:-------|
| Bars/cycle  | 1      | ~3.2   |
| Throughput  | 1x     | ~3.2x  |

### Quality Metrics

| Metric      | Score | Notes |
|:------------|:------|:------|
| Accuracy    | 8/10  | Running sums accumulate FP drift; resync every 1000 ticks |
| Timeliness  | 9/10  | No lag beyond window fill |
| Sensitivity | 7/10  | Responds to both skewness and kurtosis changes |
| Robustness  | 8/10  | NaN/Infinity guarded; non-negative by construction |

## Validation

No external library implements rolling Jarque-Bera with matching methodology. Validation relies on mathematical properties.

| Library  | Status | Notes |
|:---------|:------:|:------|
| TA-Lib   | -      | Not implemented |
| Skender  | -      | Not implemented |
| Tulip    | -      | Not implemented |
| Ooples   | -      | Not implemented |

Self-validation:

- Constant series produces JB = 0
- Linear sequence {1..20} produces JB = 1.2 (analytical: uniform excess kurtosis = -6/5)
- 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.
- PineScript reference: `lib/statistics/jb/jb.pine`