How do you define $F(X_i-)$, where $F$ is a cumulative distribution function? [closed]

Question

Let's say $X_1, X_2, \dots, X_n \stackrel{\mathrm{iid}}{\sim} F$, then the empirical likelihood function is

$$L(F) = \prod_{i=1}^n (F(X_i) - F(X_i-)).$$

How do you define $F(X_i-)$. Is it: $$\lim_{x\to X_i^-} F(x)$$

Any info appreciated.

Answer 1

Yes, you are right.

Recall that CDFs are right-continuous. So the point of taking the limit from the left is to account for any jumps (i.e. think of a discrete probability distribution). If there is a point mass at some point $x'$, then $F(x') - F(x'-) > 0$. (Draw a picture!)

Note that this definition is true in general for $x' \in \mathbb{R}$. In your example, your empirical likelihood function is a function of the random variables, and hence it is also a random variable.