I am reading the book Stochastic Simulation, Algorithms and Analysis by Asmussen. On page 39, when talking about inversion method for sampling a distribution, he gave an example:
an r.v. distributed as the overshoot X − a given X > a can be generated as $F^{-1} (U (1-F (a)) − a$,
where $U$ is a r.v. uniform in $[0,1]$.
But I think the distribution is $F^{-1} (U (1-F (a)) + F(a)) - a$. I was wondering which is correct and why?
Thanks and regards!
Let $Y=X$ with support on $[a,\infty)$. If $f_X$ is the p.d.f of $X$ and $f_Y$ is the pd.f. of $Y$ then
$f_Y(y) = \frac{f_X(y)}{1-F(a)}$ for $y>a$
Hence
$F_Y(y) = \frac{F_X(y)-F_X(a)}{1-F_X(a)}$
If $U=F_Y(Y)$ then as usual we have $U\sim Uni[0,1]$.
That is
$U=\frac{F_X(Y)-F_X(a)}{1-F_X(a)}$
Rearranging gives us
$Y=F_X^{-1}(U(1-F_X(a))+F_X(a))$
Thus the over shoot r.v. $X-a$ for $X>a$ can be generated as
$F_X^{-1}(U(1-F_X(a))+F_X(a))-a$
I agree with you.