ST 733 Practice Problems for Midterm Exam
Spring 2005
Midterm Exam on March 15, 2005
1. Suppose {Z(s); s∈ D} is a spatial process defined on some domain Dsuch that
E[Z2(s)] <∞for all s∈ D. Let the mean and covariance functions be defined
as,
µ(s) = E[Z(s)] and C(s, h) = Cov[Z(s), Z (s+h)],
respectively.
(a) What conditions on µ(s) and C(s, s +h) are needed such that the spatial
process is 2nd order stationary, i.e., weakly stationary?
(b) Let γ(s, h) = 1
2V ar[Z(s)−Z(s+h)]2denotes the semivariance function of
the spatial process. Write γ(s, h) as a function of C(s, h). Can you write
C(s, h) as a function of γ(s, h)?
(c) What conditions on µ(s) and C(s, s +h) are needed such that the spatial
process is intrinsic stationary?
(d) Show that weak stationarity implies intrinsic stationarity. Does intrinsic
stationarity imply weak stationarity?
(e) What conditions on C(s, s +h) are needed such that the spatial process is
weakly stationary and isotropic?
(f) What conditions on γ(s, s +h) are needed such that the spatial process is
intrinsic stationary and isotropic?
2. Let Z1=Z(s1), Z2=Z(s2), . . . , Zn=Z(sn) be a sample from a weakly
stationary spatial process {Z(s); s∈ D} having covariance function C(h) =
Cov[Z(s), Z (s+h)],∀s, s +h∈ D.
(a) Show that the semivariance function, γ(h) = 1
2V ar[Z(s)−Z(s+h)]2,∀s, s+
h∈ D is given by γ(h) = C(0) −C(h).
(b) Obtain an unbiased estimate of the semivariance function γ(h).
(c) Obtain an unbiased estimate of the covariance function C(h) given that
and unbiased estimate ˆ
C(0) of C(0).
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