Generalized Skewness-Kurtosis Parameters

• Generalized skewness-kurtosis parameters are extensions of classical moments that accurately capture asymmetry, tail-thickness, and peakedness under support constraints.
• They serve as robust alternatives in diverse models, including univariate distributions, multivariate projections, and quantile-based or L-moment frameworks such as the g-and-k and SkeGTD families.
• These parameters impose admissibility constraints on moment pairs and enable practical estimation techniques in financial modeling, mixture structures, and structural shape controls.

Generalized skewness-kurtosis parameters are quantities that extend, constrain, or replace the classical third and fourth standardized moments in order to describe asymmetry, tail-thickness, peakedness, and support geometry in settings where the Pearson coefficients alone are insufficient. In the literature, these parameters appear as support-aware standardized moments $D_3,D_4$, family-specific shape parameters such as $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$, quantile- and L-moment ratios, Mardia-type multivariate indices, and sub-dimensional extrema over projections or coordinate subspaces (Meer et al., 2023, Rubio et al., 2014, Wang et al., 2023). A central theme is that skewness and kurtosis are often not free coordinates: bounded support, mixture structure, latent-variable parametrization, or multivariate geometry can impose explicit admissibility constraints or require generalized measures that separate effects which classical moments conflate.

1. Classical moment parameters and the need for extension

For a real random variable $X$ with mean $\mu$ and central moments $m_n=\langle (x-\mu)^n\rangle$, the standardized moment sequence is

$D_n=\frac{m_n}{m_2^{n/2}},$

with $D_3$ the skewness and $D_4$ the kurtosis in the notation of Meer and Weeks (Meer et al., 2023). In the notation used for Beta-law modeling, the corresponding non-excess skewness and kurtosis are

$$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$

where $\sigma^2=E[(X-\mu)^2]$ (Ausloos et al., 2018). Several families instead use excess kurtosis, such as $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$0, while retaining the usual standardized third moment $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$1 (Rubio et al., 2014).

The need for generalized parameters arises in several distinct ways. For bounded distributions, not every $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$2 pair is admissible, because support restrictions induce lower and upper bounds on feasible moments (Meer et al., 2023). In flexible univariate families, one parameter may control “main-body” skewness while another controls “tail skewness” or tail thickness, so a single $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$3 or $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$4 is only a summary index of a richer shape decomposition (Rubio et al., 2014). In multivariate settings, Mardia’s global measures compress the entire distribution into two scalars and may fail to reflect sub-dimensional features, motivating projection-based generalizations (Chowdhury et al., 2021). Robust alternatives arise when ordinary moments are unavailable, unstable, or deliberately avoided, leading to quantile-based or L-moment-based skewness and kurtosis parameters (Rubio et al., 2014, Lian et al., 2024).

2. Support-constrained skewness and kurtosis

For distributions supported on $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$5 with $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$6, Meer and Weeks show that the skewness $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$7 is bounded below by the coefficient of variation $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$8: $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$9 with equality for a two-point law supported on $X$0 (Meer et al., 2023). The proof proceeds by a bidisperse extremal ansatz and an extension to arbitrary positive-support distributions through a Rohatgi–Szekely decomposition; convex mixtures preserve the inequality.

The same logic extends to arbitrary lower and upper bounds. If the support is $X$1, define

$X$2

and then

$X$3

If the support is $X$4, define

$X$5

and obtain

$X$6

When both bounds hold, $X$7, the admissible skewness interval is simultaneously bounded by

$X$8

with $X$9 (Meer et al., 2023).

Kurtosis is constrained in parallel. Pearson’s inequality gives

$\mu$0

for every distribution. Combining this with the positive-support skewness bound yields the two-branch result

$\mu$1

Analogous piecewise bounds hold for singly bounded and doubly bounded supports, with the doubly bounded case also imposing an upper bound on $\mu$2 and forbidding $\mu$3 (Meer et al., 2023).

These results directly delimit the feasible $\mu$4-region. Because $\mu$5 itself is confined between lower and upper support-dependent bounds, and $\mu$6 is bounded below and, in some cases, above, the admissible region in the $\mu$7-plane is “triangular” or “trapezoidal.” A common misconception is therefore corrected: skewness and kurtosis are not arbitrary coordinates once the support is bounded. Meer and Weeks further conjecture analogous bounds for higher standardized moments $\mu$8, with numerical tests for $\mu$9 showing that randomly generated bounded distributions lie within bidisperse-predicted bounds (Meer et al., 2023).

3. Structural shape parameters in flexible univariate families

In many parametric families, generalized skewness-kurtosis parameters are structural controls rather than the standardized moments themselves. The double two-piece (DTP) family is defined from a symmetric unimodal base density $m_n=\langle (x-\mu)^n\rangle$0 and introduces five interpretable parameters $m_n=\langle (x-\mu)^n\rangle$1. Here $m_n=\langle (x-\mu)^n\rangle$2 induces “main-body” skewness, while $m_n=\langle (x-\mu)^n\rangle$3 induces “tail skewness.” The four-parameter subfamilies are the two-piece scale (TPSC) model with $m_n=\langle (x-\mu)^n\rangle$4 and the two-piece shape (TPSH) model with $m_n=\langle (x-\mu)^n\rangle$5 (Rubio et al., 2014).

The $m_n=\langle (x-\mu)^n\rangle$6-and-$m_n=\langle (x-\mu)^n\rangle$7 and generalized $m_n=\langle (x-\mu)^n\rangle$8-and-$m_n=\langle (x-\mu)^n\rangle$9 families are defined through their quantile functions rather than closed-form densities. In these models, $D_n=\frac{m_n}{m_2^{n/2}},$0 largely governs skewness through the factor $D_n=\frac{m_n}{m_2^{n/2}},$1, while $D_n=\frac{m_n}{m_2^{n/2}},$2 or $D_n=\frac{m_n}{m_2^{n/2}},$3 governs tail-thickness and hence kurtosis through $D_n=\frac{m_n}{m_2^{n/2}},$4 or $D_n=\frac{m_n}{m_2^{n/2}},$5. For small parameters, the paper gives the approximations $D_n=\frac{m_n}{m_2^{n/2}},$6 in the purely $D_n=\frac{m_n}{m_2^{n/2}},$7-skew case and $D_n=\frac{m_n}{m_2^{n/2}},$8 at $D_n=\frac{m_n}{m_2^{n/2}},$9 (Prangle, 2017).

The Skewed Generalized $D_3$0 distribution, $D_3$1, makes this separation explicit: $D_3$2 is the skewness parameter, while $D_3$3 and $D_3$4 are shape parameters that jointly control tail-heaviness and peakedness. The $D_3$5-th moment exists only when $D_3$6. The paper states that $D_3$7 increases in $D_3$8, $D_3$9 decreases as $D_4$0 or $D_4$1 increase, $D_4$2 as $D_4$3 or $D_4$4, and $D_4$5 as $D_4$6 (Lian et al., 2024).

Family	Generalized parameters	Role
DTP / TPSC / TPSH	$D_4$7	Separate main-body skewness from tail skewness
$D_4$8-and-$D_4$9, generalized $$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$0-and-$$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$1	$$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$2	$$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$3 controls skew; $$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$4 or $$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$5 controls tail-thickness
SkeGTD	$$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$6	$$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$7 controls asymmetry; $$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$8 control tail-heaviness and peakedness

A recurring point across these families is that parameters named “skewness” or “shape” need not equal Pearson skewness or kurtosis numerically. In DTP, classical $$S=\frac{E[(X-\mu)^3]}{\sigma^3},\qquad K=\frac{E[(X-\mu)^4]}{\sigma^4},$$9 are functions of half-tail moments $\sigma^2=E[(X-\mu)^2]$0, and if these are unavailable in closed form, one may use numerical quadrature or Monte Carlo, or instead adopt the Arnold–Groeneveld skewness $\sigma^2=E[(X-\mu)^2]$1 and the Critchley–Jones functional skewness

$\sigma^2=E[(X-\mu)^2]$2

which are always well-defined and do not require moments (Rubio et al., 2014).

4. Multivariate generalizations

The standard multivariate generalization is due to Mardia. For $\sigma^2=E[(X-\mu)^2]$3 with mean $\sigma^2=E[(X-\mu)^2]$4, covariance $\sigma^2=E[(X-\mu)^2]$5, and third-order cumulant tensor $\sigma^2=E[(X-\mu)^2]$6, Mardia’s multivariate skewness is

$\sigma^2=E[(X-\mu)^2]$7

while multivariate kurtosis is

$\sigma^2=E[(X-\mu)^2]$8

For the unified skew-$\sigma^2=E[(X-\mu)^2]$9 (SUT) distribution, Wang et al. express these indices in terms of a SUN-scale mixture representation. If

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$00

then

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$01

and

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$02

Consequently, $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$03 drives skewness through $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$04, while the degrees of freedom $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$05 act through the mixing moments $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$06; as $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$07, the relevant factors diverge and both skewness and kurtosis diverge, whereas $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$08 recovers the SUN case (Wang et al., 2023).

The univariate reduction clarifies the relationship to ordinary moments. When $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$09, Wang et al. obtain

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$10

so Mardia’s univariate skewness is the square of classical skewness, while Mardia’s kurtosis reduces to ordinary excess kurtosis (Wang et al., 2023).

A broader skew-elliptical analysis studies eight measures of multivariate skewness together with Mardia’s kurtosis: Mardia, Malkovich–Afifi, Isogai, Song, Balakrishnan–Brito–Quiroz, Móri–Rohatgi–Székely, Kollo, and Srivastava. The framework uses a canonical form in which at most one component of the skewness vector is nonzero, enabling explicit formulas in terms of radial moments $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$11 and a scalar $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$12. These measures are affine-invariant, except Song’s, which is location-scale invariant, and they vanish when $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$13, that is, in the purely elliptical case. The same paper states the inequality

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$14

for Mardia’s indices (Zuo et al., 2023).

5. Sub-dimensional, quantile-based, and L-moment generalizations

Global multivariate indices do not reveal where non-Gaussianity is located. Sub-dimensional Mardia measures address this by evaluating skewness and kurtosis on projected data. For a $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$15-dimensional subspace $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$16, the projected observations $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$17 yield sub-dimensional measures $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$18 and $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$19, and one then defines

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$20

The maximizers identify the subspace bearing the most extreme non-Gaussian feature. The classical Mardia measures arise as the special case $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$21, and the paper emphasizes that the classical summaries can miss skewness or kurtosis confined to a lower-dimensional face (Chowdhury et al., 2021).

Robust generalizations proceed differently. In the SkeGTD framework, the first four L-moments $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$22 generate the ratios

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$23

described as “generalized skewness” and “generalized kurtosis.” Closed-form series representations for $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$24 are given in the paper, and the resulting ratios are presented as robust alternatives to classical moment-based $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$25 (Lian et al., 2024).

Quantile-based generalization is equally important in families whose moments are cumbersome or unstable. In DTP distributions, the Arnold–Groeneveld and Critchley–Jones measures are available independently of the existence or tractability of higher moments (Rubio et al., 2014). A plausible implication is that “generalized skewness-kurtosis parameters” do not denote a single invariant pair of numbers, but a class of descriptors chosen to match support, robustness requirements, dimensionality, and model structure.

6. Estimation, inversion, and applications

Generalized skewness-kurtosis parameters are often operationalized through inversion or calibration. For the Beta$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$26 law, explicit formulas relate the theoretical skewness $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$27 and kurtosis $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$28 to $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$29. Hanson’s method-of-moments inversion writes

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$30

then determines $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$31, and finally recovers $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$32 as the two roots of $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$33. The same work fits rank-ordered provincial skewness and kurtosis by a discrete Lavalette law and maps the fitted exponents to $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$34 and $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$35, which the authors connect to a Yule–Simon or preferential-attachment mechanism (Ausloos et al., 2018).

Inference is more demanding in quantile-defined families. For the $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$36-and-$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$37 and generalized $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$38-and-$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$39 models, the density is unavailable in closed form, so likelihood evaluation requires numerical inversion of the quantile function. The paper therefore discusses numerical root-finding, Approximate Bayesian Computation, finite-difference stochastic approximation for maximum likelihood, and adaptive Metropolis MCMC (Prangle, 2017). For SkeGTD, three estimation routes are given: maximum-likelihood via an EM algorithm with Newton–Raphson updates, L-moment estimation through numerical inversion of $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$40, and a two-step estimation scheme that combines a robust mode estimator, sign-count information for $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$41, and moment or L-moment matching for $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$42 (Lian et al., 2024).

Generalized skewness and kurtosis also enter probabilistic inequalities. For martingale differences $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$43 with skewness $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$44 and kurtosis $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$45, Bentkus and Juškevičius define variance-substitute functions

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$46

and use $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$47, $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$48, or their minimum with $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$49 to sharpen Hoeffding-type tail bounds. The resulting Bernoulli-proxy inequalities extend to martingales, supermartingales, and maximal inequalities, and the paper states that up to the universal factor $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$50 the Bernoulli-sum tail is the final answer (Bentkus et al., 2011).

In financial modeling, the generalized Jarrow–Rudd tree introduces a skew random walk with

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$51

so that the model parameter $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$52 governs one-step asymmetry. The paper gives leading-order natural-measure skewness and excess kurtosis as

$g,k,h,r,\alpha,\beta,\delta_1,\delta_2$53

and calibrates implied $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$54 surfaces from option prices after first estimating $g,k,h,r,\alpha,\beta,\delta_1,\delta_2$55 from spot-return time series (Hu et al., 2021). This use is structurally different from Pearson-moment estimation: the generalized parameters are embedded directly in a complete-market tree and then backed out from observed market data.

Across these settings, generalized skewness-kurtosis parameters serve three distinct functions. They can be admissibility constraints on standardized moments under bounded support, interpretable shape controls in parametric families, or diagnostic summaries over multivariate, projected, quantile-based, or robust representations. The common feature is not a single formula, but a systematic extension of skewness and kurtosis beyond the unconstrained univariate Pearson paradigm.