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Lecture Note: Econ 703 (week 2)

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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


Bias should be:



Balance Bias and Precision

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


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


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