6.6 Guyon (1982). This exercise looks in more detail at the unbiased sample covariance function used in Section 6.5. Suppose D is a rectangular lattice of length n in each direction so that it contains |D| = nd data sites. For simplicity, consider the lag h = [

1 0 ... 0

]

representing one step along the first coordinate axis. Assume the data xt, t ∈ D, come from a stationary random field with the known mean 0 and with the covariance function {????h ∶

h ∈ ℤd}. Consider the unbiased and biased sample covariance functions s

(u)

h = 1

|Dh|

∑

t∈Dh xtxt+h, s

(b)

h = 1

|D|

∑

t∈Dh xtxt+h.

These definitions are almost the same as in Eqs. (6.8)–(6.10), except they have been centered at 0 rather than the sample mean to make the calculations simpler.

(a) Show that the size of Dh is |Dh| = (n − 1)nd−1.

(b) Show that the unbiased covariance function s

(u)

h in (6.8) is unbiased, i.e.

E{s

(u)

h } = ????h.

(c) Hence, show that the biased sample covariance s

(b)

h in (6.10) is biased, E{s

(b)

h } = |Dh|

|D|

????h =

(

1 − 1 n

)

????h.

(d) Under mild regularity conditions (e.g. Section 5.5.3) it can be shown that s

(u)

h is asymptotically normally distributed for large n, nd∕2

(s

(u)

h − ????h) ∼ N(0, ????2 h)

for some variance ????2 h, say, not depending on n. Hence, deduce that asymptotic distribution of (6.10) is nd∕2

(s

(b)

h − ????h) ∼ N(−nd∕2−1

????h, ????2 h).

(e) In dimensions d = 1, 2, 3, deduce that for this distribution

(i) if d = 1, the bias n−1∕2????h is negligible in comparison to the standard deviation ????h.

(ii) if d = 2, the bias n0????h = ????h has the same order as the standard deviation ????h.

(iii) if d = 3, the bias n1∕2????h dominates the standard deviation