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The Student t random-effect model is useful when most studies are comparable but occasional genuine departures are more frequent than a normal distribution hierarchy allows. Its heavier tails permit a study effect far from $\mu$ without forcing a large common between-study heterogeneity scale on every study. In the Gaussian scale mixture representation, a small study-specific ${\lambda }_j$ lowers its precision parameter and weakens its shrinkage.

Use this as robust partial pooling when occasional atypical effects are plausible. Known systematic population or design differences should still be modeled explicitly; a heavy tail cannot identify or correct within-study bias by itself.

Two useful features of autoregressive conditional heteroscedasticity are persistent changes in conditional scale and excess unconditional kurtosis. Its time-varying conditional variance can explain volatility clustering, where large absolute returns occur in groups even when signed returns have little autocorrelation. A homoscedastic autoregressive moving-average model has a fixed innovation variance.

Also, a conditional normal distribution with a random scale is a Gaussian scale mixture. Its unconditional kurtosis can exceed $3$, or its fourth moment can be infinite. A Gaussian autoregressive moving-average model remains jointly Gaussian and cannot reproduce this effect. In the particular lag-two ARCH process, persistence of the squared scale occurs within each parity subsequence; the two parity subsequences are independent.

Recognize the integrand as a Gaussian scale mixture. If $R$ has the unit Rayleigh distribution and $Z$ is an independent standard normal distribution variable, then the conditional density of $RZ$ given $R=r$ is $(\sqrt{2\pi }r{)}^{-1}{e}^{-x^2/(2r^2)}$. Multiplying by the radius density givesThus use $U_1$ for the independent radius and $U_2,U_3$ for the normal output of the Box-Muller transform:The mixture argument also proves that $g$ integrates to one, by the Tonelli theorem. As a check, $R^2$ is exponential of rate $1/2$, so the characteristic function of $RZ$ is $\mathbb{E}{e}^{-t^2{R}^2/2}=1/(1+t^2)$. This is the Rayleigh-normal scale mixture, with Laplace distribution density $e^{-|x|}/2$.

A Student t random-effect model assigns Student's t-distributions to exchangeable study effects, allowing heavier tails than a normal distribution hierarchy. An equivalent Gaussian scale mixture is ${\lambda }_j\sim {\chi }_{\nu }^2$, ${\beta }_j\mid {\lambda }_j,\mu ,\psi \sim N(\mu ,\nu {\psi }^2/{\lambda }_j)$, with independent latent draws. For $\nu >2$ the variance is $\nu {\psi }^2/(\nu -2)$, so $\psi$ is a scale rather than a standard deviation. Small latent precisions weaken shrinkage for atypical studies while retaining partial pooling for the rest.