## 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