Spatial Survey Data Modeling - Sampling Spatial Units for Agricultural Surveys - page 304

non-sampled N n units. Similarly, the vector of constant ʳ can be partitioned into

s t .
Now the estimated target can be expressed as

s t

ʳ ¼ ʳ ; ʳ

t Y ¼ ʳ t

s t Y s ʳ

s Y s þ ʳ

ð 12 : 3 Þ

t Y s Y s þ ʳ t

which is a realization of the random variable ʳ ¼ ʳ

s Y s . Because we

ʳ t y s after we have selected the sample, the estimation problem reduces to
predicting the unknown quantity ʳ

know

s y s (see Sect. 1.3 for a general formulation of

this question).
Therefore, a linear estimator of

t Y can be defined

ʸ ¼ ʳ

ʸ ¼

g s Y s ; ð 12 : 4 Þ

t is a vector of coefficients. In this way, the error of

where g s ¼ ð g 1 g 2 ... g n Þ

the estimator ʸ ¼

g s Y s is

Y s ʳ

ʸ ʸ ¼

g s Y s ʳ

t Y

g s ʳ

t
s

t

a t Y s ʳ

t

¼

s Y s ¼ s Y s ;

ð 12 : 5 Þ

t Y

where a ¼ ð g s ʳ s Þ . As demonstrated by Valliant et al. ( 2000 ), estimating ʳ

using g t Y s is equivalent to estimating

s Y s using a t Y s .
Generally, we assume that the covariates of Model ( 12.2 ) are known for each
unit in the population. In some particular cases, this assumption can be relaxed to
knowing only the population totals of the components of X (Valliant 2009 ).
The matrices X and V can be re-expressed

t

ʳ

;

X s
X s

V ss V ss
V ss V ss

X ¼ V ¼ ; ð 12 : 6 Þ

where X s is n

q , X s is ( N n ) q , V ss is n n , V ss is ( N n )

( N

n ), V ss is n

V ss . Finally, we assume that V ss is positive definite. ( N n ), and V ss ¼

ʸ ¼

g s Y s is unbiased (or, equivalently, prediction unbiased or

The estimator

¼

, if E ξ ʸ ʸ model unbiased) for 0, see Eq. ( 1.31 ).

The error variance (or the prediction variance) of ʸ ¼

ʸ under a model ξ

g s Y s under a model ξ is

then

2

E ξ ʸ ʸ : ð 12 : 7 Þ

The BLUP estimator under Model ( 12.2 ) is obtained by minimizing the error
variance in Eq. ( 12.7 ), that is (Royall 1976 ),