3. Asymptotic Properties

Question 1

What does it mean that asymptotic properties pertain to large samples?

Answer 1

They follow certain behaviours and two important ones are consistency (estimator conerges to true value as sample size increase) and asymptotic normality (distribution of estimator converges to a normal distribution as sample size goes to infinity)

Question 2

Explain LLN and CLT

Answer 2

LLN tells us that the estimated mean (parameter) becomes closer to the true population mean as sample size increase. CLT tells us that the distribution of an estimator regardless of its shape or distribution will be normally distributed as the sample size goes to infinity.

Question 3

How do you calculate the empirical mean and what is its distribution it converges to?

Answer 3

Square root of T (m hat - m) ->
N(0, Σ)

Question 4

How is the empirical CDF calculated and what distribution does it converge to?

Answer 4

square root of T (F hat (ζi) - F(ζi​)) -> N(0,Fi​(ζi​)(1−Fi​(ζi​)))

Question 5

What is the kernel density estimator and the distribution it converges to?

Answer 5

Square root of Th (F hat (ζi) - F(ζi​)) -> N(0, fi(ζi) integral +- infinity K^2(u)du)

Question 6

What is the difference in the conversion rate of empirical mean and kernel density estimator?

Answer 6

The empirical mean has a parametric rate of conversion square root of T while the kernel density estimator converges at a rate of square root of T*h

Question 7

What are the differences and pros and cons of parametris vs nonparametrics estimators?

Answer 7

Parametric estimators assume a distribution and might therefore be misspecified, but they are in general more efficient than non-parametic if correct. Non-parametric on the other hand does not assume a distribution and therefore avoids the problems of misspecification but they might also loose some effiency.

Question 8

Explain the steps of hypothesis testing

Answer 8

1) State null and alternative hypothesis
2) Choose significance level α (usually 5%)
3) Calculate test statistic
4) Determine critical values and confidence intervals (critical value is found through tables based on the significance level)
5) Calculate P value (this is the area in the rejection region which is found through area tables associated to the critical values. You would find the critical values and then calculate the area (1 - area value in table) for one side (take x2 if its a two sided test)
6) Make decision: if the p value is smaller than significance level you reject and if larger or equal you fail to reject the null