VE.Jk.CBS.SYG.RegCo.Hajek {samplingVarEst}R Documentation

The Campbell-Berger-Skinner unequal probability jackknife variance estimator for the estimator of the regression coefficient using the Hajek point estimator (Sen-Yates-Grundy form)

Description

Computes the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator for the estimator of the regression coefficient using the Hajek (1971) point estimator. It uses the Sen (1953); Yates-Grundy(1953) variance form.

Usage

VE.Jk.CBS.SYG.RegCo.Hajek(VecY.s, VecX.s, VecPk.s, MatPkl.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.

MatPkl.s

matrix of the second-order inclusion probabilities; its number of rows and columns equals nn, 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.

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 Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator (implemented by the current function):

V^(β^Hajek)=12kslsπklπkπlπkl(εkεl)2\hat{V}(\hat{\beta}_{Hajek}) = \frac{-1}{2}\sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} (\varepsilon_k - \varepsilon_l)^{2}

where

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. The Sen-Yates-Grundy form for the Campbell(1980); Berger-Skinner(2005) unequal probability jackknife variance estimator is proposed in Escobar-Berger (2013) under less-restrictive regularity conditions.

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

References

Campbell, C. (1980) A different view of finite population estimation. Proceedings of the Survey Research Methods Section of the American Statistical Association, 319–324.

Berger, Y. G. and Skinner, C. J. (2005) A jackknife variance estimator for unequal probability sampling. Journal of the Royal Statistical Society B, 67, 79–89.

Escobar, E. L. and Berger, Y. G. (2013) A jackknife variance estimator for self-weighted two-stage samples. Statistica Sinica, 23, 595–613.

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.

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.

See Also

VE.Jk.CBS.SYG.RegCoI.Hajek
VE.Jk.Tukey.RegCo.Hajek
VE.Jk.CBS.HT.RegCo.Hajek
VE.Jk.B.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
#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 regression coeff. point estimator using y1
VE.Jk.CBS.SYG.RegCo.Hajek(y1[s==1], x[s==1], pik.U[s==1], pikl.s)
#Computes the var. est. of the regression coeff. point estimator using y2
VE.Jk.CBS.SYG.RegCo.Hajek(y2[s==1], x[s==1], pik.U[s==1], pikl.s)

[Package samplingVarEst version 1.5 Index]