Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 253

relevant to the uncertainty of the estimates, but there has been a relatively small
amount of research into variance estimators with nonresponse-adjusted weights
(Brick and Montaquila 2009 ).

10.5 Variance Estimation

The sampling error is the first and basic index of the quality of the estimates,
because it represents the uncertainty associated with an estimate as a result of the
observations being performed on a sample of n units rather than the whole popu-
lation of N units. It is typically measured in terms of the variance of the estimates
across samples. For a linear estimator, the estimation of this design-based variance
(where it is easy to calculate the inclusion probabilities) involves obtaining an exact
expression of the design unbiased estimators (see Eqs. ( 1.27 ) and ( 1.30 ) of Sect.
1.2 ).
However, this direct approach may not be straightforward for a complex sample
design, with multiple stages of sampling, stratifications at various stages, and
possibly varying inclusion probabilities π k s. Moreover, in survey sampling we

often require that parameters are estimated using a nonlinear function of the
observed data. Ratios, correlations, and regression coefficients are just a few
examples of such parameters. Exact expressions of the HT variance estimator can
only be used for parameters that are a linear function of the population data, for
examples, a total or a mean. These results are generally not available when dealing
with nonlinear functions of the observed data. Thus, it is quite difficult to find
unbiased estimates of their sampling error.
One interesting alternative is to approximate the estimator using a linear func-
tion of the observations, i.e., a series expansion up to the first term. Once linearized,
the appropriate variance for the specific sampling design can be applied to the
approximation. This leads to a biased, but typically consistent, estimator of the
variance of the nonlinear estimator (Binder 1983 ; Wolter 2007 ).
Now, assume that we are interested in a population parameter that is a generic
linear function of the observed total of g survey variables. We consider this
situation because it is directly relevant, and because many other nonlinear estima-
tors can be approximately reduced to this simple linear expression using a Taylor
series expansion.
Consider a nonlinear estimator

ʸ

that is explicitly a continuously differentiable

function of g population totals

ʸ ¼ f y 1 ; t y 2 ; ...; t y g : ð 10 : 25 Þ

We can approximate its sample variance using the Taylor series expansion up to the
first term, which can be expressed as a function of the estimates of the variance ( V )
and covariance ( C ˆ ) (S¨rndal et al. 1992 , p. 172). That is,