VE.Hajek.Total.NHT | R Documentation |
Computes the Hajek (1964) variance estimator for the Narain (1951); Horvitz-Thompson (1952) point estimator for a population total.
VE.Hajek.Total.NHT(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 total of the variable y:
t = ∑_{k\in U} y_k
the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of t is given by:
\hat{t}_{NHT} = ∑_{k\in s} \frac{y_k}{π_k}
where π_k denotes the inclusion probability of the k-th element in the sample s. For large-entropy sampling designs, the variance of \hat{t}_{NHT} is approximated by the Hajek (1964) variance:
V(\hat{t}_{NHT}) = \frac{N}{N-1}≤ft[∑_{k\in U}\frac{y_k^2}{π_k}(1-π_k)-dG^2\right]
with d=∑_{k\in U}π_k(1-π_k) and G=d^{-1}∑_{k\in U}(1-π_k)y_k.
The variance V(\hat{t}_{NHT}) can be estimated by the variance estimator (implemented by the current function):
\hat{V}(\hat{t}_{NHT}) = \frac{n}{n-1}≤ft[∑_{k\in s}≤ft(\frac{y_k}{π_k}\right)^2(1-π_k)-\hat{d}\hat{G}^2\right]
where \hat{d}=∑_{k\in s}(1-π_k) and \hat{G}=\hat{d}^{-1}∑_{k\in s}(1-π)y_k/π_k.
Note that the Hajek (1964) variance approximation is 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.
Hajek, J. (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491–1523.
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.
Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169–175.
VE.HT.Total.NHT
VE.SYG.Total.NHT
data(oaxaca) #Loads the Oaxaca municipalities dataset pik.U <- Pk.PropNorm.U(373, oaxaca$SURFAC05) #Reconstructs the 1st order incl. probs. s <- oaxaca$sSURFAC #Defines the sample to be used y1 <- oaxaca$POP10 #Defines the variable of interest y1 y2 <- oaxaca$HOMES10 #Defines the variable of interest y2 #Computes the (approximate) var. est. of the NHT point est. from y1 VE.Hajek.Total.NHT(y1[s==1], pik.U[s==1]) #Computes the (approximate) var. est. of the NHT point est. from y2 VE.Hajek.Total.NHT(y2[s==1], pik.U[s==1])
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