Generalized normal distribution

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The generalized normal distribution, also known as the generalized Gaussian distribution or exponential power distribution, is a family of symmetric continuous probability distributions defined on the real line, characterized by three parameters: a location parameter

\mu

(the mean), a scale parameter

\alpha > 0

, and a shape parameter

\beta > 0

that governs the tail heaviness and peakedness around the mean.^[1] Its probability density function is

f(x; \mu, \alpha, \beta) = \frac{\beta}{2\alpha \Gamma(1/\beta)} \exp\left( -\left( \frac{|x - \mu|}{\alpha} \right)^\beta \right),

where

\Gamma

denotes the gamma function, allowing the distribution to flexibly model data with varying degrees of kurtosis from sub-Gaussian (light tails when

\beta > 2

) to super-Gaussian (heavy tails when

\beta < 2

).^[1] Originally proposed by Subbotin in 1923 as a generalization satisfying axioms similar to those used by Gauss for the normal distribution, it encompasses special cases including the Laplace distribution (when

\beta = 1

) and the standard normal distribution (when

\beta = 2

, up to scale adjustment).^[2]^[1] Key properties of the generalized normal distribution include a mean of

\mu

, a variance of

\alpha^2 \frac{\Gamma(3/\beta)}{\Gamma(1/\beta)}

, and higher moments that depend on the shape parameter

\beta

, enabling precise control over skewness (which is zero due to symmetry) and kurtosis (which is 3 for

\beta = 2

, approaching 9/5 (1.8) as

\beta \to \infty

and infinity as

\beta \to 0^+

).^[3]^[4] Parameter estimation typically involves methods like maximum likelihood or moment matching, with the distribution proving useful in scenarios requiring robust modeling of non-normal data.^[2] An asymmetric variant exists, incorporating separate shape parameters for the left and right tails to handle skewness, though the symmetric form remains the most commonly applied.^[5] In applications, the generalized normal distribution is prominent in signal and image processing for modeling wavelet coefficients and impulsive noise, where its tail flexibility outperforms the normal distribution in capturing heavy-tailed phenomena.^[5] It also appears in finance for risk modeling, operations research for forecasting wait times, and machine learning for robust estimation in independent component analysis (ICA) algorithms like EFICA.^[6]^[5] These uses leverage its ability to approximate a wide range of empirical distributions while maintaining tractable analytical properties, such as the characteristic function involving modified Bessel functions.^[2]

Symmetric Case

Probability Density Function

The symmetric generalized normal distribution, also known as the exponential power distribution, is a three-parameter family defined by the location parameter

\mu \in \mathbb{R}

(mean), the scale parameter

\alpha > 0

, and the shape parameter

\beta > 0

that controls tail heaviness.^[1] The probability density function (PDF) is

f(x; \mu, \alpha, \beta) = \frac{\beta}{2\alpha \Gamma(1/\beta)} \exp\left( -\left( \frac{|x - \mu|}{\alpha} \right)^\beta \right),

where

\Gamma(\cdot)

denotes the gamma function. This form is continuous and symmetric around

\mu

, integrating to 1 over

\mathbb{R}

. The normalizing constant arises from the integral of the exponential power form over both tails, each contributing

\alpha \Gamma(1/\beta)/\beta

.^[1]^[2] The shape parameter

\beta

determines the distribution's peakedness and tails:

\beta = 2

recovers the normal distribution (up to scale),

\beta < 2

yields heavier tails (leptokurtic), and

\beta > 2

lighter tails (platykurtic). As

\beta \to \infty

, it approaches a uniform distribution, while

\beta \to 0^+

produces increasingly heavy tails.^[2]

Moments and Characteristic Function

The symmetric generalized normal distribution, with probability density function

f(x) = \frac{\beta}{2\alpha \Gamma(1/\beta)} \exp\left( -\left| \frac{x - \mu}{\alpha} \right|^\beta \right)

f(x)=2αΓ(1/β)βexp(−

αx−μ

β) for $x \in \mathbb{R}$ , $\alpha > 0$ , $\beta > 0$ , and location parameter $\mu \in \mathbb{R}$ , possesses central moments that reflect its symmetry around $\mu$ .^[7] Due to symmetry, the skewness is zero, and all odd-order central moments vanish: $\mu_k = E[(X - \mu)^k] = 0$ for odd $k$ . For even $k$ , the central moments are given by $\mu_k = \alpha^k \frac{\Gamma((k+1)/\beta)}{\Gamma(1/\beta)}$ . In particular, the mean is $E[X] = \mu$ .^[7]^[2] The variance is $\sigma^2 = \alpha^2 \frac{\Gamma(3/\beta)}{\Gamma(1/\beta)}$ . For example, when $\beta = 2$ , the distribution corresponds to a normal distribution with variance $\alpha^2 / 2$ ; when $\beta = 1$ , it is a Laplace distribution with variance $2\alpha^2$ . The kurtosis is $\frac{\Gamma(5/\beta) \Gamma(1/\beta)}{[\Gamma(3/\beta)]^2}$ , yielding an excess kurtosis of $\frac{\Gamma(5/\beta) \Gamma(1/\beta)}{[\Gamma(3/\beta)]^2} - 3$ . This excess kurtosis ranges from $-1.2$ (approaching the uniform distribution as $\beta \to \infty$ ) to $\infty$ (as $\beta \to 0^+$ , indicating increasingly heavy tails).^[7]^[2] The characteristic function is \phi(t) = E[e^{itX}] = e^{i t \mu} \sqrt{\pi} \, \Gamma(1/\beta) \,\, _1\Psi_1 \begin{pmatrix} (1/\beta, & 2/\beta) \\ (1/2, & 1) \end{pmatrix} \left( -\frac{(\alpha t)^2}{4} \right), where $_1\Psi_1$ denotes the Fox-Wright function. This closed-form expression facilitates analysis of sums and convolutions involving the distribution.^[7]

Parameter Estimation

Parameter estimation for the symmetric generalized normal distribution involves fitting the three parameters—location

\mu

, scale

\alpha

, and shape

\beta

—to data with symmetric but potentially non-normal characteristics. Unlike the asymmetric case, the symmetry simplifies computations, though the shape parameter

\beta

can lead to non-convex likelihoods requiring careful initialization. Maximum likelihood estimation (MLE) is commonly used, maximizing the log-likelihood

\ell(\mu, \alpha, \beta) = n \log(\beta) - n \log(2\alpha \Gamma(1/\beta)) - \sum_{i=1}^n (|x_i - \mu| / \alpha)^\beta

, often via numerical optimization like Newton-Raphson iterations. Initial values can be set using sample mean for

\mu

, a robust scale estimate for

\alpha

, and

\beta_0 \approx m_1 / \sqrt{m_2}

β0≈m1/m2

where $m_1$ and $m_2$ are first and second sample absolute moments.^[2] The method of moments provides an alternative, equating sample moments to theoretical ones: the first central moment to $\mu$ , the second to the variance $\alpha^2 \Gamma(3/\beta)/\Gamma(1/\beta)$ , and using higher even moments (e.g., fourth) to solve for $\beta$ via kurtosis matching. This yields a system of nonlinear equations solved numerically, offering simplicity but potential bias for small samples or extreme $\beta$ . Quantile-based methods, such as matching symmetric quantiles, can also estimate $\beta$ robustly, especially for heavy-tailed data. Challenges include sensitivity to outliers when $\beta < 2$ , addressed by robust variants or bootstrapping; goodness-of-fit is evaluated using Kolmogorov-Smirnov tests or Q-Q plots against the fitted CDF. Simulations show MLE performs well for large samples, while moment methods are faster for moderate $\beta$ near 2.^[2]^[8]

Applications

The symmetric generalized normal distribution is widely used in signal and image processing to model wavelet coefficients and impulsive noise, where its adjustable tails capture heavy-tailed phenomena better than the normal distribution. For example, it has been applied in texture retrieval, face recognition via Gabor filters, and image compression, leveraging the shape parameter

\beta

to fit empirical distributions of coefficients.^[2]^[5] In communications, it models noise in channels like ultra-wideband (UWB) and underwater acoustics, enabling robust detection and equalization by accounting for impulsive interference.^[2] It also appears in power systems for forecasting hourly peak load demand, providing flexible kurtosis control for symmetric but peaked load profiles.^[2] In machine learning, it supports robust estimation in independent component analysis (ICA) algorithms, such as EFICA, by assuming non-Gaussian sources with varying tail heaviness.^[6] These applications benefit from the distribution's tractable moments and characteristic function, facilitating analytical derivations in filtering and regression tasks.^[2]

Asymmetric Case

Probability Density Function

The asymmetric generalized normal distribution, also known as the asymmetric exponential power distribution in some contexts, belongs to a four-parameter family defined by the location parameter

\mu \in \mathbb{R}

, the left scale parameter

\alpha_1 > 0

, the right scale parameter

\alpha_2 > 0

, and the common shape parameter

\beta > 0

. The probability density function (PDF) is piecewise and given by

f(x; \mu, \alpha_1, \alpha_2, \beta) = \frac{\beta}{(\alpha_1 + \alpha_2) \Gamma\left(1/\beta\right)} \begin{cases} \exp\left( -\left[ \frac{\mu - x}{\alpha_1} \right]^\beta \right) & \text{if } x < \mu, \\ \exp\left( -\left[ \frac{x - \mu}{\alpha_2} \right]^\beta \right) & \text{if } x \geq \mu, \end{cases}

f(x;μ,α1,α2,β)=(α1+α2)Γ(1/β)β⎩

⎨

⎧exp(−[α1μ−x]β)exp(−[α2x−μ]β)if x<μ,if x≥μ, where $\Gamma(\cdot)$ denotes the gamma function. This form ensures continuity at $x = \mu$ and integrates to 1 over $\mathbb{R}$ , with the normalizing constant derived from the integrals over each side equaling $\alpha_1 \Gamma(1/\beta)/\beta$ and $\alpha_2 \Gamma(1/\beta)/\beta$ , respectively. The parameters $\alpha_1$ and $\alpha_2$ introduce asymmetry by allowing different decay rates for the left tail (controlled by $\alpha_1$ ) and the right tail (controlled by $\alpha_2$ ), while $\beta$ governs the overall tail heaviness: $\beta = 2$ yields a piecewise Gaussian form, $\beta < 2$ produces heavier tails than the normal, and $\beta > 2$ lighter tails. When $\alpha_1 = \alpha_2$ , the distribution reduces to the symmetric generalized normal distribution.^[9]^[10] A related form allows for side-specific shape parameters (separate $\beta_1$ and $\beta_2$ ) alongside varying scales, providing even greater flexibility for modeling differing tail behaviors on each side, though the common- $\beta$ version with distinct scales remains widely used for its simplicity.

Moments and Properties

The asymmetric generalized normal distribution exhibits distinct statistical properties due to the differing scale parameters

\alpha_1

(for the left tail,

x < \mu

) and

\alpha_2

(for the right tail,

x \geq \mu

), while sharing the common shape parameter

\beta > 0

. These parameters introduce non-zero skewness and influence higher-order moments, differentiating it from the symmetric case where

\alpha_1 = \alpha_2

. The mean is given by

E[X] = \mu - (\alpha_1 - \alpha_2) \frac{\Gamma(2/\beta)}{\Gamma(1/\beta)},

where

\Gamma

denotes the gamma function. This expression reflects the asymmetry: when

\alpha_1 > \alpha_2

, the longer left tail shifts the mean below the location parameter

\mu

, and vice versa. The term

\Gamma(2/\beta)/\Gamma(1/\beta)

arises from the expected value of the standardized generalized normal variable and scales the shift proportionally to the scale difference.^[11] The variance lacks a simple closed form but can be expressed as

\text{Var}(X) = \frac{\alpha_1^3 + \alpha_2^3}{\alpha_1 + \alpha_2} \frac{\Gamma(3/\beta)}{\Gamma(1/\beta)} - (\alpha_1 - \alpha_2)^2 \left( \frac{\Gamma(2/\beta)}{\Gamma(1/\beta)} \right)^2.

It comprises contributions from both tails, weighted by powers of the scales, with the gamma ratio capturing the shape-dependent dispersion. For

\beta = 2

(recovering a piecewise Gaussian), this simplifies to a form involving the difference in variances, but numerical evaluation is often required for general

\beta

due to the gamma functions. When asymmetry is pronounced (

\alpha_1 \neq \alpha_2

), the variance exceeds that of the symmetric counterpart with equivalent average scale.^[11] Skewness is inherently non-zero and given by

\gamma_1 = \frac{E[(X - E[X])^3]}{\text{Var}(X)^{3/2}} = \frac{\alpha_1 - \alpha_2}{\sigma^3} \left[ \frac{3 (\alpha_1^3 + \alpha_2^3)}{\alpha_1 + \alpha_2} \frac{\Gamma(2/\beta) \Gamma(3/\beta)}{\Gamma(1/\beta)^2} - 2 (\alpha_1 - \alpha_2)^2 \frac{\Gamma(2/\beta)^3}{\Gamma(1/\beta)^3} - (\alpha_1^2 + \alpha_2^2) \frac{\Gamma(4/\beta)}{\Gamma(1/\beta)} \right],

where

\sigma^2 = \text{Var}(X)

. The sign of

\gamma_1

depends on

\alpha_1 - \alpha_2

: positive if

\alpha_1 < \alpha_2

(right-skewed) and negative otherwise. The formula involves differences of gamma function products, highlighting the asymmetry's impact on the third central moment; its magnitude increases with the scale disparity and decreases with

\beta

approaching 2, where skewness vanishes in the limit of symmetry.^[11] The mode always coincides with the location parameter

\mu

, as the probability density function increases towards

\mu

from the left and decreases away from

\mu

to the right. For general parameters, the density is continuous at

\mu

but has differing left- and right-hand behaviors.^[10] Kurtosis exceeds that of the symmetric generalized normal and is more complex, expressed via the fourth central moment:

\gamma_2 = \frac{E[(X - E[X])^4]}{\text{Var}(X)^2} - 3 = \frac{1}{\sigma^4} \left[ \frac{\alpha_1^5 + \alpha_2^5}{\alpha_1 + \alpha_2} \frac{\Gamma(5/\beta)}{\Gamma(1/\beta)} - 4 (\alpha_1 - \alpha_2)^2 \frac{ (\alpha_1^2 + \alpha_2^2) \Gamma(2/\beta) \Gamma(4/\beta) }{ \Gamma(1/\beta)^2 } - 6 \frac{\alpha_1^3 + \alpha_2^3}{\alpha_1 + \alpha_2} \frac{ \Gamma(2/\beta)^2 \Gamma(3/\beta) }{ \Gamma(1/\beta)^3 } + 3 (\alpha_1 - \alpha_2)^4 \frac{ \Gamma(2/\beta)^4 }{ \Gamma(1/\beta)^4 } \right] - 3.

This leptokurtic property intensifies with asymmetry and smaller

\beta

, as the tail-specific contributions amplify heavy tails compared to the symmetric case.^[11] The characteristic function lacks a simple closed form due to the piecewise nature of the PDF and requires series expansions or numerical integration for evaluation; alternatively, the moment-generating function can be derived via power series involving the raw moments. Higher-order raw moments can be computed using conditional expectations on the left and right tails.^[11] The cumulative distribution function (CDF) and quantile function generally require numerical methods, as no elementary antiderivative exists for the PDF. The CDF is

F(x) = \begin{cases} \frac{\alpha_1}{(\alpha_1 + \alpha_2) \Gamma(1/\beta)} \Gamma\left( \frac{1}{\beta}, \left( \frac{\mu - x}{\alpha_1} \right)^\beta \right) & x < \mu, \\[1em] \frac{\alpha_1}{\alpha_1 + \alpha_2} + \frac{\alpha_2}{(\alpha_1 + \alpha_2) \Gamma(1/\beta)} \gamma\left( \frac{1}{\beta}, \left( \frac{x - \mu}{\alpha_2} \right)^\beta \right) & x \geq \mu, \end{cases}

F(x)=⎩

⎨

⎧(α1+α2)Γ(1/β)α1Γ(β1,(α1μ−x)β)α1+α2α1+(α1+α2)Γ(1/β)α2γ(β1,(α2x−μ)β)x<μ,x≥μ, where $\Gamma(s, z)$ and $\gamma(s, z)$ are the upper and lower incomplete gamma functions, respectively. Quantiles are obtained by inverting this numerically, often via root-finding algorithms, due to the incomplete gamma's transcendental nature.^[11]

Parameter Estimation

Parameter estimation for the asymmetric generalized normal distribution, also known as the asymmetric exponential power distribution (AEPD), involves fitting four parameters—typically location

\mu

, scale

\sigma

, shape

\beta

, and asymmetry

\kappa

—to data exhibiting differing tail behaviors on either side of the location parameter. This added complexity compared to the symmetric case arises from the need to capture both kurtosis and skewness, often requiring numerical methods due to the lack of closed-form solutions. Maximum likelihood estimation (MLE) is a primary approach, where the log-likelihood function separates contributions from observations below and above

\mu

, reflecting the distinct left- and right-tail densities. Optimization proceeds over the four parameters using numerical solvers or the expectation-maximization (EM) algorithm, which handles the non-convexity and ensures convergence even in challenging cases; however, identifiability issues emerge when

\beta

is small, as the distribution approaches a uniform limit, leading to flat likelihood surfaces. The method of moments provides an alternative, matching sample moments to theoretical counterparts: the first moment to the mean, the second to the variance, and the third to skewness, which incorporates the asymmetry parameter and is derived from the distribution's properties. This results in a system of nonlinear equations solved numerically, offering robustness in moderate samples but potential bias in small ones due to higher-moment sensitivity. Quantile-based estimation, particularly via L-moments, leverages asymmetric quantiles to estimate the scale and shape parameters separately for left and right tails, providing unbiased estimators robust to outliers and suitable for heavy-tailed data; L-moment ratios, such as the coefficient of L-skewness, directly inform the asymmetry

\kappa

. Estimation faces challenges including overparameterization, where the four parameters can lead to multiple solutions fitting the data equally well, especially in finite samples, increasing variance compared to the three-parameter symmetric MLE. Sensitivity to outliers is pronounced in the tails, as the shape parameter

\beta

amplifies deviations, necessitating robust preprocessing or penalized likelihood variants. Post-estimation, goodness-of-fit is assessed using adapted Anderson-Darling or Kolmogorov-Smirnov tests, which compare empirical cumulatives to the fitted AEPD quantile function, with the Anderson-Darling emphasizing tail fit crucial for asymmetry; entropy-based variants further evaluate departure from the assumed form. Simulations indicate L-moments often outperform MLE in small samples for tail parameter recovery, while EM-MLE excels in large datasets for precise likelihood-based inference.

Applications

The asymmetric generalized normal distribution has found applications in finance for modeling asset returns that exhibit left-skewness and heavy tails, such as those observed in stock indices. For instance, scale mixtures of the distribution have been used to fit daily returns of the S&P 500 and Shanghai Stock Exchange Composite Index over periods spanning 1998 to 2023, capturing leptokurtosis and downside risk through flexible shape parameters that allow heavier left tails (corresponding to α₁ > α₂).^[12] This approach outperforms simpler models by accommodating the non-normal characteristics of financial time series, enabling better risk assessment and forecasting.^[12] In environmental engineering, particularly wind power analysis, variants like the mixed skew generalized error distribution—closely related to the asymmetric generalized normal—model forecast errors in wind speed data to quantify frequency regulation potential. Applied to SCADA data from a 1.5 MW turbine, it effectively captures skewness and thick tails in error distributions across wind speed intervals, improving uncertainty estimates for grid stability compared to Gaussian mixtures.^[13] Biomedical applications include regression modeling of physiological data with asymmetry, such as lean body mass in athletes, where the asymmetric exponential power distribution (an equivalent form) provides a superior fit to skewed datasets over normal or skew-normal alternatives.^[14] Additionally, it has been fitted to heart failure datasets, demonstrating enhanced performance in goodness-of-fit metrics for data with high skewness and excess kurtosis, aiding in survival analysis or growth curve modeling.^[15] In image processing and surface metrology, the distribution models non-symmetric noise and roughness profiles in machined surfaces, such as those from abrasive waterjet milling. It predicts skewness (Rsk) and kurtosis (Rku) indicators with low error for skewed profiles (e.g., 0.457% error in Rsk for certain cases), though it may overestimate kurtosis below 3, offering a flexible alternative to symmetric models for asymmetric surface textures.^[16] Case studies highlight its utility: fitting stock returns reveals superior AIC/BIC scores for mixtures capturing financial asymmetry, while surface roughness predictions validate its use in engineering quality control, and wind error modeling enhances renewable energy reliability assessments.^[12]^[16]^[13] A key advantage is its ability to independently control skewness and kurtosis via asymmetry and shape parameters, providing a broader range of tail behaviors than the skew-normal distribution, which exhibits more restricted kurtosis flexibility and symmetric tail influences.^[15]^[16] This makes it particularly suitable for scenarios requiring distinct left- and right-tail modeling, unlike skew-normal distributions with fixed kurtosis constraints.^[15]

Relations and Extensions

Connections to Other Distributions

The generalized normal distribution, in its symmetric case, exhibits several limit behaviors depending on the shape parameter β. When β = 2, it reduces to the standard normal distribution.^[2] For β = 1, it coincides with the Laplace distribution.^[2] As β approaches infinity, the distribution converges to a uniform distribution over the interval [-α, α], where α is the scale parameter. Under unit variance normalization, this interval is [-√3, √3].^[2] The symmetric generalized normal distribution can be represented as a scale mixture of normal distributions, where the mixing distribution is a generalized gamma distribution on the scale parameter.^[17] This mixture representation facilitates Bayesian inference and highlights its flexibility in modeling varying tail behaviors through the choice of mixing parameters.^[17] For β < 2, the generalized normal distribution relates to symmetric stable distributions through subordination in Lévy processes, where a Brownian motion is subordinated by a stable subordinator, yielding processes with similar heavy-tailed marginals.^[2] Additionally, the probability density function of the generalized normal distribution, up to normalization, serves as the characteristic function of a symmetric α-stable distribution with index α = β, underscoring its role as a positive-definite function associated with infinitely divisible distributions.^[2] As β approaches 0, the generalized normal distribution develops extremely heavy tails, enabling it to model data with significant outliers while maintaining symmetry.^[5] This limit emphasizes its utility for leptokurtic phenomena beyond the normal case.^[5] The generalized normal distribution is equivalently known as the exponential power distribution in an alternative parameterization, where the density is expressed as proportional to exp(-|x - μ|^p / (2σ^p)) with p = β, offering computational advantages in certain optimization contexts.

Infinite Divisibility

A probability distribution is infinitely divisible if, for every positive integer

n

, it can be expressed as the distribution of the sum of

n

independent and identically distributed random variables.^[18] The symmetric generalized normal distribution, with probability density function

f(x) = \frac{\beta}{2\alpha \Gamma(1/\beta)} \exp\left( - \left| \frac{x - \mu}{\alpha} \right|^\beta \right)

f(x)=2αΓ(1/β)βexp(−

αx−μ

β) for $\beta > 0$ , $\alpha > 0$ , and $\mu \in \mathbb{R}$ , is infinitely divisible if and only if $\beta \in (0,1] \cup \{2\}$ .^[2] For $\beta = 2$ , it reduces to the normal distribution, which is infinitely divisible.^[18] For $\beta = 1$ , it is the Laplace distribution, also infinitely divisible.^[2] The proof relies on properties of the characteristic function $\phi(t)$ and the density. For $\beta \in (0,1]$ , the density (after standardization) is completely monotone on $(0, \infty)$ , implying infinite divisibility via representation as a scale mixture of Gaussians or direct verification that $\log \phi(t)$ admits the Lévy-Khinchine form.^[2] For $\beta = 2$ , the Gaussian case follows from the classical Lévy-Khinchine theorem with zero Lévy measure and quadratic cumulant. For $\beta > 2$ , $\phi(t)$ has real zeros, which infinitely divisible characteristic functions cannot possess.^[2] For $\beta \in (1,2) \setminus \{2\}$ , non-infinite divisibility follows from failure of self-decomposability and mismatch with extended generalized gamma convolution properties.^[2] In the infinitely divisible cases, the Lévy-Khinchine representation applies, with the cumulant function $\psi(t) = -\log \phi(t)$ given by a Gaussian component plus an integral over the Lévy measure $\nu(dx)$ . For $\beta \in (0,1]$ , the Lévy measure is absolutely continuous with explicit density $f_\theta(x) = -\frac{x}{\pi(1+x^2)} \int_0^\infty (\log \phi_\beta(t))' \sin(tx) \, dt$ for $x \neq 0$ , where $\phi_\beta(t)$ involves the gamma function in its normalization but is computed via cosine transforms.^[2] For $\beta = 2$ , the measure is degenerate at zero. This property enables representations as limits of compound Poisson processes, facilitating applications in risk theory and Lévy process modeling where jumps (for $\beta \leq 1$ ) capture heavy tails relevant to financial extremes.^[2]

Kullback-Leibler Divergence

The Kullback–Leibler (KL) divergence quantifies the information loss when approximating one probability distribution with another, defined as

D_{\mathrm{KL}}(P \parallel Q) = \int p(x) \log \frac{p(x)}{q(x)} \, dx

for continuous distributions

P

and

Q

with densities

p

and

q

. In the context of the generalized normal distribution (GND), also parameterized as the γ-order GND in some literature, the KL divergence serves to measure differences between GNDs or between a GND and related distributions like the Student's t-distribution, aiding in applications such as model selection and information theory analysis.^[19] Toulias and Kitsos (2021) derived a closed-form expression for the KL divergence between a multivariate γ-order GND

X \sim N_p^\gamma(\mu_1, \Sigma_1)

and a multivariate Student's t-distribution

Y \sim t_p(v, \mu_2, \Sigma_2)

, valid when

\gamma > v/(v-1)

D_{\mathrm{KL}}(X \parallel Y) = -\frac{1}{2} \log \left( \frac{|\Sigma_1|}{|\Sigma_2|} \right) + \frac{1}{2} \operatorname{tr} \left( \Sigma_2^{-1} \Sigma_1 \right) - \frac{p}{2} + \frac{v+p}{2} \log \left( \frac{\Gamma(1/2)}{\Gamma(v/2)} \right) + \text{additional terms involving the mean difference and gamma functions}.

This expression underscores the asymmetry of the KL divergence, with the reverse

D_{\mathrm{KL}}(Y \parallel X)

yielding a different form; when

\gamma = v/(v-1)

, the divergence tends to infinity, reflecting incompatibility in tail behaviors. For the special case

\gamma = 2

, the GND reduces to the multivariate normal, and the KL simplifies to the divergence between a multivariate normal and a multivariate Student's t-distribution, which approaches the standard normal-to-normal formula as the degrees of freedom

v \to \infty

D_{\mathrm{KL}}(N(\mu_1, \Sigma_1) \parallel N(\mu_2, \Sigma_2)) = \frac{1}{2} \left[ \operatorname{tr}(\Sigma_2^{-1} \Sigma_1) + (\mu_1 - \mu_2)^\top \Sigma_2^{-1} (\mu_1 - \mu_2) - p - \log \frac{|\Sigma_1|}{|\Sigma_2|} \right].

^[19] To address the asymmetry, symmetrized variants have been explored for pairs of γ-order GNDs. The Jeffreys distance, defined as

J(P, Q) = D_{\mathrm{KL}}(P \parallel Q) + D_{\mathrm{KL}}(Q \parallel P)

, provides a symmetric measure; for two γ-GNDs with identical means but different scales, it involves logarithmic ratios of determinants and traces adjusted by γ-dependent normalizing constants. Similarly, the geometric-KL distance

G(P, Q) = D_{\mathrm{KL}}(P \parallel G) + D_{\mathrm{KL}}(Q \parallel G)

, where

G

is the geometric mean distribution, and the harmonic-KL distance exhibit finite values under equal scales but diverge otherwise, with limits analyzed as γ approaches boundary values like 1 (Laplace-like) or infinity (uniform-like). These symmetrized forms facilitate comparative studies of shape parameters in GND families.^[19] Further evaluations confirm that the KL divergence for the γ-order GND family aligns with the normal case at γ = 2 and extends properties from the Euclidean logarithmic Sobolev inequality, enabling bounds on information measures across varying γ and scale parameters. Such analyses highlight the GND's flexibility in capturing symmetric and heavy-tailed behaviors relative to the normal distribution.^[20]

Generalizations

The generalized normal distribution, originally introduced by Subbotin in 1923 as a flexible family satisfying certain axioms akin to those of the normal distribution, has undergone significant extensions over the decades to accommodate more complex data structures and asymmetries.^[2] Subsequent developments, such as the asymmetric exponential power distribution proposed by Zhu and Zinde-Walsh in 2009, which uses separate shape parameters for the left and right tails (β_left and β_right), have broadened its applicability to non-symmetric phenomena. These evolutions trace a path from Subbotin's symmetric core to modern parametric families that enhance tail flexibility and multivariate capabilities.^[21]^[22] Multivariate generalizations of the distribution extend its univariate form to vector-valued random variables, often adopting elliptical contours for symmetry or non-elliptical structures for added flexibility. The multivariate generalized Gaussian distribution (MGGD), for instance, is parameterized by a mean vector, a scatter matrix, and a shape parameter that controls super- or sub-Gaussian behavior, making it suitable for modeling correlated data in signal processing and beyond.^[23] Earlier work, such as the multivariate θ-generalized normal distributions proposed in 1973, introduced canonical forms that generalize the multivariate normal while preserving desirable properties like infinite divisibility under certain conditions.^[24] These extensions maintain the core exponential power structure but incorporate covariance to capture dependencies among dimensions.^[25] Skewed variants further generalize the distribution by allowing distinct tail behaviors, as seen in the asymmetric exponential power distribution (AEPD), which employs separate β parameters for each tail to model skewness without assuming symmetry.^[10] A recent advancement, the novel skewed generalized normal distribution (NSGN) introduced in 2024, builds on this as a two-component mixture model that nests several classical distributions and excels in capturing high kurtosis and skewness, particularly for heavy-tailed data.^[22] Such forms address limitations of the symmetric generalized normal by providing greater control over asymmetry, though they introduce additional parameters that can elevate estimation complexity.^[26] Bivariate asymmetric extensions often rely on direct parametric formulations or copula constructions to induce dependence while preserving marginal asymmetries. For example, scale mixtures of Kotz-type distributions, which generalize the bivariate normal and incorporate shape parameters for asymmetry, offer a framework for modeling joint skewed behaviors in two dimensions.^[27] Copula-based approaches, such as those combining asymmetric margins with bivariate copulas like the skew-t, enable flexible dependence structures without restricting to elliptical symmetry.^[28] Hybrids involving generalized gamma convolutions (GGCs) represent another extension, where the generalized normal serves as a limiting case or component in convolution classes that include infinitely divisible distributions like the lognormal.^[29] These convolutions, originally explored by Thorin in 1977, blend gamma-scale mixtures with power-law tails to yield broader families suitable for positive-valued processes.^[26] Despite these advances, generalizations with increased parameters—such as separate tail shapes or multivariate covariances—pose risks of overfitting, particularly in finite samples, where maximum likelihood estimation may yield unstable fits without regularization.^[10] This underscores the need for careful model selection in applications demanding high flexibility.^[30]

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