Maximum likelihood estimator for binomial model

Question

The main problem I'm having is that I'm getting $\hat{p}=\frac{\bar{x}}{n}$, not $\frac{x}{n}$.

For some reason, many of the derivations of the MLE for the binomial leave out the product and summation signs. When I do it without the product and summation signs, I get $\frac{x}{n}$, but leaving them in I get the following:

$$L=\prod_{i=1}^{n}{n \choose x_i}p^{x_i}(1-p)^{n-k_i}$$ $$\ell=\sum_{i=1}^{n}ln{n \choose x_i}+ln(p)\sum_{i=1}^{n}x_i +ln(1-p)\sum_{i=1}^{n}(n-x_i)$$

$$\frac{\partial \ell}{\partial p}=\frac{\sum_{i=1}^{n} x_i}{p}-\frac{\sum_{i=1}^{n}(n-x_i)}{1-p}$$

$$\sum_{i=1}^{n}x_i-\sum_{i=1}^{n}x_ip=\sum_{i=1}^{n}(n-x_i)p$$

$$p=\frac{\sum_{i=1}^{n}x_i}{\sum_{i=1}^{n}(n-x_i)+\sum_{i=1}^{n}x_i}$$

$$p=\frac{\sum_{i=1}^{n}x}{n\cdot n}$$

$$p=\frac{\bar{x}}{n}$$

What am I doing wrong? Any help is appreciated.

Answer 1

You're confusing multiple binomial experiments with a single binomial experiment. Imagine you flip a coin 10 times and want to estimate the probability of Heads. If you observe 3 Heads, you predict $\hat{p} = \frac{3}{10}$. This is a sum of bernoullis, i.e. a single binomial experiment.

You're describing a sum of binomials, which corresponds to e.g. repeating your 10 flip experiment 5 times and observing:

$$X_1 = 3H$$ $$X_2 = 4H$$ $$X_3 = 2H$$ $$X_4 = 3H$$ $$X_5 = 4H$$

But if you think about it, doing 5 separate experiments in which you flip the coin 10 times is the same as doing a single experiment where you flip the coin 50 times. In this "condensed" experiment, we would estimate:

$$\hat{p} = \frac{16}{50} = \frac{\frac{16}{5}}{10} = \frac{\bar{x}}{n}$$

Answer 2

First of all, if I may be a bit pedantic: you don't calculate the MLE "for the Binomial" because we usually don't estimate distributions, but rather parameters of distributions. In your case you wish to estimate the "p parameter" of the Binomial, i.e. the probability of "success" in the underlying Bernoulli trials.

The estimate $\hat{p} = \frac{\bar{x}}{n}$ seems correct to me, because the MLE of the population mean $\mu$ is the sample mean $\bar{x}$, and the population mean of the Binomial is $\mu = n p$ where $n$ is the number of Bernoulli trials what I assume to be known in this case.