Description Usage Arguments Details Value References Examples
View source: R/jip_approximations.R
Approximations of jointinclusion probabilities by means of firstorder inclusion probabilities.
1  jip_approx(pik, method)

pik 
numeric vector of firstorder inclusion probabilities for all population units. 
method 
string representing one of the available approximation methods. 
Available methods are "Hajek"
, "HartleyRao"
, "Tille"
,
"Brewer1"
,"Brewer2"
,"Brewer3"
, and "Brewer4"
.
Note that these methods were derived for highentropy sampling designs,
therefore they could have low performance under different designs.
Hájek (1964) approximation [method="Hajek"
] is derived under Maximum Entropy sampling design
and is given by
π(ij) = π(i) π(j) (1  ( 1π(i) )( 1 π(j) ) ) /d
where d = ∑ π(i)(1π(i))
Hartley and Rao (1962) proposed the following approximation under
randomised systematic sampling [method="HartleyRao"
]:
*see pdf version of documentation*
Tillé (1996) proposed the approximation π(ij) = β_i β_j,
where the coefficients β are computed iteratively through the
following procedure [method="Tille"
]:
β(0) = π, i = 1, ..., N
β(2k1) = ( (n1)π )/(∑β(2k2)  β(2k2))
β(2k) = β(2k1) ( n(n1) / ( (∑β(2k1))^2  ∑( β(2k1)^2 ) ) )^{(1/2)}
with
Finally, Brewer (2002) and Brewer and Donadio (2003) proposed four approximations, which are defined by the general form
π(ij) = π(i)π(j) [c(i) + c(j) ]/2
where the c_i determine the approximation used:
Equation (9) [method="Brewer1"
]:
c(i) = [n1] / [nπ(i) ]
Equation (10) [method="Brewer2"
]:
c(i) = [n1] / [n ∑_U π(i)^2 / n ]
Equation (11) [method="Brewer3"
]:
c(i) = [n1] / [n 2π(i) + ∑_U π(i)^2 / n ]
Equation (18) [method="Brewer4"
]:
c(i) = [n1] / [n π(i)(2n 1)/(n1) + ∑_U π(i)^2 / (n1) ]
A symmetric matrix of inclusion probabilities, which diagonal is the vector of firstorder inclusion probabilities.
Hartley, H.O.; Rao, J.N.K., 1962. Sampling With Unequal Probability and Without Replacement. The Annals of Mathematical Statistics 33 (2), 350374.
Hájek, J., 1964. Asymptotic Theory of Rejective Sampling with Varying Probabilities from a Finite Population. The Annals of Mathematical Statistics 35 (4), 14911523.
Tillé, Y., 1996. Some Remarks on Unequal Probability Sampling Designs Without Replacement. Annals of Economics and Statistics 44, 177189.
Brewer, K.R.W.; Donadio, M.E., 2003. The High Entropy Variance of the HorvitzThompson Estimator. Survey Methodology 29 (2), 189196.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17  ### Generate population data 
N < 20; n<5
set.seed(0)
x < rgamma(N, scale=10, shape=5)
y < abs( 2*x + 3.7*sqrt(x) * rnorm(N) )
pik < n * x/sum(x)
### Approximate jointinclusion probabilities 
pikl < jip_approx(pik, method='Hajek')
pikl < jip_approx(pik, method='HartleyRao')
pikl < jip_approx(pik, method='Tille')
pikl < jip_approx(pik, method='Brewer1')
pikl < jip_approx(pik, method='Brewer2')
pikl < jip_approx(pik, method='Brewer3')
pikl < jip_approx(pik, method='Brewer4')

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