SU3 - Elements of Finite Sample Properties, Asymptotic Theory, Confidence Intervals and Hypothesis Testing

Question 1

An estimate is a ?, but an estimate is a ?

Answer 1

An estimate is a number, but an estimator is a random variable

Question 2

What are finite sample properties?

Answer 2

Properties of an estimator when the sample size is not arbitrarily large

Question 3

What is an unbiased estimator?

Answer 3

An estimator that does not over or under-estimate the true population parameter
if we could draw repeated random samples on Y from the population and then compute an estimate repeatedly, the average of these estimates should be close to θ.

Question 4

What is i.i.d?

Answer 4

Independently and identically distributed

Question 5

Var (Y bar) = variance / n

What happens to Var (Y bar) when sample size (n) increases?

Answer 5

Var (Y bar) (bias) goes towards 0.

Increasing sample size will lead to a decrease in sampling variance of the sample mean

Question 6

If there are two unbiased estimators, how to see which is more efficient?

Answer 6

Relative efficiency is used. Smaller variance = more efficient

Question 7

What is used to compare estimators that are bias?

Answer 7

Using Mean Squared Error.

Large MSE means variance or bias is large.

Question 8

what are asymptotic properties?

Answer 8

Statistical properties when n is very very large.

It enables us to answer the following questions:
Does the variance of some unbiased estimator decrease as n↑?
Does the estimator become more precise as n↑?
Does the bias of an estimator shrink towards zero as n↑?
What is the distribution of the estimator when n is large?

Question 9

𝑷(|𝜃𝑛−𝜃|>𝜀)→0 as 𝑛→∞

What is consistency?

Answer 9

if 𝜃n is consistent, it will become more improbable for the error |𝜃𝑛−𝜃| to exceed 𝜀
If (𝜃𝑛) is unbiased and 𝑉𝑎𝑟(𝜃𝑛)→0 as 𝑛→∞, then (𝜃𝑛) is consistent.

Question 10

what is plim?

Answer 10

“probability limit” of an estimator. It is the value that the estimator converges to in probability when the sample size becomes arbitrarily large.

Question 11

What is slutsky theorem?

Answer 11

If 𝑝𝑙𝑖𝑚(𝑇𝑛)=𝛼and 𝑝𝑙𝑖𝑚(𝑈𝑛)=𝛽, then
–
𝒑lim(𝑇𝑛+𝑈𝑛)=𝑝𝑙𝑖𝑚𝑇𝑛+𝑝𝑙𝑖𝑚𝑈𝑛=𝛼+𝛽
–
𝒑lim(𝑇𝑛𝑈𝑛)=𝛼𝛽
–
𝒑lim(𝑇𝑛/𝑈𝑛)=𝛼/𝛽provided 𝛽≠0.

Question 12

What is the central limit theorem?

Answer 12

the sample average, when standardized, has an asymptotically standard normal distribution, provided the population mean and variance exist

Question 13

what is weak law of large numbers?

Answer 13

𝒑lim(𝑌𝑛)=𝜇

In other words, LLN states that the sample average converges in probability to its expectation.

Question 14

Difference between consistency and asymptotic normality?

Answer 14

Consistency refers to the convergence of an estimator in probability to a single point. Asymptotic normality refers to the convergence of the distribution of an estimator to the normal distribution.

Question 15

What is an estimator?

Answer 15

an estimator of θ is a rule that assigns each possible outcome of the sample a value of θ. It is a function of random variables. Plugging the sample into an estimator gives us an estimate.

Question 16

If it’s unbiased, must it be consistent as well?

Answer 16

Unbiased estimators are not necessarily consistent. Likewise, biased estimators are not necessarily inconsistent.

Question 17

What is asymptotic normality?

Answer 17

If an estimator does not have a normal distribution when the sample size is finite, but has a distribution that resembles the normal as the sample size
increases, we say that it is asymptotically normal.

Question 18

Between unbiased estimators, which is most efficient?

One with smallest variance or one with biggest variance?

Answer 18

Smallest

Question 19

Reason why MSE can increase?

Answer 19

1) large variance

2) large bias

Question 20

A positively biased estimator always produces estimates that are greater than the parameter that we are interested in estimating. Yes or no?

Answer 20

Not always, only on average

Question 21

As the sample size increases, the variance shrinks towards zero When the sample size is infinitely large, the variance is zero. Why?

Answer 21

This makes sense because if you have all the sample, the estimator should have no uncertainty

Question 22

What is the sampling distribution?

Answer 22

the distribution of the estimator

Question 23

Why is an estimator a random variable?

Answer 23

This is because it can take on a range of values depending on the sample.

Question 24

What is Chebyshev’s inequality?

Answer 24

If 𝜃𝑛 is unbiased and 𝑉𝑎𝑟𝜃𝑛→0as 𝑛→∞, then 𝜃𝑛 is consistent.

Question 25

What is the formula for MSE?

Answer 25

MSE(Y) = Bias(Y)^2 + Var(Y)

Bias(Y) also known as expected value

Question 26

Three things to construct a confidence interval?

Answer 26

• Point estimate,
• Standard deviation (or standard error, which is an estimate of the standard deviation),
• Critical values.

Question 27

If our test statistic has a p-value of 0.02, what does it mean?

Answer 27

There is only a 2% chance that our null hypothesis is true. The smaller the p-value, the less likely our null is true.

This also means there is a 2% chance of rejecting the null wrongly.

Question 28

if p > a, do we reject the null?

Answer 28

we do not reject the null hypothesis based on our chosen level of significance 𝛼

Question 29

When is the t-test/z test used to find a confidence interval?

Answer 29

Standard deviation known > z test

Standard deviation unknown > t test

Question 30

What are the two types of errors in hypothesis testing?

Answer 30

Type 1: rejecting null hypothesis H0 when H0 is true

Type 2: rejecting alternative hypothesis H1 when H1 is true

Question 31

What is the power of the test?

Answer 31

Probability of accepting H1 when H1 is true

Question 32

What are the steps to conduct a hypothesis test?

Answer 32

1) specify the null and alternative hypothesis
2) construct the test statistic
3) specify the level of significance
4) based on 3, state the critical values or rejection region
5) reject the null hypothesis if p<a></a>