Statistics 102H: MLE's (Statistical Inference by S. D. Lilvey

Administrivia

• data analysis due monday
• If you have any questions about how your data analysis, include them with what you turn in and I'll email you an answer.
• I wrote up details of what the rest of the projct should entail.

Example 1: binomial (cont'd)

Example 2: normal theory

• God it is ugly!
• Take logs. Ah, a bit prettier
• Ignore constants. Thats more like it!
• Take derivative
• Complete the square also: this is where leasts squares comes from

Calculating the MLE

• Book uses D for multi-derivative. we will stick with single derivatives in class.
• Two term taylor series: log likelihood = l(theta₀) + ...
• DEEP THEOREM (LeCam): This is a VERY good approximation. Almost exact up to several standard errors. (This is what I studied in my thesis. I didn't prove it. I just tried to understand it.)
• So maximum is at -b/2a. Or take dervatives. Or use Newton's method.

First useful example

• Suppose there are outliers in the data
• You can handle it using ideas of the first part of hte course
• Now lets do it carefully.
• Need a distribution that generates outliers: Cauchy
• Compute the score
• Notice: large values are down-weighted! Just like you do when you remove them!

Homework

• Read pages 68 - 73 of handout. Hopefully this time through they will make more sense. Start looking at the exercises at the end of the section.