Chapter 4: Estimation

Question 1

State the sample mean distribution

Answer 1

X̄ ≈ N(E{X̄}, var{X̄}) by CLT

with E{X̄} = μ and var{X̄} = 1/N sum_-(N-1)^(N-1) (1 - |τ|/N) sτ (by diagonal sums or row sums of covariance matri)

Question 2

State the Cesaro summability theorem

Answer 2

If sum_τ=-∞ ^∞ sτ converges to a limit (or sum_τ=-∞ ^∞ |sτ| by triangle inequality) then N var{X̄} = sum_-(N-1)^(N-1) (1 - |τ|/N) sτ converges to the same limit

Question 3

What is the unbiased estimator of autocovariance

Answer 3

ŝτ (u) = 1/(N-|τ|) sum_t=1^(N-|τ|) (Xt - X̄)(Xt+|τ| - X̄)

and E(ŝτ (u)) = sτ - unbiased when μ is known

Question 4

What is the biased estimator of autocovariance

Answer 4

ŝτ (p) = 1/N sum_t=1^(N-|τ|) (Xt - X̄)(Xt+|τ| - X̄)

and E(ŝτ (p)) = (1-|τ|/N) sτ - biased

Question 5

What are the advantages of biased ŝτ (p)

Answer 5

• for many processes MSE(ŝτ (p)) < MSE(ŝτ (u)) (eg. PS2)
• as |τ| -> N-1, ŝτ (p) decreases nicely (knowing sτ->0 as |τ| -> ∞)
• {ŝτ (p)} is positive semidefinite (knowing {sτ} has to be)

Question 6

What is the periodogram spectral estimator

Answer 6

ŝ(p) (f) = sum_τ=-∞ ^∞ ŝτ (p) exp(- i 2π f τ) ŝτ (p) for |f| < 1/2 = 1/N | sum_t=1^N Xt exp(- i 2π f t) |^2

where ŝτ (p) = 1/N sum_t=1^(N-|τ|) Xt Xt+|τ| (biased estimator at μ=0

Question 7

What is the direct spectral estimator

Answer 7

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Question 8

State the Fejer’s kernel and its 5 properties

Answer 8

• F(f) -> N as f->0