# Scale Mixture Models

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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

$\tau$ 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.

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

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