Let $x$ describe a continuous predictor variable (e.g. age). Let $Y$ be a random variable (e.g. height) which is some function of $x$.

The data consists of $n$ points, each a combination of $x$ and $y$ (e.g. $data_i = (age_i, height_i)$).

The distribution of $Y$ may be non-normal, with some level of skewness. Not only the mean of $Y$, but also its variance and skewness may be functions of $x$ (e.g. adults are taller than children on average, but also more diverse in height and with more extremely tall outliers).

I know that GAMLSS can be used to describe the distribution of $Y$, whereby each parameter of a chosen distribution is modelled as a function of $x$. These functions can be given as polynomials or as splines, possibly smoothed using some penalisation.

$$Y \sim \mathcal{D}(\mu, \sigma)\\ g_1(\mu) = s_1(x)\\ g_2(\sigma) = s_2(x)$$

However, these represent functions of parameters of a chosen distribution. Is it also possible to obtain functions of the moments (e.g. mean, variance, skewness) of the data? Without assuming a form for the distribution.

$$mean(Y) = s_3(x)\\ var(Y) = s_4(x)\\ skew(Y) = s_5(x)$$

I guess a moving average and likewise, a moving variance and moving skewness, will give a similar function of $x$. But I would like to make use of GAMLSS's optimisation (e.g. ML, GAIC or GCV) including some penalisation for overfitting. If this exists. If that even makes sense without defining a distribution first.