jip_approx: Approximate Joint-Inclusion Probabilities In rhobis/jipApprox: Approximate Inclusion Probabilities for Survey Sampling

Description

Approximations of joint-inclusion probabilities by means of first-order inclusion probabilities.

Usage

 `1` ```jip_approx(pik, method) ```

Arguments

 `pik` numeric vector of first-order inclusion probabilities for all population units. `method` string representing one of the available approximation methods.

Details

Available methods are `"Hajek"`, `"HartleyRao"`, `"Tille"`, `"Brewer1"`,`"Brewer2"`,`"Brewer3"`, and `"Brewer4"`. Note that these methods were derived for high-entropy 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 - ( 1-π(i) )( 1 -π(j) ) ) /d

where d = ∑ π(i)(1-π(i))

Hartley and Rao (1962) proposed the following approximation under randomised systematic sampling [`method="HartleyRao"`]:

Tillé (1996) proposed the approximation π(ij) = β_i β_j, where the coefficients β are computed iteratively through the following procedure [`method="Tille"`]:

1. β(0) = π, i = 1, ..., N

2. β(2k-1) = ( (n-1)π )/(∑β(2k-2) - β(2k-2))

3. β(2k) = β(2k-1) ( n(n-1) / ( (∑β(2k-1))^2 - ∑( β(2k-1)^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) = [n-1] / [n-π(i) ]

• Equation (10) [`method="Brewer2"`]:

c(i) = [n-1] / [n- ∑_U π(i)^2 / n ]

• Equation (11) [`method="Brewer3"`]:

c(i) = [n-1] / [n- 2π(i) + ∑_U π(i)^2 / n ]

• Equation (18) [`method="Brewer4"`]:

c(i) = [n-1] / [n- π(i)(2n -1)/(n-1) + ∑_U π(i)^2 / (n-1) ]

Value

A symmetric matrix of inclusion probabilities, which diagonal is the vector of first-order inclusion probabilities.

References

Hartley, H.O.; Rao, J.N.K., 1962. Sampling With Unequal Probability and Without Replacement. The Annals of Mathematical Statistics 33 (2), 350-374.

Hájek, J., 1964. Asymptotic Theory of Rejective Sampling with Varying Probabilities from a Finite Population. The Annals of Mathematical Statistics 35 (4), 1491-1523.

Tillé, Y., 1996. Some Remarks on Unequal Probability Sampling Designs Without Replacement. Annals of Economics and Statistics 44, 177-189.

Brewer, K.R.W.; Donadio, M.E., 2003. The High Entropy Variance of the Horvitz-Thompson Estimator. Survey Methodology 29 (2), 189-196.

Examples

 ``` 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 joint-inclusion 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') ```

rhobis/jipApprox documentation built on Sept. 26, 2018, 5:20 p.m.