### Maximum Likelihood Estimator Examples

To judge a ML estimator is efficient (CRLB), we can use the following theorem.

## Example 1: 丟銅板 (Binomial distribution).

### 如果 flip a coin 10 times and get 7 heads.  What’s the ML estimation of the probability of heads?

The ML estimator is unbiased!

Is it efficient?

$\frac{n}{p}-\frac{N-n}{1-p} = \frac{n-Np}{p(1-p)}$

I(p) = N/p(1-p)  g(n) = n/N  ==>  It is efficient

where $E(n^2)=N^2 p^2 + Np(1-p)$

ML estimator is efficient!

Apply the other theory to check!!

## Example 2b: Multivariate Poisson random variable arrivial rate

It seems there is no close form for the above equation.  We can use iterative algorithm to get the answer.  It turns out to be EM (expectation maximization) algorithm; or Richardson-Lucy algorithm in optical literature.

## Example 3a:  Mean is unknown, variance is known

score = 1/sig^2 (sum(xj) – Nu) = N/sig^2( sum(xj)/N – u)

I = N/sig^2  g = sum(xj)/N

It is unbiased and efficient!!

## Example 3b: Mean and variance are unknowns

The mean is unbiased but the variance is biased!!  Therefore, the ML estimator is biased!!   But it is asympotically unbiased!

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