Expression of estimated standart deviation in GAMLSS model

Question

I'm trying to replicate the results of the North Carolina Birth Data Analysis in the paper A Bivariate Copula Additive Model for Location, Scale and Shape by Giamperro Marra and Rosalba Radice (https://arxiv.org/abs/1605.07521). They provided the code in supplementary material in the paper but I don't know where does this expression of standart deviation came from

polys.map(NC.polys, sqrt(pi^2*sigma21/3), rev.col = FALSE,
main = expression(paste(hat(sigma)," of bwgram")),
scheme = "topo", cex.main = 1.7)

$$\hat{\sigma} = \sqrt{\pi^2 \sigma^2_1/3}$$

That sigma hat is a parameter estimated from a GAMLSS.

Thank you

Answer 1

See Table 2 in the paper. The equation you quote is not found in the paper or code. The table states that the variance of a logistic-distributed variable is:

$$ \mathbb{V}(Y_m) = \frac{\pi^2\sigma^2}{3} $$

You state that $\hat{\sigma}$ (implying the thing on the left hand side of the equation) is the parameter estimated by the model. This is incorrect. The model estimates a scale parameter $\sigma$, the estimated value of which we would write $\hat{\sigma}$, and hence we would have

$$ \widehat{\mathbb{V}(Y_m)} = \frac{\pi^2\hat{\sigma}^2}{3} $$

I.e. the estimated variance of the logistic distributed variable is given by plugging the estimated scale parameter $\hat{\sigma}^2$ into the equation for the variance of this distribution.

This is typical of many distributions that can be fitted in GAMLSS models or related. We often aren't modelling the "mean" (expectation, $\mathbb{E}(Y_m)$) or the variance directly in such a model but instead we model location ($\mu$), scale ($\sigma$), and shape parameters of the distribution, which are then used to to estimate mean, variance, kurtosis, etc. of the distribution. The table in the paper shows how expectation and variance of a range of distributions depend on the estimated LSS parameters given the parameterisation of said distribution in the model/package.