Spatial Survey Data Modeling - Sampling Spatial Units for Agricultural Surveys - page 305

h i

ʸ opt ¼ ʳ

s X s βþ

X s β

t t

V ss V ss Y s

s Y s þ ʳ

; ð 12 : 8 Þ

where β¼ A s X s V 1 X s V 1

ss Y s with A s ¼ ss X s . In this case, the predictor is unbiased

so the prediction variance is

2 ¼ ʳ A 1 s X s t

E ξ ʸ opt ʸ Var ξ ʸ opt ʸ

V ss V 1 V ss V 1

¼

s X s

t

ʳ s

ss X s ss X s

ʳ s :

t
s V ss

V ss V 1

þ ʳ

ss V ss

ð 12 : 9 Þ

t
s Y s )

ʳ

Note that the BLUP is equivalent to a weighted sum of the sample units (i.e.,

plus a predictor of the weighted sum for the non-sample units (i.e.,

h i ). The optimum value of a is

s X s βþ V ss V 1

ss Y s X s β

ʳ

ʳ s :

V 1 X s A 1 s X s X s V 1

a opt ¼

ss V ss þ ss V ss

ð 12 : 10 Þ

If the sample and non-sample units are not correlated (i.e., V ss ¼

0 ), the BLU

predictor and the error variance are much simpler

ʸ opt ¼ ʳ

s X s β

t
s Y s þ ʳ

t

ð 12 : 11 Þ

and

¼ ʳ ʳ s :

Var ξ ʸ opt ʸ

t
s V ss þ

X s A s X s ð 12 : 12 Þ

This hypothesis of having no correlation between sample and non-sample units is
often reasonable in populations where single-stage sampling is appropriate and
units are not spatial and/or time series, i.e., there is no neighborhood effect.
In some circumstances, the BLUPs reduce to well-known estimators. For exam-
ple, consider the model Y k ¼ ʼ þ ʵ k with uncorrelated

ʵ k s an d ʵ k ð 0 ; ˃

2

Þ .

X s Y k =

β ¼

2 I, and

Consider Model ( 12.2 ) with β¼ʼ , X ¼ I, V ¼ ˃ Y s ¼ n ,

where I ¼ diag 11 ð ... 1 Þ N . In this case, the BLUP is

X s Y k þ X s Y s ¼

T 0 ¼

NY s : ð 12 : 13 Þ

The error variance of this estimator is

Var ξ T 0 N 2 1

2

T ¼ ð f Þ˃ = n ; ð 12 : 14 Þ

with f ¼ n / N . Note that Eq. ( 12.14 ) is also the design-based variance formula

for SRS.