Suppose the cumulative distribution function $F$ is given but not invertible to use the inverse transform sampling technique (to compute $X=F^{-1}(Y)$). Do we have other alternative methods? I would appreciate to know the name of all possible methods…
Quantiles – Inverse Transform Sampling: Handling Non-Invertible CDF
cumulative distribution functionquantiles
Related Solutions
There's no closed form expression for the inverse cdf of a normal (a.k.a. the quantile function of a normal). It looks like this:
There are various ways to express the function (e.g. as an infinite series or as a continued fraction), and numerous approximations (which is how computers are able to "calculate" it).
Reasonably accurate approximations are tedious to write and not especially enlightening (except in so far as the general forms convey a little insight into common ways of approximating functions you can't easily obtain in closed form).
If you regard $\text{erf}^{-1}$ or $\text{erf}$ itself as a special function, then it could be written as a function of one of those, but one could as well call $\Phi^{-1}$ a special function and be done in one step.
Yes you can, but it is not very efficient. If $X$ follows standard normal distribution, then
$$ Y = F^{-1}_Y (\Phi(X)) $$
follows the distribution described by cumulative distribution function $F_Y$, where $\Phi$ is standard normal CDF.
See also How does the inverse transform method work? and Help me understand the quantile (inverse CDF) function threads, and my answer in Efficiently sampling a thresholded Beta distribution with the discussion in comments relating to expensiveness of inverse transform sampling.
Best Answer
In low dimensions a good alternative is to use rejection sampling from the pdf $f_X$ (in high dimensions this becomes very inefficient).
Say $f_X$ is your pdf for some random variable $X$, which you want to sample from in the interval $I=[x_\mathrm{min}, x_\mathrm{max}]$. Then you can draw samples $x_i$ uniformly from $I$ and accept/reject them with the probability $f_X(x_i)$, i.e. you draw another uniformly distributed random number $u_i\in[0, \max(f_X(x)|x \in I)]$ and if $u_i \lt f_X(x_i)$ you accept that sample point otherwise you reject it.