Please read this short snippet on importance sampling, and explain to me why both $F_{3n}$ and $F_{4n}$ are estimators of $F(x)$. I get the first two estimators, but why the last 2?
The context is that we estimate $F(x)$ for some distribution by instead simulating $x_1, ..., x_n$ from a distribution $P'$ with respect to which $F$ is absolutely continuous with density $p(x)$.
In the last two estimators the denominator is close to 1. Here's why: By construction, the connection between $F$ and the density $p$ is $$ F(x)=\int_{-\infty}^x p(t)dP'(t). $$ Now if $X_i$ has distribution $P'$ then the expectation of $L_i:=p(X_i)$ is $$ \int p(t)dP'(t) = F(\infty)=1. $$ So the quantity $\frac1n\sum L_i$ is equal to one on average.