1. week 41 How to find estimators? There are two main methods for finding estimators: 1) Method of moments. 2) The method of Maximum likelihood. Sometimes the two methods will give the same estimator.

2. week 42 Method of Moments The method of moments is a very simple procedure for finding an estimator for one or more parameters of a statistical model. It is one of the oldest methods for deriving point estimators. Recall: the k moment of a random variable is These will very often be functions of the unknown parameters. The corresponding k sample moment is the average. The estimator based on the method of moments will be the solutions to the equation μ k = m k.

3. week 43 Examples

4. week 44 Maximum Likelihood Estimators In the likelihood function, different values of θ will attach different probabilities to a particular observed sample. The likelihood function, L(θ | x 1, …, x n ), can be maximized over θ, to give the parameter value that attaches the highest possible probability to a particular observed sample. We can maximize the likelihood function to find an estimator of θ. This estimator is a statistics – it is a function of the sample data. It is denoted by

5. week 45 The log likelihood function l(θ) = ln(L(θ)) is the log likelihood function. Both the likelihood function and the log likelihood function have their maximums at the same value of It is often easier to maximize l(θ).

6. week 46 Examples

7. week 47 Important Comment Some MLE’s cannot be determined using calculus. This occurs whenever the support is a function of the parameter θ. These are best solved by graphing the likelihood function. Example:

8. week 48 Properties of MLE The MLE is invariant, i.e., the MLE of g(θ) equal to the function g evaluated at the MLE. Proof: Examples: