Maximum Likelihood Estimate for an Unknown Distribution

Question

Suppose that $X_1,\dots,X_n$ are i.i.d. random variables having a CDF (cumulative distribution function) $F$. For each fixed $x$, I am asked to determine the maximum likelihood estimate of $F(x)$.

I am having difficulty understanding what the question is asking. For one, I do not see how to recover a probability density (or mass) function from $F$, let alone how to maximize the likelihood function without a given parameter space.

If anyone could shed some light on this it would be greatly appreciated.

Edit

I now understand that I should take the MLE will be the empirical distribution. However, I am still having difficulty proving this directly from the definition.

Answer 1

Here the CDF is the thing you are estimating. You can think of its values as an infinite number of parameters (in a constrained space that says they need to comprise a right-continuous, nondecreasing function, between zero and yada yada yada).

Let's say we get $X_1=3$ and $X_2=4.$ We need to find the CDF that maximizes the probability of this data. It's pretty clear that anything other than an atom at $3$ and an atom at $4$ is a waste of real estate. Let $p$ be the mass at $3$ and $(1-p)$ the mass at $4$. Then we want to maximize $p(1-p),$ so we get $p=1/2$ (What else could it have been?)

This generalizes to putting $1/n$ mass at each of the points $X_1,\ldots, X_n.$

Answer 2

$F(x) = P(X \leq x) = p_x$. The problem asks you to find a MLE for $p_x$ regardless of what $x$ is.

You know how to find MLEs for the $p$ of a Bernoulli distribution, correct? Can you find a sequence of Bernoulli random variables, $I_1, \ldots, I_n$, from $X_1, \ldots, X_n$? Sure you can, use an indicator function:

$$ I_j = \begin{cases} 1 & \text{ if } X_j \leq x \\ 0 & \text{ otherwise}\end{cases}$$

So then what is $P(I_j = 1)$? How does it relate to $F(x)$? Use $I_1, \ldots, I_n$ to get the MLE of $p_x$.