6.15 Let Xn = (X1, . . . ,Xn) be generated from the AR(1) process Xt = Xt1 + ut, (|| < 1), (6.538) where

6.15 Let Xn = (X1, . . . ,Xn)′ be generated from the AR(1) process Xt = θXt−1 + ut, (|θ| < 1), (6.538)

where {ut} ∼ i.i.d. N(0, 1). Then, the spectral density function of {Xt}

is fθ(λ) = (1/2π)|1 − θeiλ|−2. Here we consider the discriminant problem described by the following categories

Π1 : f(λ) = fθ(λ), Π2 : f(λ) = fμ(λ), (μ ̸= θ). (6.539)

(i) For n = 512, give the discriminant statistics I(f : g) of (6.426) in an explicit form and simplify it as much as possible.

(ii) Let μ and θ be contiguous to each other with the relation of μ =

θ + 1/

√

512 and generate 100 times iterated samples of X512 for θ =

0.3, 0.6, 0.9, respectively. Then, calculate the ratio of I(f : g) > 0 in these 100 times experiments for each θ = 0.3, 0.6, 0.9.