Lecture Note: Econ 703 (week 2)

Delta method (Univariate form) ​

Extended from CLT: from CLT we know $\sqrt{n}(X_n-\mu )\stackrel{d}{\to }N(0,var(X_n))$

by continuous mapping theorem we have $\sqrt{n}(g(X_n)-g(\mu ))\stackrel{d}{\to }N(0,var(X_n)(g^{\prime }(\mu {)}^2))$

for any function $g$ which has first derivative, and $g$ should be continuous and $var(X_n)(g^{\prime }(\mu {)}^2))$ 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 $\theta$ we wanna propose $\hat{\theta }$ to estimate it.

Two factors to consider:

• Closeness: $\hat{\theta }$ is close to $\theta$
• 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 $\mathbb{P}(X_i\in [0,1])$, and define loss function as $L(\mathbb{P},a)=(a-\theta (\mathbb{P}){)}^2$

A good estimator can be:

This estimator will be better than $\overline{X_n}$ only since it uses 0.5, which contains information for the distribution $[0,1]$

Closeness ​

Different notation for estimator $\hat{\theta }$ is close to $\theta$.

Generally speaking, the following three are correlated.

• consistent

  • $\hat{\theta }\stackrel{P}{\to }\theta$ (for all $\theta ,P,\theta (P)$)
  • say $\hat{\theta }$ is consistent for $\theta$

• unbiased and the limit

  • $\mathbb{E}(\hat{\theta })\stackrel{}{\to }\theta$
  • say $\hat{\theta }$ is unbiased and the limit

• asymptotically unbiased

  • $r_n(\hat{\theta }-\theta )\stackrel{d}{\to }Z$ if $Z=0$

Precision ​