VE.Jk.B.RegCo.Hajek {samplingVarEst}R Documentation

The Berger (2007) unequal probability jackknife variance estimator for the estimator of the regression coefficient using the Hajek point estimator

Description

Computes the Berger (2007) unequal probability jackknife variance estimator for the estimator of the regression coefficient using the Hajek (1971) point estimator.

Usage

VE.Jk.B.RegCo.Hajek(VecY.s, VecX.s, VecPk.s)

Arguments

VecY.s

vector of the variable of interest Y; its length is equal to nn, the sample size. Its length has to be the same as that of VecPk.s and VecX.s. There must not be missing values.

VecX.s

vector of the variable of interest X; its length is equal to nn, the sample size. Its length has to be the same as that of VecPk.s and VecY.s. There must not be missing values.

VecPk.s

vector of the first-order inclusion probabilities; its length is equal to nn, 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.

Details

From Linear Regression Analysis, for an imposed population model

y=α+βxy=\alpha + \beta x

the population regression coefficient β\beta, assuming that the population size NN is unknown (see Sarndal et al., 1992, Sec. 5.10), can be estimated by:

β^Hajek=kswk(ykyˉ^Hajek)(xkxˉ^Hajek)kswk(xkxˉ^Hajek)2\hat{\beta}_{Hajek} = \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2}

where yˉ^Hajek\hat{\bar{y}}_{Hajek} and xˉ^Hajek\hat{\bar{x}}_{Hajek} are the Hajek (1971) point estimators of the population means yˉ=N1kUyk\bar{y} = N^{-1} \sum_{k\in U} y_k and xˉ=N1kUxk\bar{x} = N^{-1} \sum_{k\in U} x_k, respectively,

yˉ^Hajek=kswkykkswk\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}

xˉ^Hajek=kswkxkkswk\hat{\bar{x}}_{Hajek} = \frac{\sum_{k\in s} w_k x_k}{\sum_{k\in s} w_k}

and wk=1/πkw_k=1/\pi_k with πk\pi_k denoting the inclusion probability of the kk-th element in the sample ss. The variance of β^Hajek\hat{\beta}_{Hajek} can be estimated by the Berger (2007) unequal probability jackknife variance estimator (implemented by the current function):

V^(β^Hajek)=ksnn1(1πk)(εkB^)2\hat{V}(\hat{\beta}_{Hajek}) = \sum_{k\in s} \frac{n}{n-1}(1-\pi_k) \left(\varepsilon_k - \hat{B}\right)^{2}

where

B^=ks(1πk)εkks(1πk)\hat{B} = \frac{\sum_{k\in s}(1-\pi_k) \varepsilon_k}{\sum_{k\in s}(1-\pi_k)}

and

εk=(1w~k)(β^Hajekβ^Hajek(k))\varepsilon_k = \left(1-\tilde{w}_k\right) \left(\hat{\beta}_{Hajek}-\hat{\beta}_{Hajek(k)}\right)

with

w~k=wklswl\tilde{w}_k = \frac{w_k}{\sum_{l\in s} w_l}

and where β^Hajek(k)\hat{\beta}_{Hajek(k)} has the same functional form as β^Hajek\hat{\beta}_{Hajek} but omitting the kk-th element from the sample ss. 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).

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

References

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.

Sarndal, C.-E. and Swensson, B. and Wretman, J. (1992) Model Assisted Survey Sampling. Springer-Verlag, Inc.

See Also

VE.Jk.B.RegCoI.Hajek
VE.Jk.Tukey.RegCo.Hajek
VE.Jk.CBS.HT.RegCo.Hajek
VE.Jk.CBS.SYG.RegCo.Hajek
VE.Jk.EB.SW2.RegCo.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
y1     <- oaxaca$POP10                       #Defines the variable of interest y1
y2     <- oaxaca$POPMAL10                    #Defines the variable of interest y2
x      <- oaxaca$HOMES10                     #Defines the variable of interest x
#Computes the var. est. of the regression coeff. point estimator using y1
VE.Jk.B.RegCo.Hajek(y1[s==1], x[s==1], pik.U[s==1])
#Computes the var. est. of the regression coeff. point estimator using y2
VE.Jk.B.RegCo.Hajek(y2[s==1], x[s==1], pik.U[s==1])

[Package samplingVarEst version 1.5 Index]