For my research I need to obtain a series of densities, however, I am encountering some problems.
The first problem is perhaps very simple, but the answer eludes me. Let's say I have an observation from a time series, $x=0.001$ together with estimated mean $\mu=0.0001$ and standard deviation $\sigma=0.4$. Using R:
dnorm(0.001, 0.0001, 0.4) z=(0.001-0.0001)/0.4 dnorm(z, 0, 1)
where z is the standardized variable. Why are the result different?
The second issue is connected to Hansen's Skew-t distribution. Let's add the skew $\lambda=0.1$ and shape $\eta=5$ parameters. When plugging in the all the parameters into the pdf (I take z which takes $\mu$ and $\sigma$ into account) I obtain $0.4832$, but when I use the
sgt package in R:
dsgt(0.001, 0.0001, 0.4, 0.1, p = 2, 5, mean.cent=TRUE, var.adj=TRUE)
Iobtain $1.075749$. Inputting the standardized variable z and $\mu = 0$ and $\sigma = 1$ into the above yields $0.5425881$.
Can you please explain the reason why the values are different?