approx_var_est {UPSvarApprox} | R Documentation |
Approximated Variance Estimators
Description
Approximated variance estimators which use only first-order inclusion probabilities
Usage
approx_var_est(y, pik, method, sample = NULL, ...)
Arguments
y |
numeric vector of sample observations |
pik |
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 |
... |
two optional parameters can be modified to control the iterative
procedures in methods |
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 packagesampling
;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:
\widehat{var}(\hat{t}_{HT}) = \sum_{i \in s}c_i e_i^2
where e_i = \frac{y_i}{\pi_i} - \hat{B}
, with
\hat{B} = \frac{\sum_{i\in s} a_i (y_i/\pi_i) }{\sum_{i\in s} a_i}
and a_i
and c_i
are parameters that define the different
estimators:
-
method="Hajek"
[Class 2]c_i = \frac{n}{n-1}(1-\pi_i) ; \quad a_i= c_i
-
method="Deville2"
[Class 2]c_i = (1-\pi_i)\Biggl\{ 1 - \sum_{j\in s}\Bigl[ \frac{1-\pi_j}{\sum_{k\in s} (1-\pi_k)} \Bigr]^2 \Biggr\}^{-1} ; \quad a_i= c_i
-
method="Deville3"
[Class 2]c_i = (1-\pi_i)\Biggl\{ 1 - \sum_{j\in s}\Bigl[ \frac{1-\pi_j}{\sum_{k\in s} (1-\pi_k)} \Bigr]^2 \Biggr\}^{-1}; \quad a_i= 1
-
method="Rosen"
[Class 2]c_i = \frac{n}{n-1} (1-\pi_i); \quad a_i= (1-\pi_i)log(1-\pi_i) / \pi_i
-
method="Brewer1"
[Class 2]c_i = \frac{n}{n-1}(1-\pi_i); \quad a_i= 1
-
method="Brewer2"
[Class 3]c_i = \frac{n}{n-1} \Bigl(1-\pi_i+ \frac{\pi_i}{n} - n^{-2}\sum_{j \in U} \pi_j^2 \Bigr) ; \quad a_i=1
-
method="Brewer3"
[Class 3]c_i = \frac{n}{n-1} \Bigl(1-\pi_i - \frac{\pi_i}{n} - n^{-2}\sum_{j \in U} \pi_j^2 \Bigr); \quad a_i = 1
-
method="Brewer4"
[Class 3]c_i = \frac{n}{n-1} \Bigl(1-\pi_i - \frac{\pi_i}{n-1} + n^{-1}(n-1)^{-1}\sum_{j \in U} \pi_j^2 \Bigr); \quad a_i=1
-
method="Berger"
[Class 3]c_i = \frac{n}{n-1} (1-\pi_i) \Biggl[ \frac{\sum_{j\in s} (1-\pi_j)}{\sum_{j\in U} (1-\pi_j)} \Biggr] ; \quad a_i=c_i
-
method="HartleyRao"
[Class 3]c_i = \frac{n}{n-1} \Bigl(1-\pi_i - n^{-1}\sum_{j \in s}\pi_i + n^{-1}\sum_{j\in U} \pi_j^2 \Bigr) ; \quad a_i=1
Some additional estimators are defined in Matei and Tillé (2005):
-
method="Deville1"
[Class 2]\widehat{var}(\hat{t}_{HT}) = \sum_{i \in s} \frac{c_i}{ \pi_i^2} (y_i - y_i^*)^2
where
y_i^* = \pi_i \frac{\sum_{j \in s} c_j y_j / \pi_j}{\sum_{j \in s} c_j}
and
c_i = (1-\pi_i)\frac{n}{n-1}
-
method="Tille"
[Class 3]\widehat{var}(\hat{t}_{HT}) = \biggl( \sum_{i \in s} \omega_i \biggr) \sum_{i\in s} \omega_i (\tilde{y}_i - \bar{\tilde{y}}_\omega )^2 - n \sum_{i\in s}\biggl( \tilde{y}_i - \frac{\hat{t}_{HT}}{n} \biggr)^2
where
\tilde{y}_i = y_i / \pi_i
,\omega_i = \pi_i / \beta_i
and\bar{\tilde{y}}_\omega = \biggl( \sum_{i \in s} \omega_i \biggr)^{-1} \sum_{i \in s} \omega_i \tilde{y}_i
The coefficients
\beta_i
are computed iteratively through the following procedure:-
\beta_i^{(0)} = \pi_i, \,\, \forall i\in U
-
\beta_i^{(2k-1)} = \frac{(n-1)\pi_i}{\beta^{(2k-2)} - \beta_i^{(2k-2)}}
-
\beta_i^{2k} = \beta_i^{(2k-1)} \Biggl( \frac{n(n-1)}{(\beta^(2k-1))^2 - \sum_{i\in U} (\beta_k^{(2k-1)})^2 } \Biggr)^{(1/2)}
with
\beta^{(k)} = \sum_{i\in U} \beta_i^{i}, \,\, k=1,2,3, \dots
-
-
method="MateiTille1"
[Class 3]\widehat{var}(\hat{t}_{HT}) = \frac{n(N-1)}{N(n-1)} \sum_{i\in s} \frac{b_i}{\pi_i^3} (y_i - \hat{y}_i^*)^2
where
\hat{y}_i^* = \pi_i \frac{\sum_{i\in s} b_i y_i/\pi_i^2}{\sum_{i\in s} b_i/\pi_i}
and the coefficients
b_i
are computed iteratively by the algorithm:-
b_i^{(0)} = \pi_i (1-\pi_i) \frac{N}{N-1}, \,\, \forall i \in U
-
b_i^{(k)} = \frac{(b_i^{(k-1)})^2 }{\sum_{j\in U} b_j^{(k-1)} } + \pi_i(1-\pi_i)
a necessary condition for convergence is checked and, if not satisfied, the function returns an alternative solution that uses only one iteration:
b_i = \pi_i(1-\pi_i)\Biggl( \frac{N\pi_i(1-\pi_i)}{ (N-1)\sum_{j\in U}\pi_j(1-\pi_j) } + 1 \Biggr)
-
-
method="MateiTille2"
[Class 3]\widehat{var}(\hat{t}_{HT}) = \frac{1}{1 - \sum_{i\in U} \frac{d_i^2}{\pi_i} } \sum_{i\in s} (1-\pi_i) \Biggl( \frac{y_i}{\pi_i} - \frac{\hat{t}_{HT}}{n} \Biggr)^2
where
d_i = \frac{\pi_i(1-\pi_i)}{\sum_{j\in U} \pi_j(1-\pi_j) }
-
method="MateiTille3"
[Class 3]\widehat{var}(\hat{t}_{HT}) = \frac{1}{1 - \sum_{i\in U} \frac{d_i^2}{\pi_i} } \sum_{i\in s} (1-\pi_i) \Biggl( \frac{y_i}{\pi_i} - \frac{ \sum_{j\in s} (1-\pi_j)\frac{y_j}{\pi_j} }{ \sum_{j\in s} (1-\pi_j) } \Biggr)^2
where
d_i
is defined as inmethod="MateiTille2"
. -
method="MateiTille4"
[Class 3]\widehat{var}(\hat{t}_{HT}) = \frac{1}{1 - \sum_{i\in U} b_i/n^2 } \sum_{i\in s} \frac{b_i}{\pi_i^3} (y_i - y_i^* )^2
where
y_i^* = \pi_i \frac{ \sum_{j\in s} b_j y_j/\pi_j^2 }{ \sum_{j\in s} b_j/\pi_j }
and
b_i = \frac{ \pi_i(1-\pi_i)N }{ N-1 }
-
method="MateiTille5"
[Class 3] This estimator is defined as inmethod="MateiTille4"
, and theb_i
values are defined as inmethod="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)