Draft:Q-distribution

• File:Symbol opinion vote.svg Comment: New references appear to be searches for other distributions? This needs more sourcing for this in particular. Smallangryplanet (talk) 15:03, 11 December 2025 (UTC)

• File:Symbol opinion vote.svg Comment: Only one source. Also it is overly complex. General readers won't have a clue what is going on here. Ktkvtsh (talk) 02:34, 6 December 2025 (UTC)

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An asymmetric and heavy tailed probability distribution, Zhao's q-distribution is a four parameter distribution for a real valued random variable. The distribution is proposed for modeling financial asset returns, which usually have heavy tails than Gaussian and also exhibit with asymmetry.

The four parameters of the q-distribution model explicitly represent the location, scaling parameters of positive and negative deviations and heaviness of tails. When one or two parameters reach their boundary, e.g. zero or infinity, the limiting distributions can be identical to one of the classical distributions. For instance,Gaussian, Laplace, logistic, and exponential are among the known special cases , which have numerous applications in many scientific fields.

Definition

The q-distribution has a probability density function

$f(x)=\frac{1}{C(\alpha ,\beta ,\theta )}{\left[(1+e^{\frac{x-\mu }{\alpha \theta }})(1+e^{-\frac{x-\mu }{\beta \theta }})\right]}^{-\theta }$

for $x\in ℝ$, where $C(\alpha ,\beta ,\theta )$ is a compensating factor and it often needs to be computed numerically except for the symmetric cases when $\theta$ is an integer.

Density function of a symmetric q-distribution versus Gaussian distribution

Density of an asymmetric q-distribution versus Gaussian distribution

The q-distribution has four parameters:

• $\mu \in ℝ$ - location parameter
• $\alpha >0$ - scaling parameter for positive domain or right tail
• $\beta >0$ - scaling parameter for negative domain or left tail
• $\theta >0$ - shape parameter that controls kurtosis

Except the location parameter, all the other three parameters are positive.

Symmetric case

A symmetric q-distribution arrives naturally when the scaling parameters $\alpha ,\beta$ are identical. In this case, the probability density function is simplified to

$f(x)=\frac{1}{\alpha \theta B(\theta ,\theta )}{\left[(1+e^{\frac{x-\mu }{\alpha \theta }})(1+e^{-\frac{x-\mu }{\alpha \theta }})\right]}^{-\theta }$

where the scaling parameters are equal and the compensator can be written as the Beta function. When the kurtosis parameter $\theta$ is an integer, the compensator can be easily calculated. For instance, when $\theta =1,B(1,1)=1$, one gets the logistic distribution. The symmetric q-distribution can be directly derived from Beta distribution though variable exchange with a logistic function.

The $q_2$ distribution

A very special case of the q-distribution is when $\theta =2$, we call it a $q_2$ distribution. It mimics the Gaussian distribution in many ways. The density function is simple and beautiful

$f(x)=\frac{3}{\alpha }{\left[(1+e^{\frac{x-\mu }{2\alpha }})(1+e^{-\frac{x-\mu }{2\alpha }})\right]}^{-2}$

which is symmetric with exponential tails. Its MLE estimator is similar to Huber's M-estimation, which is known to be robust and insensitive to outlier data points.

Other limiting cases

A few classical distributions can be derived from q-distribution. These are the boundary cases when one or more of the parameters reaches zero or infinity.

• Gaussian distribution: $\theta \to \mathrm{\infty },\alpha =\beta =\sigma /\sqrt{\theta }$

$f(x)\to \frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$

• Logistic distribution: $\theta =1,\alpha =\beta$

$f(x)\to \frac{1}{\alpha }{\left[(1+e^{\frac{(x-\mu )}{\alpha }})(1+e^{-\frac{(x-\mu )}{\alpha }})\right]}^{-1}$

When $\alpha \ne \beta$, it defines an asymmetric logistic distribution, which is new and unseen from the literature.

• Asymmetric Laplace distribution: $\theta \to 0$

$f(x)\to \frac{1}{\alpha +\beta }{e}^{-\frac{(x-\mu {)}^{+}}{\alpha }-\frac{(x-\mu {)}^{-}}{\beta }}$

This is different from the definition from Kotz et al. When $\alpha =\beta$, it becomes the Laplace distribution.

• Exponential distribution: $\theta \to 0,\beta \to 0$

$f(x)\to \frac{1}{\alpha }{e}^{-\frac{(x-\mu {)}^{+}}{\alpha }}$

Properties

The q-distribution has the following known properties:

• Skewness ranges from $-2$ to $2$.
• Kurtosis ranges from $3$ to $9$.
• Finite moments for any $n>0$

$E|X{|}^n<\mathrm{\infty }.$

• Finite expectation for $\lambda <{\alpha }^{-1}$

$E\left(e^{\lambda X}\right)<\mathrm{\infty }.$

Computational method

With a data sample $\{x_1,x_2,\dots ,x_n\}$, the parameters of q-distribution can be estimated using maximum likelihood method.

In general, for asymmetric q-distribution, there is no closed formula for the cumulative distribution function (c.d.f.) and neither the compensator $C(\alpha ,\beta ,\theta )$. However, expectation related to q-distribution can be computed effectively using Gaussian quadrature approach, because the distribution has exponential tails.

For any smooth function, the following integral can be effectively computed using Gauss–Laguerre quadrature method:

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x}G(x)=\sum _{i=1}^k{w}_{i}G(x_i)$

where $\{x_i,w_i\}$ are the quadrature abscissas and weights.

Any expectation with respect to q-density can be written into the above format by variable exchanges This makes it convenient to compute the compensator $C(\alpha ,\beta ,\theta )$ as well as the c.d.f function and any other types of expected values for the related application.

Applications

For many applications where Gaussian distribution is used, but is not perfect because the data set presents either asymmetry or long tails,  the q-distribution can jump in as a helper.

The four parameters of q-distribution have the flexibility to match moments up to the fourth order instead of the second order for Gaussian.  The distribution has potential applications in risk management, e.g. projection of tail risk, and many other scientific fields.

See also

The following classical distributions are related to the q-distribution. Their properties are well studied.

• Gaussian distribution
• Laplace distribution
• Logistic distribution
• Beta distribution