I illustrate the decrease of the maximum likelihood with a very simple case. Suppose a random binary draw ("success" or not "success") is done N times. K times there is a success. Let P be the unknown probability of success. Of course, the probability distribution of K depends on P.

By definition, the likelihood of P is:

L(P) = P^K times (1-P)^(N-K)

The maximum L(P) is at P = K/N.

Now let the number of draws be doubled, to 2N draws. And if success occurs -coincidentally- 2K times, then the maximum likelihood is at P = 2K/2N = K/N, as before. And the likelihood is:

L(P) = P^(2K) times (1-P)^(2N-2K)

The maximum is at P = 2K/2N = K/N

This L(P) is smaller than the previous L(P).

Basically all this is about the following: P^2 is smaller than P (unless P is 0 or 1).

What I called N should be R, used by the author of the question. I am sorry.

Hi, Thank you. Could you please provide me the reference where I can learn more on this?@arie ten cate