VE_Jk_Tukey_Total_Hajek: The Tukey (1958) jackknife variance estimator for the Hajek... In samplingVarEst: Sampling Variance Estimation

 VE.Jk.Tukey.Total.Hajek R Documentation

The Tukey (1958) jackknife variance estimator for the Hajek estimator of a total

Description

Computes the Quenouille(1956); Tukey (1958) jackknife variance estimator for the Hajek (1971) estimator of a total.

Usage

VE.Jk.Tukey.Total.Hajek(VecY.s, VecPk.s, N, FPC= TRUE)

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 that 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. N the population size. It must be an integer or a double-precision scalar with zero-valued fractional part. This information is also utilised for the finite population correction; see FPC below. FPC logical value. If an ad hoc finite population correction FPC=1-n/N is to be used. The default is TRUE.

Details

For the population total of the variable y:

t = ∑_{k\in U} y_k

the approximately unbiased Hajek (1971) estimator of t is given by:

\hat{t}_{Hajek} = N \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{t}_{Hajek} can be estimated by the Quenouille(1956); Tukey (1958) jackknife variance estimator (implemented by the current function):

\hat{V}(\hat{t}_{Hajek}) = ≤ft(1-\frac{n}{N}\right)\frac{n-1}{n}∑_{k\in s} ≤ft( \hat{t}_{Hajek(k)}-\hat{t}_{Hajek} \right)^2

where

\hat{t}_{Hajek(k)} = N \frac{∑_{l\in s, l\neq k} w_l y_l}{∑_{l\in s, l\neq k} w_l}

We are implementing the Tukey (1958) jackknife variance estimator using the ‘ad hoc’ finite population correction 1-n/N (see Shao and Tu, 1995; Wolter, 2007). If FPC=FALSE, then the term 1-n/N is omitted from the above formula.

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

References

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.

Quenouille, M. H. (1956) Notes on bias in estimation. Biometrika, 43, 353–360.

Shao, J. and Tu, D. (1995) The Jackknife and Bootstrap. Springer-Verlag, Inc.

Tukey, J. W. (1958) Bias and confidence in not-quite large samples (abstract). The Annals of Mathematical Statistics, 29, 2, p. 614.

Wolter, K. M. (2007) Introduction to Variance Estimation. 2nd Ed. Springer, Inc.

VE.Jk.CBS.HT.Total.Hajek
VE.Jk.CBS.SYG.Total.Hajek
VE.Jk.B.Total.Hajek
VE.Jk.EB.SW2.Total.Hajek

Examples

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
#Computes the var. est. of the Hajek total point estimator using y1
VE.Jk.Tukey.Total.Hajek(y1[s==1], pik.U[s==1], N)
#Computes the var. est. of the Hajek total point estimator using y2
VE.Jk.Tukey.Total.Hajek(y2[s==1], pik.U[s==1], N, FPC= FALSE)


samplingVarEst documentation built on Jan. 14, 2023, 5:08 p.m.