Principles of point estimation

Question 1

Define statistical model

Answer 1

A statistical model for X is any family {P_ θ : θ ∈ Θ} of probability distributions P_ θ for the distribution of X. The set Θ is called the parameter space.

Question 2

What does it mean to say that a model is identifiable?

Answer 2

P_{ θ1} = P_{ θ2}

implies θ1 = θ2 for all θ1; θ2 ∈ Θ.

Question 3

What is an exponential family?

Answer 3

pg 5

Question 4

Define statistic and sampling distribution

Answer 4

A statistic is any function T(X) of the observed data X. The distribution of T(X) is called the sampling distribution of the statistic.

Question 5

Define bias and asymptotically unbiased

Answer 5

b( θ^) = E[ θ^] - theta

E_ θ[ θˆn] → θ as n → ∞

Question 6

What is the mean squared error of an estimator?

Answer 6

MSE(theta^) = E_ θ[ ( θ^ - θ)2] = var_ θ( θ^) + b_ θ( θ^)2

Question 7

Define standard error of an estimator

Answer 7

se(θ^) = sqrt(var(θ^))

Question 8

Define kth population moment and kth sample moment

Answer 8

μ_k = μk (θ) = E_θ[X^k]
μ^_k = (1/n) sum_{i=1}^n X_i^k

Question 9

What does it mean to say a statistic T(X) is sufficient for θ?

Answer 9

If the conditional distribution of X given T(X) does not depend on θ, i.e.
P(X ∈ A|T = t) = free of θ

Question 10

State the factorization criterion for sufficiency

Answer 10

pg 9

Question 11

What does it mean to say T(X) is minimal? State thm 1.2

Answer 11

A sufficient statistic T(X) is minimal if it is a function of every other sufficient statistic,
i.e. if T’(X) is also sufficient, then T’(X) = T’(Y) ⇒ T(X) = T(Y)
the theorem is on pg 11

Question 12

State the Rao-Blackwell theorem

Answer 12

pg 12