Estimators Asymptotics and Comparisons

Question 1

What does a sequence of Bin(n, p/n) RVs coverge too

Answer 1

A sequence of Bin(n, p/n) converges to a

Poisson RV as n increases

Question 2

Name two types of convergence

Answer 2

Convergence in probability and in distribution

Question 3

Which type of convergence is stronger

Answer 3

Convergence in probability is stronger than convergence in distribution so convergence in probability implies convergence in distribution

Question 4

Which convergence implies another

Answer 4

Convergence in probability implies convergence in distribution

Question 5

What inequality so we use to show convergence in probability

Answer 5

Chebyshevs

Question 6

State the WLLN

Answer 6

Let X1, X2, . . . be a sequence of IID random variables with E [Xi ] = μ and Var (Xi ) = σ^2 .

Then Xbar_n)=1/n(sum of Yis up to n)
is convergent in probability to μ

Question 7

What is a consistent estimator

Answer 7

Means the estimator converges in probability to the population parameter. Consistency is generally regarded as bare minimum property an estimator
should satisfy.
• Note: A biased estimator may still be consistent

Question 8

Is the sample mean consistent

Answer 8

Yes, consistent to/for μ

Question 9

Define the variance of an estimator

Answer 9

The variance of an estimator, already defined, measures how variable the
values of the estimator are

Question 10

Define the standard error

Answer 10

Let ˆθ be an estimator for θ. The standard error of θestimator is defined as the square
root of its variance

Question 11

What is the MSE of an estimator in terms of other measures

Answer 11

MSE = Var + Bias^2

Question 12

If an estimator is unbiased what can we say about the MSE

Answer 12

MSE= Variance of estimator

Question 13

If we sum m exponential RVs with rate lamda what is their distribution

Answer 13

Gamma(m, lamda)