VE.Jk.B.Mean.Hajek | R Documentation |
Computes the Berger (2007) unequal probability jackknife variance estimator for the Hajek (1971) estimator of a mean.
VE.Jk.B.Mean.Hajek(VecY.s, VecPk.s)
VecY.s |
vector of the 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 mean of the variable y:
\bar{y} = \frac{1}{N} ∑_{k\in U} y_k
the approximately unbiased Hajek (1971) estimator of \bar{y} is given by:
\hat{\bar{y}}_{Hajek} = \frac{∑_{k\in s} w_k y_k}{∑_{k\in s} w_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{\bar{y}}_{Hajek} can be estimated by the Berger (2007) unequal probability jackknife variance estimator (implemented by the current function):
\hat{V}(\hat{\bar{y}}_{Hajek}) = ∑_{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{\bar{y}}_{Hajek}-\hat{\bar{y}}_{Hajek(k)}\right)
with
\tilde{w}_k = \frac{w_k}{∑_{l\in s} w_l}
and
\hat{\bar{y}}_{Hajek(k)} = \frac{∑_{l\in s, l\neq k} w_l y_l}{∑_{l\in s, l\neq k} w_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.
Hajek, J. (1971) Comment on An essay on the logical foundations of survey sampling by Basu, D. in Foundations of Statistical Inference (Godambe, V.P. and Sprott, D.A. eds.), p. 236. Holt, Rinehart and Winston.
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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$POP10 #Defines the variable of interest y1 y2 <- oaxaca$POPMAL10 #Defines the variable of interest y2 #Computes the var. est. of the Hajek mean point estimator using y1 VE.Jk.B.Mean.Hajek(y1[s==1], pik.U[s==1]) #Computes the var. est. of the Hajek mean point estimator using y2 VE.Jk.B.Mean.Hajek(y2[s==1], pik.U[s==1])
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