VE.Jk.Tukey.RegCo.Hajek: The Tukey (1958) jackknife variance estimator for the estimator of the regression coefficient using the Hajek point estimator

Description

Computes the Quenouille(1956); Tukey (1958) jackknife variance estimator for the estimator of the regression coefficient using the Hajek (1971) point estimator.

Usage

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

Value

The function returns a value for the estimated variance.

Arguments

VecY.s: vector of the variable of interest Y; its length is equal to $n$, 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 $n$, 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 $n$, 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 utilised for the finite population correction only; 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.

Author

Emilio Lopez Escobar.

Details

From Linear Regression Analysis, for an imposed population model $$y=\alpha + \beta x$$ the population regression coefficient $\beta$, assuming that the population size $N$ is unknown (see Sarndal et al., 1992, Sec. 5.10), can be estimated by: $$\hat{\beta}_{Hajek} = \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2}$$ where $\hat{\bar{y}}_{Hajek}$ and $\hat{\bar{x}}_{Hajek}$ are the Hajek (1971) point estimators of the population means $\bar{y} = N^{-1} \sum_{k\in U} y_k$ and $\bar{x} = N^{-1} \sum_{k\in U} x_k$, respectively, $$\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}$$ $$\hat{\bar{x}}_{Hajek} = \frac{\sum_{k\in s} w_k x_k}{\sum_{k\in s} w_k}$$ and $w_k=1/\pi_k$ with $\pi_k$ denoting the inclusion probability of the $k$-th element in the sample $s$. The variance of $\hat{\beta}_{Hajek}$ can be estimated by the Quenouille(1956); Tukey (1958) jackknife variance estimator (implemented by the current function): $$\hat{V}(\hat{\beta}_{Hajek}) = \left(1-\frac{n}{N}\right)\frac{n-1}{n}\sum_{k\in s} \left( \hat{\beta}_{Hajek(k)}-\hat{\beta}_{Hajek} \right)^2$$ where $\hat{\beta}_{Hajek(k)}$ has the same functional form as $\hat{\beta}_{Hajek}$ but omitting the $k$-th element from the sample $s$. 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.

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.

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

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.

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
x     <- oaxaca$HOMES10                     #Defines the variable of interest x
#Computes the var. est. of the regression coeff. point estimator using y1
VE.Jk.Tukey.RegCo.Hajek(y1[s==1], x[s==1], pik.U[s==1], N)
#Computes the var. est. of the regression coeff. point estimator using y2
VE.Jk.Tukey.RegCo.Hajek(y2[s==1], x[s==1], pik.U[s==1], N, FPC= FALSE)

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