Scale Mixture Models

Posted on Mar 25, 2022

This note is for scale mixture models.

Scale parameter in mixture models

Given the mixing parameter $\tau > 0$, $S$ and $X$ are independent statistics such that $\tau S/\sigma$ and $\tau X/\sigma$ have densities

$g(v)I_{(0, \infty)}(v) \quad \text{and}\quad h(x;\lambda)I_{(k(\lambda), \infty)}(x)$

$\tau$ is assumed to have a distribution function $G(\cdot)$

The paper consider two cases:

• consider estimation of $\sigma$ when $G(\cdot)$ is a known (complete specified) nondegenerate distribution
• investigate the robustness of the improved estimators of $\sigma$ obtained under the degenerate model

what if $G(\cdot)$ is unknown

Scale mixtures of normal distributions

$e$ is variance mixture of i.i.d. normal distributions, i.e., its density is given by

$f(e;\theta^2) = \int_0^\infty \frac{\tau^{n/2}}{(2\pi)^{n/2}\theta^n}\exp\left(-\frac{\tau}{2\theta^2}\Vert e\Vert^2\right)dG(\tau)$

Scale mixtures of exponential distributions

$f(x_1,\ldots,x_n;\mu,\sigma) = \int_0^\infty \frac{\tau^n}{\sigma^n}\exp\left\{-\frac{\tau}{\sigma}\sum_{i=1}^n(x_i-\mu)\}I_{(\mu,\infty)}(x_{(1)})dG(\tau)$

mixsmsn: Finite Mixture of Scale Mixture of Skew-Normal

In a word, each component of the finite mixture is assumed to follow some scale mixture distribution.

• SMSN: scale mixtures of skew-normal distribution
• FMSMSN: finite mixture of SMSN, where the $i$-th component of the mixture is a SMSN distribution.
$SMSN(y\mid \mu,\sigma^2,\lambda,\nu) = 2\int_0^\infty \phi(y\mid \mu, u^{-1}\sigma^2) \Phi(u^{1/2}\lambda\sigma^{-1}(y-\mu))dH(u\mid\nu)$

choices of $H(\cdot\mid \nu)$

• univariate normal
• univariate skew-normal
• univariate skew-Student-t
• univariate skew-slash
• univariate skew-contaminated normal

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