VE.EB.SYG.Total.Hajek: The Escobar-Berger unequal probability replicate variance estimator for the Hajek (1971) estimator of a total (Sen-Yates-Grundy form)

Description

Computes the Escobar-Berger (2013) unequal probability replicate variance estimator for the Hajek estimator of a total. It uses the Sen (1953); Yates-Grundy(1953) variance form.

Usage

VE.EB.SYG.Total.Hajek(VecY.s, VecPk.s, MatPkl.s, N,
                      VecAlpha.s = rep(1, times=length(VecPk.s)))

Value

The function returns a value for the estimated variance.

Arguments

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 the length of VecPk.s. There must not be missing values.
VecPk.s: vector of the first-order inclusion probabilities; its length is equal to $n$, the sample size. Values in VecPk.s must be greater than zero and less than or equal to one. There must not be missing values.
MatPkl.s: matrix of the second-order inclusion probabilities; its number of rows and columns is equal to $n$, the sample size. Values in MatPkl.s must be greater than zero and less than or equal to one. There must not be missing values.
N: the population size. It must be an integer or a double-precision scalar with zero-valued fractional part.
VecAlpha.s: vector of the $\alpha_k$ values; its length is equal to $n$, the sample size. Values in VecAlpha.s can be different for each unit and they must be greater or equal to zero. Escobar-Berger (2013) showed that this replicate variance estimator is valid for $\alpha_k\geq 0$. In particular, they suggest using $\alpha_k=1$ for all units in the sample (the default for VecAlpha.s if omitted in the function call). Using $\alpha_k>1$ results in approximating the Demnati-Rao (2004) linearisation variance estimators. There must not be missing values.

Author

Emilio Lopez Escobar.

Details

For the population total of the variable $y$: $$t = \sum_{k\in U} y_k$$ the approximately unbiased Hajek (1971) estimator of $t$ is given by: $$\hat{t}_{Hajek} = N \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}$$ where $w_k=1/\pi_k$ and $\pi_k$ denotes the inclusion probability of the $k$-th element in the sample $s$. The variance of $\hat{t}_{Hajek}$ can be estimated by the Escobar-Berger (2013) unequal probability replicate variance estimator (implemented by the current function): $$\hat{V}(\hat{t}_{Hajek}) = \frac{-1}{2}\sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} (\breve{\nu}_k - \breve{\nu}_l)^{2}$$ where $$\breve{\nu}_k = w_k^{\alpha_k} \left(\hat{t}_{Hajek}-\hat{t}_{Hajek,k}^{*}\right)$$ for some $\alpha_k\geq0$ (suggested to be 1, see below comments) and with $$\hat{t}_{Hajek,k}^{*} = N \frac{\sum_{l\in s} w_l y_l - w_k^{1-\alpha_k} y_k}{\sum_{l\in s} w_l - w_k^{1-\alpha_k}} $$ Regarding the value of $\alpha_k$, Escobar-Berger (2013) show that $\hat{V}(\hat{t}_{Hajek})$ is valid for $\alpha_k\geq0$ but conclude that $\alpha_k>0$ should be used as $\alpha_k=0$ corresponds to a naive biased and unstable jackknife. They recommend $\alpha_k=1$ or $\alpha_k>1$. If $\alpha_k=1$, $\hat{V}(\hat{t}_{Hajek})$ reduces to the Escobar-Berger (2011) jackknife. Using $\alpha_k>1$ results in approximating the empirical influence function, i.e. the Gateaux (1919) derivative, or Demnati-Rao (2004) linearisation variance estimators. The larger the $\alpha_k$, the closer the approximation. Further, Escobar-Berger (2013) give an intuitive explanation of the replication method from a jackknife and bootstrap perspective.

References

Demnati, A. and Rao, J. N. K. (2004) Linearization variance estimators for survey data. Survey Methodology, 30, 17--26.

Escobar, E. L. and Berger, Y. G. (2011) Jackknife variance estimation for functions of Horvitz-Thompson estimators under unequal probability sampling without replacement. In Proceeding of the 58th World Statistics Congress. Dublin, Ireland: International Statistical Institute.

Escobar, E. L. and Berger, Y. G. (2013) A new replicate variance estimator for unequal probability sampling without replacement. Canadian Journal of Statistics 41, 3, 508--524.

Gateaux, R. (1919) Fonctions d'une infinite de variables indeependantes. Bulletin de la Societe Mathematique de France, 47, 70--96.

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.

Sen, A. R. (1953) On the estimate of the variance in sampling with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 5, 119--127.

Yates, F. and Grundy, P. M. (1953) Selection without replacement from within strata with probability proportional to size. Journal of the Royal Statistical Society B, 15, 253--261.

Examples

Run this code

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
N       <- dim(oaxaca)[1]                     #Defines the population size
y1      <- oaxaca$POP10                       #Defines the variable of interest y1
y2      <- oaxaca$POPMAL10                    #Defines the variable of interest y2
Alpha.s <- rep(2, times=373)                  #Defines the vector with Alpha values
#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 Hajek total point estimator using y1
VE.EB.SYG.Total.Hajek(y1[s==1], pik.U[s==1], pikl.s, N)
#Computes the var. est. of the Hajek total point estimator using y2
VE.EB.SYG.Total.Hajek(y2[s==1], pik.U[s==1], pikl.s, N, Alpha.s)

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