numeric vector of first-order inclusion probabilities of length N,
the population size, or n, the sample size
depending on the chosen method (see Details for more information)
method
string indicating the desired approximate variance estimator.
One of "Deville1", "Deville2", "Deville3", "Hajek", "Rosen", "FixedPoint",
"Brewer1", "HartleyRao", "Berger", "Tille", "MateiTille1", "MateiTille2",
"MateiTille3", "MateiTille4", "MateiTille5", "Brewer2", "Brewer3", "Brewer4".
sample
Either a numeric vector of length equal to the sample size with
the indices of sample units, or a boolean vector of the same length of pik, indicating which
units belong to the sample (TRUE if the unit is in the sample,
FALSE otherwise.
Only used with estimators of the third class (see Details for more information).
...
two optional parameters can be modified to control the iterative
procedures in methods "MateiTille5", "Tille" and
"FixedPoint": maxIter sets the maximum number
of iterations to perform and eps controls the convergence error
Details
The choice of the estimator to be used is made through the argument method,
the list of methods and their respective equations is presented below.
Matei and Tillé (2005) divides the approximated variance estimators into
three classes, depending on the quantities they require:
First and second-order inclusion probabilities:
The first class is composed of the Horvitz-Thompson estimator (Horvitz and Thompson 1952)
and the Sen-Yates-Grundy estimator (Yates and Grundy 1953; Sen 1953),
which are available through function varHT in package sampling;
Only first-order inclusion probabilities and only for sample units;
Only first-order inclusion probabilities, for the entire population.
Haziza, Mecatti and Rao (2008) provide a common form to express most of the
estimators in class 2 and 3:
var(t^HT)=∑i∈sciei2
where ei=πiyi−B^, with
B^=∑i∈sai∑i∈sai(yi/πi)
and ai and ci are parameters that define the different
estimators:
method="MateiTille5" [Class 3]
This estimator is defined as in method="MateiTille4", and the bi
values are defined as in method="MateiTille1"
Value
a scalar, the estimated variance
References
Matei, A.; Tillé, Y., 2005. Evaluation of variance approximations and estimators
in maximum entropy sampling with unequal probability and fixed sample size.
Journal of Official Statistics 21 (4), 543-570.
Haziza, D.; Mecatti, F.; Rao, J.N.K. 2008.
Evaluation of some approximate variance estimators under the Rao-Sampford
unequal probability sampling design. Metron LXVI (1), 91-108.
Examples
### Generate population data ---
N <- 500; n <- 50
set.seed(0)
x <- rgamma(500, scale=10, shape=5)
y <- abs( 2*x + 3.7*sqrt(x) * rnorm(N) )
pik <- n * x/sum(x)
s <- sample(N, n)
ys <- y[s]
piks <- pik[s]
### Estimators of class 2 ---
approx_var_est(ys, piks, method="Deville1")
approx_var_est(ys, piks, method="Deville2")
approx_var_est(ys, piks, method="Deville3")
approx_var_est(ys, piks, method="Hajek")
approx_var_est(ys, piks, method="Rosen")
approx_var_est(ys, piks, method="FixedPoint")
approx_var_est(ys, piks, method="Brewer1")
### Estimators of class 3 ---
approx_var_est(ys, pik, method="HartleyRao", sample=s)
approx_var_est(ys, pik, method="Berger", sample=s)
approx_var_est(ys, pik, method="Tille", sample=s)
approx_var_est(ys, pik, method="MateiTille1", sample=s)
approx_var_est(ys, pik, method="MateiTille2", sample=s)
approx_var_est(ys, pik, method="MateiTille3", sample=s)
approx_var_est(ys, pik, method="MateiTille4", sample=s)
approx_var_est(ys, pik, method="MateiTille5", sample=s)
approx_var_est(ys, pik, method="Brewer2", sample=s)
approx_var_est(ys, pik, method="Brewer3", sample=s)
approx_var_est(ys, pik, method="Brewer4", sample=s)