Lecture Note: Econ 703 (week 2)

Delta method (Univariate form) ​

Extended from CLT: from CLT we know

by continuous mapping theorem we have

for any function which has first derivative, and should be continuous and should be non-zero.

Former notes include core math tools for this course.

Notes after will introduce Theory of Statistics.

How to pick a good estimator ​

Let's say for we wanna propose to estimate it.

Two factors to consider:

• Closeness: is close to
• Precision: $\hat{\theta} $ should be precise

Closeness/Bias ​

Bias should be:

Precision ​

variance:

Balance Bias and Precision ​

Use the square loss form (MSE, mean square error):

Example ​

Suppose we know , and define loss function as

A good estimator can be:

This estimator will be better than only since it uses 0.5, which contains information for the distribution

Closeness ​

Different notation for estimator is close to .

Generally speaking, the following three are correlated.

• consistent

  • (for all )
  • say is consistent for

• unbiased and the limit

  • say is unbiased and the limit

• asymptotically unbiased

  • if

Precision ​