Solved – Distributions similar to Normal distribution

You can also use heavy tail Lambert W x Gaussian random variables Y with tail parameter $delta geq 0$ and $alpha geq 0$. Similar to the $t_{nu}$ distribution, the Normal distribution is nested for $delta = 0$ (in this case the input $X$ equals output $Y$). In R you can simulate, estimate, plot, etc. several Lambert W x F distributionswith the LambertW package.

In this similar post I fix the input variance $sigma_X = 1$ and vary $delta$ from $0$ to $2$. As the variance of $Y$ depends on $delta$ ($sigma_Y$ increases with $delta$ and does not exist for $delta geq 0.5$), the comparison of densities are not at the same variance.

However, you want to compare the actual distribuation at the same (finite) variance, so we need to

• keep $delta < 0.5$ as otherwise $var(Y) rightarrow infty$ or undefined;
• and compute the corresponding input standard deviation $sigma_X = sigma_X(delta)$ so that $sigma_Y = sigma_Y(sigma_X, delta) = 1$ for any given $delta$.

The following plot shows densities at varying $delta$; as $delta$ increases the density becomes more peaked/concentrated around $0$.

library(LambertW) library(RColorBrewer) # several heavy-tail parameters (delta < 0.5 so that variance exists) delta.v <- seq(0, 0.45, length = 10) x.grid <- seq(-3, 3, length = 201) col.v <- colorRampPalette(c("blue", "red"))(length(delta.v))  pdf.vals <- matrix(NA, ncol = length(delta.v), nrow = length(x.grid)) for (ii in seq_along(delta.v)) {   # compute sigma_x such that sigma_y(delta) = 1   sigma.x <- delta_01(delta.v[ii])["sigma_x"]   theta.01 <- list(delta = delta.v[ii], beta = c(0, sigma.x))   pdf.vals[, ii] <- dLambertW(x.grid, "normal", theta = theta.01) } matplot(x.grid, pdf.vals, type = "l", col = col.v, lwd = 2,          ylab = "", xlab = "") grid() legend("topleft", paste(delta.v), col = col.v, title = expression(delta),         lwd = 3, lty = seq_along(delta.v)) 

And similar to post on t distribution peak height at $0$:

plot(delta.v, pdf.vals[x.grid == 0, ] / dnorm(0), pch = 19, lwd = 10,      col = col.v, ylab = "", xlab = expression(delta), xlim = c(0, 0.5)) mtext("Relative peak height n (Normal(0, 1) = 1.0)", side = 2, line = 2) grid() abline(h = 1, lty = 2, col = "darkgreen")