The sinh-arcsinh (SHASH) distribution has a pdf as follows:

$f(x) = {\delta cosh(\omega)\over \sqrt{1+({x-\theta \over \sigma})^2}}\phi[sinh(\omega)]$ where $\omega=\gamma+\delta sinh^{-1}({x-\theta \over \sigma})$

and $\phi()$ is the standard normal pdf.

Note: $-\infty<\gamma,x,\theta<\infty; 0<\delta, \sigma$

Does anyone know what the expected value and variance of it is? I can see it is closely related to the [Johnson SU][1] distribution which has the mean, variance, median, and quantiles analytically defined. It also has the Normal distribution as a special case when $\gamma = 0$ and $ \delta = 1$. I know some distributions don't have these available in closed form analytic expressions....but I don't know if that's the case here. If available, it would be very helpful if someone knows what they are!

Update: It appears the moments are [published][2], but with the caveat that they depend on the modified Bessel function of the second kind ($P_\nu)$ below. E.g.

$E(X_{\gamma, \delta})=-sinh(\gamma/ \delta)P_{1/\delta}$

$Var(X_{\gamma, \delta})={1 \over 2}(cosh({2\gamma \over \delta})P_{2/\delta} - 1) - \mu_{\gamma, \delta}^2$

However -- this doesn't take into account the center, scale in the full version ($\theta, \sigma$). Can someone help modify the mean and var with center and scale included? [1]: https://en.wikipedia.org/wiki/Johnson%27s_SU-distribution [2]: https://www.jstor.org/stable/pdf/27798865.pdf?refreqid=excelsior%3Aa9989a2b34d375e9cb7fb3e0290c5911

The sinh-arcsinh (SHASH) distribution has a pdf as follows:

$f(x) = {\delta cosh(\omega)\over \sqrt{1+({x-\theta \over \sigma})^2}}\phi[sinh(\omega)]$ where $\omega=\gamma+\delta sinh^{-1}({x-\theta \over \sigma})$

and $\phi()$ is the standard normal pdf.

Note: $-\infty<\gamma,x,\theta<\infty; 0<\delta, \sigma$

Does anyone know what the expected value and variance of it is? I can see it is closely related to the Johnson SU distribution which has the mean, variance, median, and quantiles analytically defined. It also has the Normal distribution as a special case when $\gamma = 0$ and $ \delta = 1$. I know some distributions don't have these available in closed form analytic expressions....but I don't know if that's the case here. If available, it would be very helpful if someone knows what they are!

Update: It appears the moments are published, but with the caveat that they depend on the modified Bessel function of the second kind ($P_\nu)$ below. E.g.

$E(X_{\gamma, \delta})=-sinh(\gamma/ \delta)P_{1/\delta}$

$Var(X_{\gamma, \delta})={1 \over 2}(cosh({2\gamma \over \delta})P_{2/\delta} - 1) - \mu_{\gamma, \delta}^2$

However -- this doesn't take into account the center, scale in the full version ($\theta, \sigma$). Can someone help modify the mean and var with center and scale included?

The sinh-arcsinh (SHASH) distribution has a pdf as follows:

$f(x) = {\delta cosh(\omega)\over \sqrt{\sigma^2+(x-\theta)^2}}\phi[sinh(\omega)]$ where $\omega=\gamma+\delta sinh^{-1}({x-\theta \over \sigma})$

and $\phi()$ is the standard normal pdf.

Note: $-\infty<\gamma,x,\theta<\infty; 0<\delta, \sigma$

Does anyone know what the expected value and variance of it is? I can see it is closely related to the Johnson SU distribution which has the mean, variance, median, and quantiles analytically defined. It also has the Normal distribution as a special case when $\gamma = 0$ and $ \delta = 1$. I know some distributions don't have these available in closed form analytic expressions....but I don't know if that's the case here. If available, it would be very helpful if someone knows what they are!

The sinh-arcsinh (SHASH) distribution has a pdf as follows:

$f(x) = {\delta cosh(\omega)\over \sqrt{1+({x-\theta \over \sigma})^2}}\phi[sinh(\omega)]$ where $\omega=\gamma+\delta sinh^{-1}({x-\theta \over \sigma})$

and $\phi()$ is the standard normal pdf.

Note: $-\infty<\gamma,x,\theta<\infty; 0<\delta, \sigma$

Does anyone know what the expected value and variance of it is? I can see it is closely related to the [Johnson SU][1] distribution which has the mean, variance, median, and quantiles analytically defined. It also has the Normal distribution as a special case when $\gamma = 0$ and $ \delta = 1$. I know some distributions don't have these available in closed form analytic expressions....but I don't know if that's the case here. If available, it would be very helpful if someone knows what they are!

Update: It appears the moments are [published][2], but with the caveat that they depend on the modified Bessel function of the second kind ($P_\nu)$ below. E.g.

$E(X_{\gamma, \delta})=-sinh(\gamma/ \delta)P_{1/\delta}$

$Var(X_{\gamma, \delta})={1 \over 2}(cosh({2\gamma \over \delta})P_{2/\delta} - 1) - \mu_{\gamma, \delta}^2$

However -- this doesn't take into account the center, scale in the full version ($\theta, \sigma$). Can someone help modify the mean and var with center and scale included? [1]: https://en.wikipedia.org/wiki/Johnson%27s_SU-distribution [2]: https://www.jstor.org/stable/pdf/27798865.pdf?refreqid=excelsior%3Aa9989a2b34d375e9cb7fb3e0290c5911