VE.Jk.B.Ratio | R Documentation |
Computes the Berger (2007) unequal probability jackknife variance estimator for the estimator of a ratio of two totals/means.
VE.Jk.B.Ratio(VecY.s, VecX.s, VecPk.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 |
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 Berger (2007) unequal probability jackknife variance estimator (implemented by the current function):
\hat{V}(\hat{R}) = ∑_{k\in s} \frac{n}{n-1}(1-π_k) ≤ft(\varepsilon_k - \hat{B}\right)^{2}
where
\hat{B} = \frac{∑_{k\in s}(1-π_k) \varepsilon_k}{∑_{k\in s}(1-π_k)}
and
\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}
Note that this variance estimator implicitly utilises the Hajek (1964) approximations that are designed for large-entropy sampling designs, large samples, and large populations, i.e., care should be taken with highly-stratified samples, e.g. Berger (2005).
The function returns a value for the estimated variance.
Emilio Lopez Escobar.
Berger, Y. G. (2005) Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365–373.
Berger, Y. G. (2007) A jackknife variance estimator for unistage stratified samples with unequal probabilities. Biometrika 94, 953–964.
Hajek, J. (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491–1523.
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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 #Computes the var. est. of the ratio point estimator using y1 VE.Jk.B.Ratio(y1[s==1], x[s==1], pik.U[s==1]) #Computes the var. est. of the ratio point estimator using y2 VE.Jk.B.Ratio(y2[s==1], x[s==1], pik.U[s==1])
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