VE.Jk.CBS.HT.Ratio | R Documentation |
Computes the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator for the estimator of a ratio of two totals/means. It uses the Horvitz-Thompson (1952) variance form.
VE.Jk.CBS.HT.Ratio(VecY.s, VecX.s, VecPk.s, MatPkl.s)
VecY.s |
vector of the numerator variable of interest; its length is equal to n, the sample size. Its length has to be the same as that of |
VecX.s |
vector of the denominator variable of interest; its length is equal to n, the sample size. Its length has to be the same as that of |
VecPk.s |
vector of the first-order inclusion probabilities; its length is equal to n, the sample size. Values in |
MatPkl.s |
matrix of the second-order inclusion probabilities; its number of rows and columns equals n, the sample size. Values in |
For the population ratio of two totals/means of the variables y and x:
R = \frac{∑_{k\in U} y_k/N}{∑_{k\in U} x_k/N} = \frac{∑_{k\in U} y_k}{∑_{k\in U} x_k}
the ratio estimator of R is given by:
\hat{R} = \frac{∑_{k\in s} w_k y_k}{∑_{k\in s} w_k x_k}
where w_k=1/π_k and π_k denotes the inclusion probability of the k-th element in the sample s. The variance of \hat{R} can be estimated by the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator (implemented by the current function):
\hat{V}(\hat{R}) = ∑_{k\in s}∑_{l\in s} \frac{π_{kl}-π_kπ_l}{π_{kl}} \varepsilon_k \varepsilon_l
where
\varepsilon_k = ≤ft(1-\tilde{w}_k\right) ≤ft(\hat{R}-\hat{R}_{(k)}\right)
with
\tilde{w}_k = \frac{w_k}{∑_{l\in s} w_l}
and
\hat{R}_{(k)} = \frac{∑_{l\in s, l\neq k} w_l y_l/∑_{l\in s, l\neq k} w_l}{∑_{l\in s, l\neq k} w_l x_l/∑_{l\in s, l\neq k} w_l} = \frac{∑_{l\in s, l\neq k} w_l y_l}{∑_{l\in s, l\neq k} w_l x_l}
The function returns a value for the estimated variance.
Emilio Lopez Escobar.
Campbell, C. (1980) A different view of finite population estimation. Proceedings of the Survey Research Methods Section of the American Statistical Association, 319–324.
Berger, Y. G. and Skinner, C. J. (2005) A jackknife variance estimator for unequal probability sampling. Journal of the Royal Statistical Society B, 67, 79–89.
Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.
VE.Lin.HT.Ratio
VE.Lin.SYG.Ratio
VE.Jk.Tukey.Ratio
VE.Jk.CBS.SYG.Ratio
VE.Jk.B.Ratio
VE.Jk.EB.SW2.Ratio
VE.EB.HT.Ratio
VE.EB.SYG.Ratio
data(oaxaca) #Loads the Oaxaca municipalities dataset pik.U <- Pk.PropNorm.U(373, oaxaca$HOMES00) #Reconstructs the 1st order incl. probs. s <- oaxaca$sHOMES00 #Defines the sample to be used y1 <- oaxaca$POP10 #Defines the numerator variable y1 y2 <- oaxaca$POPMAL10 #Defines the numerator variable y2 x <- oaxaca$HOMES10 #Defines the denominator variable x #This approximation is only suitable for large-entropy sampling designs pikl.s <- Pkl.Hajek.s(pik.U[s==1]) #Approx. 2nd order incl. probs. from s #Computes the var. est. of the ratio point estimator using y1 VE.Jk.CBS.HT.Ratio(y1[s==1], x[s==1], pik.U[s==1], pikl.s) #Computes the var. est. of the ratio point estimator using y2 VE.Jk.CBS.HT.Ratio(y2[s==1], x[s==1], pik.U[s==1], pikl.s)
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