\&r(X2 ) 2 var( i fi )+var(Ai ) 2 var(Jf2 ) 
var(Jf#2 ) 
[var(^i) + var(^2)] 2 
var(2ii ) var(^2 ) 
(17) 
[var(Xi) + var(j2 )], 
Gz var(^i). 
(18) 
The mean (Xi+X2)/2, is equivalent to equation 
(13) with Oi = 1/2. The mean is unbiased, as 
shown in (15). Variance of the mean is 
(l/2) 2 varUi) + (l/2) 2 var(l2 ) = 
variai) v&r(X2) [var(Ai) - var(^2)] 2 
+ (19) 
lvar(^i ) + var(2fe)] 4 [var(li) + var(Jz)]. 
Under heterogeneity, var(fi ) does not equal 
var(l2)t and the variance of the mean in (19) is 
larger than the variance of the composite 
estimate in (17). Therefore, the composite 
estimate is more efficient than the mean. Under 
homogeneity, var(Jii) equals va.r(X2), and the 
variance of the mean in (19) is identical to the 
variance of the composite estimate in (17). 
5.3 Estimated stratum status, all sample units 
The composite estimate of stratum status in (13) 
uses only the sample units from two cells. 
However, there are n units sampled in the 
stratum, which can be incorporated by 
sequentially applying the composite estimator. 
The composite estimate from the first two cells 
x%2 in (13) is combined with the estimate from 
the third sample unit /3 using the variance 
v&r(X%2) of the composite in equation (16): 
Xt3 = (1-&)I#2 + 03*3, (20) 
G3 = var(*#2) / [var(*#2) + var(*3)], (21) 
var(**s) = (I-G3) 2 var(A*2) + ( <33 ) 2 var(*3). (22) 
Then, composite estimate *#3 is combined with the 
estimate from the fourth sample unit. This is 
repeated until estimates from all n cells are 
combined into a single estimate for status of the 
entire stratum. The sequence is inconsequential. 
As in equation (15), the final composite estimate 
is unbiased. As in (17) and (19), the composite 
estimate using all n cells has smaller variance 
than the mean, and the composite estimate is 
identical to the mean under homogeneity. 
5.4 Lack of independence among sample units 
Sequential application of the composite estimator 
will produce a minimum variance estimate if all 
random deviations ft! are mutually independent. 
However, spatial autocorrelation and other 
factors (Section 3.6) can cause dependence among 
deviations. Consider two sample unit estimates 
(Xi,X2) that are combined to produce an estimate 
of stratum status, i.e., equation (13). If 
estimates Ai and A2 are dependent, variance of 
the combined estimate A#2 in equation (13) is 
var(A#2) = (1-Oz) 2 var(Ai) + (Gz) 2 varili) 
+ 2 (l-&)(Gz) cov(Ai,j2), (23) 
where cov(*i,*2) is the covariance between sample 
unit estimates *1 and *2. If these two estimates 
are combined using the composite estimator, as in 
equations (13), (14), and (16), then the 
composite estimate is unbiased, as shown in 
equation (15), even if the errors are correlated. 
Variance of the mean will be larger than the 
variance of the composite estimate when estimates 
Xi and *2 are not independent (Section 5.5), 
unless there is a strong negative covariance 
between estimates Xi and *2 such that 
variai )+var(*2) is less than -2cov(^i, *2 ). 
However, cov(*i,*2) is frequently positive. 
Spatial patterns in landscapes tend to produce 
positive covariances between proximate sample 
units, as will propagated errors for stratum 
level calibration models (Section 3.6) and 
prediction models (Section 7.2). 
The composite estimator can be sequentially 
applied, as described in (20) through (22). 
However, the final composite estimate will not be 
a minimum variance estimate. A more general 
formulation of the composite estimator (Maybeck 
1979) will have minimum variance with correlated 
errors using the following weight in (13): 
var(*i) + cov(*i,*2) 
Qi (24) 
var(*i) + var(*2) + 2 cov(*i,*2). 
However, this is unsatisfactory for sequential 
composite estimation with all n estimates; the 
final estimate depends upon the sequence in which 
the n estimates are combined. Section 5.5 
presents a possible solution, using a vector 
weight analogous to the scalar weight in (24). 
5.5 Vector weighting in the composite estimator 
Let X=(Ji ¡*2 ! ... ¡Jn ) ’ be the (n x 1) vector of 
estimates from n sample units; W=( №!№!•••! Mi) ’ 
be the (n x 1) vector of deviations of n sample 
units from the stratum status X\ C be the (n x n) 
estimated covariance matrix for sample unit 
deviations, E[W W’] = C, where iith element of C 
is var( Xi) and the ijth element is cov(Xi , Xi ); 
and 1 be a (n x 1) vector of ones. Vector 
representation of equation (1) is 
l = 1 X + W. (25) 
For n=2, the unbiased composite estimator in (13) 
and (16) can be expressed in matrix from as 
I#2 = G’ X, (26) 
var(Af#2) = G' C G, (27) 
where the (2x1) vector G = [(1-02)102]’ 
contains the weight applied to each estimate Afi