23.1 Estimates of R(r) 
For the discrete parameter process x; , 
R(r) = E [(X&:-4) X14 -—Mrr)] 
=F [X; Xr] 5 (2.20) 
when £; = Uy, = E[X;,] = 0. 
2.3.2 Unbiased Estimate 
Instead of averaging the ensemble, take the time average as 
N-Irl 
1 
FO) ae 2 (X,—X) Xr —X) (2.21) 
WHET 7a) () Se-t-teil et + (N-1). 
If we ignore the effect of estimating u by X,then R(r) is an unbiased estimate of 
R(r), because 
i 1 N-irl 
BLS) Ora > EX: -wXur-w) 
—Irl =i 
N-trl 
= SY Rr) = SRR) (2.22) 
N-Irl = N-tIrl 
2.3.3 Biased Estimate 
On the contrary, 
r N-—rl 
RO) = = XW) Xr — 1) (2.23) 
t=] 
is a biased estimate, because 
n | fe 1 Ir 
E[R =— = — (N-Irl) R(r) = —_— 
[R)] + Re "7 rl) Rr) (: re 
= R(r) ue R(r) 2.24 
a N ’ ( . ) 
11 
