Estimators

Question 1

If E(X) is known or assumed to be μ, what is E(X̄)?

Answer 1

μ (no further assumptions needed)

Question 2

What is the method of moments?

Answer 2

Using an estimator derived from a sample as the estimated population parameter (also called analogue estimation)

Question 3

How is an estimator usually denoted?

Answer 3

A variable with a hat

Question 4

What is required to make an interval estimate?

Answer 4

Knowing the sampling distribution of the estimator

Question 5

What is the 95% confidence interval for the estimator for the mean of a random sample from a normal distribution?

Answer 5

The CI for sqrt(n)(mu hat_n - mu)/σ is [-1.96, 1.96] which can be rearranged to get the CI for the estimator alone

Question 6

What is a pivotal quantity?

Answer 6

A function of the data and some unknown parameters whose distribution is known

Question 7

What is the bias of an estimator?

Answer 7

The difference between its expected value and true value
bias(theta hat) = E(theta hat) - theta

Question 8

Why is (mu hat)_n an unbiased estimator of mu?

Answer 8

If X_i for i = 1, …, n are IID as N(μ, σ²) then (Mu hat)_n = 1/n * Σ_i=1ⁿX_i ~ N(μ, σ²/n) therefore also has expectation mu and is therefore unbiased

Question 9

What is an example of an unbiased but implausible estimator?

Answer 9

(N hat) = 2X - 1 for a distribution with P(X = x) = 1/N for x = 1, 2, …, N
The expectation of the distribution is (N+1)/2 so the estimator is unbiased but a sample may show that N >= a and the estimator could still give (N hat) < a which would clearly be incorrect

Question 10

What is an efficient estimator?

Answer 10

The unbiased estimator with the lowest variance

Question 11

What is the mean squared error of an estimator?

Answer 11

MSE(theta hat) = E((theta hat - theta)² = Var(theta hat) + (bias(theta hat))²
Offers tradeoff between bias and variability

Question 12

What is the sample variance from a random sample with E(X_i) = μ and Var(X_i) = σ²?

Answer 12

S_n²(X₁, X₂, …, X_n) = 1/n * Σ_i=1ⁿ(X_i - X̄_n)² with X̄_n = 1/n * Σ_i=1ⁿX_i so S_n² = 1/n * (Σ_i=1ⁿX_i² - nX̄_n²)

Question 13

What is E(X_i²)?

Answer 13

Var(X_i) + (E(X_i))² = σ² + μ²

Question 14

What is E(S_n²)?

Answer 14

E(X_i²) - E(X̄_n²) = σ² + μ² - (σ²/n + μ²) = σ²(n-1)/n

Question 15

Is the sample variance an unbiased estimator?

Answer 15

No because it’s expectation is not exactly equal to σ², however as n approaches infinity the bias approaches zero (so asymptotically unbiased)

Question 16

What is an unbiased estimator for sample variance?

Answer 16

n/(n-1) * S_n² sometimes denoted S_n-1²