The image below displays an approach of using an auxiliary variable to map the parametric curves of a standard normal pdf and cdf.

In Equation (1), z as r.v. is clearly one-dimensional. However, after defining z as an auxiliary variable, it is 2-dimensional

I'm a little bit confused on why they insert the auxiliary variable into the pdf function $f_Z$.

Why is the parametric equation defined as $f_Z(z, f_Z(z))$ and not $(z, f_Z(z))$?

The authors later use this approach to define parametric curves of the pdf after transforming quantiles of the standard normal distribution with the tukey g- and h- transformation, because the pdf does not exist in closed form.

enter image description here

Reference: Headrick, T. C., Kowalchuk, R. K., & Sheng, Y. (2008). Parametric probability densities and distribution functions for Tukey g-and-h transformations and their use for fitting data. Applied Mathematical Sciences, 2(9), 449-462.

  • $\begingroup$ I'm not familiar with this particular problem. In general, you add auxiliary variables to simplify the likelihood or so you can sample from a simple conditional distribution. Maybe you can see if adding auxiliary variables in your example simplifies the likelihood or makes it easier to maximize/sample from. $\endgroup$
    – Eli
    Commented Jul 13, 2022 at 12:48
  • $\begingroup$ How does this explain the function definition of $f$ in (3)? $\endgroup$ Commented Jul 14, 2022 at 8:30
  • $\begingroup$ I don’t know. Like I said, I’m not familiar with this topic. Just thought that might point you in the right direction. $\endgroup$
    – Eli
    Commented Jul 15, 2022 at 13:25


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