Why Can't We (Directly) Simulate Data From a Semi-Parametric Distribution? I was reading this paper over here (https://journals.sagepub.com/doi/pdf/10.1177/1536867X1201200407). In the first paragraph on the second page, the authors write the following line:
"Furthermore, despite the Cox model (Cox 1972) being the most commonly used method of
survival analysis, it is not possible to simulate from a semiparametric model."
I was trying to understand this point - why is it not possible to simulate (data) from a semiparametric model?
After thinking about this question for a while, here is a possible answer I came up with. The "Probability Integral Transform" (https://en.wikipedia.org/wiki/Probability_integral_transform) allows us to simulate data from any probability distribution, assuming that we can simulate data from a Uniform Probability Distribution (I think another way to interpret this statement is : "points that were randomly sampled from any Cumulative Probability Distribution will be Uniformly Distributed" ). However, in a semiparametric model (i.e. a "semiparametric probability distribution") - some aspect of the model remains unspecified : this means that the "Probability Integral Transform" is inapplicable to a semiparametric model and results in there being no "straightforward" (i.e. direct) way to simulate data from a semiparametric model.
Is my reasoning correct?
Thanks!

A stronger statement is true. It's not possible to simulate from any model! Simulations must draw from a distribution, and a model is a set of distributions.
In order to simulate from a distribution in a model, that distribution must be specified. In a parametric model, this can be immediately accomplished by specifying parameter values. However in a semiparametric or nonparametric model, this is more challenging since an "infinite dimensional parameter" must be specified. In the case of a Cox model, the baseline hazard $\lambda: \mathbb{R}^+ \to \mathbb{R}^+$ must be specified.
In the article you mention, the authors are concerned with specifying a value of $\lambda$ which is realistic. Their remark is due to the completely unspecified nature of $\lambda$ not giving any clues for how to specify it.
