General Jarque-Berra Test (GJBT) Overview

• GJBT is an omnibus goodness-of-fit test that extends the classical Jarque–Bera test to any parametric family with finite moments up to order eight.
• It employs a covariance-weighted quadratic form of deviations in standardized cumulants, including skewness, kurtosis, and higher moments, for enhanced sensitivity.
• The test framework uses the functional empirical process and delta-method expansions to derive an asymptotically chi-square statistic, improving calibration and power for various distributions.

The General Jarque–Bera Test (GJBT) is an omnibus goodness-of-fit test built from standardized higher-order cumulants, or moments, and their asymptotic covariance structure. It generalizes the classical Jarque–Bera (JB, 1987) normality test in two directions: it allows the null hypothesis to be any parametric family with finite moments up to at least order eight, and it replaces the unweighted sum of squared deviations of sample skewness and kurtosis by a covariance-weighted quadratic form in deviations of a vector of standardized cumulants. In the formulation developed by Lo, Thiam, and Haidara, and extended in later work, the test is derived by the functional empirical process and delta-method expansions; in the skew normal specialization of Ba, Gning, Da, Sow, and Lo, the order-$p=2$ version is used, based on skewness and kurtosis, with null-specific covariance calibration (Lo et al., 2014, Ba et al., 24 Jul 2025).

1. Origins and relation to the classical Jarque–Bera statistic

The classical JB statistic for testing normality is

$$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$

where $S$ is sample skewness and $K$ is sample kurtosis. Under normality, JB converges to a chi-square with $2$ degrees of freedom. In the notation of the general theory, if $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$ denotes the $r$-th sample central moment and $\beta_r=m_r/m_2^{r/2}$ the standardized sample central moment, then $\gamma_1=\beta_3$ and $\gamma_2=\beta_4$ correspond to sample skewness and kurtosis (Lo et al., 2014).

A central point of the generalized framework is that the classical JB weights $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$0 and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$1 arise as asymptotic variances under the Gaussian null. For $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$2, the influence functions are proportional to $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$3 for skewness and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$4 for kurtosis, yielding $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$5, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$6, and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$7 by parity and orthogonality under the Gaussian measure (Lo et al., 2014). The same development shows that the constants $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$8 and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$9 depend on the normal law’s sixth and eighth moments, $S$0 and $S$1, so the JB test “really depends on the first eight moments” of the target distribution, not only on skewness and kurtosis (Lo et al., 2014).

The GJBT recovers JB when the null family is Gaussian and when the vector is restricted to skewness and excess kurtosis, with covariance matrix $S$2 or, in the skew-normal paper’s $S$3 convention for $S$4, $S$5 (Ba et al., 24 Jul 2025). This identity places JB as a special case of a broader Wald-type chi-square testing framework rather than as an isolated normality diagnostic.

2. General formulation for arbitrary distribution functions

In the general setting, let $S$6 denote the $S$7-th cumulant of a distribution and $S$8 its variance. The standardized cumulants are defined by

$S$9

In particular, $K$0 is skewness and $K$1 is excess kurtosis (Ba et al., 24 Jul 2025).

For a target distribution function $K$2 with at least $K$3 finite moments for some integer $K$4, Lo, Thiam, and Haidara consider the vector

$K$5

of dimension $K$6, where the $K$7 are the theoretical standardized moments under $K$8. The functional empirical process

$K$9

provides asymptotic linear expansions for the empirical standardized moments. Specifically, for each $2$0,

$2$1

where $2$2 and $2$3 are explicit polynomial functions depending on the target moments up to order $2$4, and the covariance matrix $2$5 is assembled from expectations such as $2$6, $2$7, and $2$8 (Lo et al., 2014).

The resulting generalized chi-square statistic is

$2$9

and under $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$0, $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$1 (Lo et al., 2014). In the more compact formulation used in the skew-normal specialization, if $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$2 collects sample standardized cumulants and $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$3 their theoretical null values, then

$m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$4

with $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$5 the asymptotic covariance matrix under the null, and $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$6 when $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$7 has dimension $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$8 (Ba et al., 24 Jul 2025).

This framework makes explicit that GJBT is not restricted to normality and is not restricted to the third and fourth standardized moments. More general versions may include higher standardized cumulants, for example up to order eight, which further increase sensitivity to departures from the null; however, even the $m_r=(1/n)\sum_{i=1}^n(X_i-\bar X)^r$9 case requires moments up to order eight for valid covariance calculations (Ba et al., 24 Jul 2025).

3. Two-dimensional GJBT and its skew normal specialization

The paper “A Jarque–Bera test for skew normal data” particularizes the GJBT to the skew normal family and uses its order $r$0 version, based on skewness and kurtosis (Ba et al., 24 Jul 2025). Let

$r$1

with theoretical counterparts $r$2 and $r$3. Under finite eighth moment,

$r$4

for a covariance matrix $r$5 derived via influence functions $r$6 and $r$7 built from polynomials $r$8 and null moments $r$9 up to $\beta_r=m_r/m_2^{r/2}$0 (Ba et al., 24 Jul 2025).

The associated GJBT statistic is

$\beta_r=m_r/m_2^{r/2}$1

Writing the inverse of a $\beta_r=m_r/m_2^{r/2}$2 matrix explicitly yields

$\beta_r=m_r/m_2^{r/2}$3

In the symmetric Gaussian case, $\beta_r=m_r/m_2^{r/2}$4, $\beta_r=m_r/m_2^{r/2}$5, $\beta_r=m_r/m_2^{r/2}$6, and $\beta_r=m_r/m_2^{r/2}$7, so the statistic reduces exactly to JB (Ba et al., 24 Jul 2025).

For the skew normal law $\beta_r=m_r/m_2^{r/2}$8, the density is

$\beta_r=m_r/m_2^{r/2}$9

where $\gamma_1=\beta_3$0 and $\gamma_1=\beta_3$1 are the standard normal pdf and cdf. A convenient reparametrization uses

$\gamma_1=\beta_3$2

with $\gamma_1=\beta_3$3 (Ba et al., 24 Jul 2025).

When $\gamma_1=\beta_3$4 and $\gamma_1=\beta_3$5,

$\gamma_1=\beta_3$6

and the standardized skewness and excess kurtosis are

$\gamma_1=\beta_3$7

These are the values used for $\gamma_1=\beta_3$8 and $\gamma_1=\beta_3$9 in the test (Ba et al., 24 Jul 2025).

The higher moments required by the covariance calculations are obtained constructively from

$\gamma_2=\beta_4$0

with $\gamma_2=\beta_4$1. From known moments of the half-normal $\gamma_2=\beta_4$2 and normal $\gamma_2=\beta_4$3, the authors derive non-centered moments $\gamma_2=\beta_4$4 for $\gamma_2=\beta_4$5 by binomial expansion, and then obtain centered moments and cumulants. The paper states that closed-form expressions for $\gamma_2=\beta_4$6 up to $\gamma_2=\beta_4$7 are algebraically lengthy, and uses Monte Carlo to obtain asymptotic covariance entries under a given $\gamma_2=\beta_4$8 (Ba et al., 24 Jul 2025).

4. Null hypotheses, parameter handling, and asymptotic calibration

The GJBT accommodates both simple and composite null hypotheses. Under the general theory, the null may be normality or membership in a specified parametric family, in which case the theoretical standardized cumulants depend on the parameter vector and are plugged into the test statistic (Ba et al., 24 Jul 2025).

For the skew normal specialization, the paper distinguishes two nulls. Under the family-fit null $\gamma_2=\beta_4$9, the theoretical pair $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$00 is computed at $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$01 and, if needed, at $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$02 and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$03, and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$04 is formed with $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$05 computed under $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$06. Under the normality null $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$07, one has $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$08, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$09, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$10, and the statistic reduces to JB (Ba et al., 24 Jul 2025).

In the paper’s specialization, the authors focus on the standardized case $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$11, note invariance to location and scale, set $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$12 and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$13 to the $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$14 values, and use the two-dimensional GJBT with covariance matrix $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$15 under that $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$16 (Ba et al., 24 Jul 2025). Under $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$17 with known $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$18,

$$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$19

so critical values and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$20-values are obtained from the chi-square distribution with $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$21 degrees of freedom (Ba et al., 24 Jul 2025).

For composite nulls with unknown parameters, one may estimate $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$22 by maximum likelihood or method of moments, plug the estimates into the null values $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$23 and into $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$24, or consistently estimate $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$25 directly. The skew-normal paper states that, under standard regularity conditions, Slutsky’s theorem implies that the limit remains $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$26 when consistent plug-in estimates are used (Ba et al., 24 Jul 2025). In the broader framework of Lo, Thiam, and Haidara, if $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$27 is unknown and consistently estimated by $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$28, then

$$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$29

with

$$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$30

where $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$31 is the covariance when $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$32 is known, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$33 is the asymptotic covariance of $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$34, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$35 is the Jacobian of the target moment map, and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$36 is the cross-covariance matrix between moment influence functions and the estimator’s influence function (Lo et al., 2014). In that Wald formulation, the degrees of freedom remain equal to the number of tested moment restrictions.

5. Implementation, power, and finite-sample behavior

A practical implementation for skew normal data proceeds as follows. If testing $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$37 with specified $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$38, and possibly specified $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$39 and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$40, the data may be transformed to standardized form because the test is invariant to location and scale; if parameters are unknown, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$41 may be estimated by maximum likelihood or method of moments. One then computes $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$42, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$43, together with $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$44 and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$45 as sample kurtosis and sample skewness. Under $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$46, the theoretical null values are obtained from $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$47, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$48, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$49, and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$50. The covariance matrix $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$51 for $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$52 is then computed or estimated; the paper provides explicit influence functions and uses Monte Carlo to estimate $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$53 at a fixed $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$54 by simulating large samples. Finally, one forms

$$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$55

and computes the $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$56-value by $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$57 (Ba et al., 24 Jul 2025).

The assumptions are the standard ones emphasized in both the general and skew-normal papers: i.i.d. observations, existence of at least the first eight moments for the $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$58 case, and smoothness conditions for the functional empirical process and delta-method expansions (Lo et al., 2014, Ba et al., 24 Jul 2025). The skew normal family satisfies the moment requirement, but heavy-tailed data may invalidate the asymptotics (Ba et al., 24 Jul 2025).

The reported simulation evidence in the skew-normal paper is specific. When testing “data are $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$59”, the mean $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$60-value is large, often above $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$61, even for very small $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$62, including $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$63 in the single digits, over a range of $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$64 values; this is presented as evidence that the test accepts the true model. For testing normality, $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$65, against $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$66, the test rejects $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$67 for moderate to large $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$68 at relatively modest sample sizes. The reported indicative sample sizes needed to reject normality increase sharply as $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$69 becomes small: for $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$70, a few thousand observations suffice in simulations; for $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$71, about $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$72; and for $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$73, extremely large $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$74, with simulations suggesting even $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$75 may be needed (Ba et al., 24 Jul 2025).

These findings are consistent with the formulas for $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$76 and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$77: as $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$78, the skew normal law approaches the Gaussian law, and skewness and excess kurtosis move slowly away from $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$79. A plausible implication is that the two-dimensional GJBT, while well calibrated for the skew-normal null, inherits the difficulty of any skewness–kurtosis-based procedure when alternatives are arbitrarily close to symmetry.

6. Comparison with JB, higher-moment extensions, and methodological caveats

Relative to the classical JB test, GJBT extends the null from Gaussianity to arbitrary distribution functions or parametric families with sufficient finite moments, and replaces the Gaussian weighting scheme by the correct covariance under the target model (Lo et al., 2014). In the skew normal context, this matters because the covariance matrix $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$80 generally has off-diagonal terms, so skewness and kurtosis are correlated under the null. The skew-normal paper therefore emphasizes that using the $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$81-based covariance yields a better-calibrated test than naively plugging skewness and kurtosis into JB with Gaussian weights (Ba et al., 24 Jul 2025).

The general high-moment construction also permits enlarging the test vector beyond skewness and kurtosis. In the notation of Lo, Thiam, and Haidara, taking

$$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$82

produces a chi-square test with $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$83 degrees of freedom. For a normal target and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$84, the framework includes $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$85 in addition to $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$86 and $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$87, and simulation studies in that paper report that the generalized test with $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$88 rejects double-exponential and double-gamma alternatives more often, and with smaller $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$89, than JB (Lo et al., 2014). The broader significance is that GJBT can detect alternatives that match low-order shape coefficients yet differ in higher moments.

A common misconception is that JB tests only skewness and kurtosis in a narrow sense. The general theory explicitly rejects that interpretation: even the classical weights depend on sixth and eighth moments under the null, and the generalized framework turns that dependence into an explicit covariance-based construction (Lo et al., 2014). Another methodological issue concerns the “samples duplication method” proposed in the skew-normal paper. The paper distinguishes between valid duplication in the form of generating extra i.i.d. observations from the $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$90 model to increase the sample size $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$91, and exact replication of the observed sample, where each datum is repeated $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$92 times. The first case is a standard Monte Carlo power assessment and is valid. The second case induces strong dependence, breaks the i.i.d. assumptions behind the asymptotic covariance matrix and the $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$93 calibration, and can produce artificially small $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$94-values unless the covariance and reference distribution are adjusted for the induced dependence (Ba et al., 24 Jul 2025).

For that reason, the skew-normal paper recommends awareness of the caveat, and a safer alternative stated there is a parametric bootstrap under the null: generate new i.i.d. datasets of size $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$95 from the fitted null, recompute the test statistic, and obtain a bootstrap $$\operatorname{JB} \;=\; \frac{n}{6}\,S^2 \;+\; \frac{n}{24}\,(K-3)^2,$$96-value (Ba et al., 24 Jul 2025). More generally, both sources note that direct plug-in estimates of asymptotic variances can be unstable in finite samples, especially when higher moments are used, so analytical covariance formulas or parametric bootstrap are preferable whenever available (Lo et al., 2014).

Taken together, these developments position the GJBT as a family of omnibus Wald-type goodness-of-fit tests whose classical normal-theory instance is JB, whose broader theory rests on the functional empirical process, and whose skew-normal specialization provides a concrete example of how null-specific cumulants and covariance weighting alter both calibration and power (Lo et al., 2014, Ba et al., 24 Jul 2025).