From equations (9), (10), and (11), the variance 
of the estimated status of sample unit i is 
Ht (1-Ht) H 2 + Ho (1 -Ho) ( 1-Yt ) 2 
var(li ) (12) 
Mt Mo 
The estimated variance var( AT ) in equation (12) 
is nearly zero when there is near perfect 
classification accuracy (Ht -> 1 and Ho ->0), or 
when there are a large number of plots (Mt and 
Mo) used to estimate the conditional 
misclassification probabilities Ht and Ho. 
3.5 Heterogeneity caused by calibration 
The variance var(AT) of estimated status AT for 
each sample unit i in (12) differs among sample 
units because the remotely sensed measurement IT 
will differ among sample units. Measurement 
error can produce heterogeneous variance among 
sample units. This is one motivation for 
composite estimation in Section 5. 
3.6 Lack of independence caused by calibration 
Propagated errors from the stratum level 
calibration model will cause a lack of 
independence among the estimated status of sample 
units within that stratum. From equation (8), 
the covariance between the unbiased estimates AT 
and Xj of status of sample units i and J is 
cov(Ii.Ij) = E[{ Jf Yi+Jo (1-Fi)}{Jf Yi+Jo (1-Tj )}]. 
E[Jf 2 ] = var(Wf) in (10), and E[Jo 2 ] = var(Wb) in 
(11). Since Jt and Jo are independently 
estimated from different plots, E[Jf Jo] = 0, and 
cov(AT , Aj ) = var(№)FiFj + var( Ho ) (1-Fi ) (1-Fj ). 
Errors propagated from the stratum level 
calibration model will cause dependence among 
estimated status of sample units within that 
stratum. This covariance will be positive, and 
will differ among sample units because the 
remotely sensed estimates IT and Yj differ. This 
is one motivation for Section 5.4, which 
considers dependent sample units. 
4. COMPOSITE ESTIMATORS 
Composite estimators are often used in forestry 
(Grégoire and Walters 1988), including sampling 
with partial replacement. A composite estimator 
(Fig. 1) combines two estimates, each of which is 
weighted inversely proportional to its variance 
(or in the multivariate case, its covariance 
matrix). The weights can be derived using 
maximum likelihood, minimum variance, or Bayesian 
theory. If all assumptions are reasonable, error 
in a composite estimate is less than error in 
either prior estimate. 
5. ESTIMATED STRATUM STATUS 
If the variances var(ft) for the calibrated 
estimates of sample unit status were homogeneous 
for the entire stratum, then the mean of the n 
sample units would be the minimum variance 
unbiased estimate of the stratum status. If 
var( ft ) is heterogeneous, the sample mean would 
Two independent Composite 
estimates estimate 
Fig. 1 Probability densities for two independent 
estimates. These are weighted inversely 
proportional to their variances, and combined 
into a single, more precise, composite estimate. 
remain unbiased, but would not be the minimum 
variance estimate. As an alternative, the 
composite estimator (Maybeck 1979) could weight 
the estimated status AT of each sample unit 
inversely proportional to its variance. 
5.1 Estimated stratum status, two sample units 
Estimates of stratum status from two sample units 
(AT, X2 ) can be combined using the composite 
estimator into an estimate Xt2 of stratum status 
X. The composite estimator uses weights, (1-&) 
and Oi, that are inversely proportional to the 
variances of estimates Ai and AT : 
A#2 = (l-Æ)Ii + 02 X2, (13) 
<32 = var( AT ) / i var ( Ai ) + var( AT ) ]. (14) 
Estimated variances var(•) are used rather than 
their true, but unknown variances. In this case, 
estimate A#2 in (13) would not be optimal with 
respect to minimum variance (Maybeck 1979). 
However, estimate A#2 will be nearly optimal if 
var(ATi) and var( AT ) are accurate estimates, and 
assuming estimates Ai and AT are independent. 
Even if estimate A#2 is suboptimal, A#2 is 
unbiased; from equations (1) and (13), 
I#2 = (1 -&)(A + Wi) + &U+ ft), 
= X + (1-02 ) Wi + 02ft. (15) 
Since random deviations ft and ft have expected 
values of zero in (1), and O2 is a nonrandom 
constant, then E[A#2] = A in (15), and estimate 
A* 2 is unbiased. Variance of the composite 
estimate using the first two sample units is 
var(A#2) = (1-O2) 2 var( Ai ) + (O2) 2 var(AT). (16) 
5.2 Efficiency of the composite estimate 
The statistical efficiency of the composite 
estimator is compared to the efficiency of the 
mean of estimates Ai and A2. Substituting O2 
from equation (14) into equation (16), variance 
of the composite estimate is given in (17) 
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