Package 'samplingVarEst'

Sampling Variance Estimation package

Description

The package contains functions to calculate some point estimators and estimate their variance under unequal probability sampling without replacement. Uni-stage and two-stage sampling designs are considered. The package further contains some approximations for the joint-inclusion probabilities (population and sample based formulae).

Emphasis has been put on the speed of routines as the package mostly uses C compiled code. Below there is a list of available functions. These are grouped in purpose lists, aiming to clarify their usage.

The user should pick a suitable combination of a population parameter of interest, a choice of point estimator, and a choice of variance estimator.

For these population parameters:	The available point estimators are:
total:	Est.Total.NHT
	Est.Total.Hajek
mean:	Est.Mean.NHT
	Est.Mean.Hajek
empirical cumulative distribution function:	Est.EmpDistFunc.NHT
	Est.EmpDistFunc.Hajek
ratio:	Est.Ratio
correlation coefficient:	Est.Corr.NHT
	Est.Corr.Hajek
regression coefficients:	Est.RegCoI.Hajek
	Est.RegCo.Hajek

For these point estimators:	The available variance estimators for uni-stage samples are:
Est.Total.NHT:	VE.HT.Total.NHT
	VE.SYG.Total.NHT
	VE.Hajek.Total.NHT
Est.Total.Hajek:	VE.Jk.Tukey.Total.Hajek
	VE.Jk.CBS.HT.Total.Hajek
	VE.Jk.CBS.SYG.Total.Hajek
	VE.Jk.B.Total.Hajek
	VE.EB.HT.Total.Hajek
	VE.EB.SYG.Total.Hajek
Est.Mean.NHT:	VE.HT.Mean.NHT
	VE.SYG.Mean.NHT
	VE.Hajek.Mean.NHT
Est.Mean.Hajek:	VE.Jk.Tukey.Mean.Hajek
	VE.Jk.CBS.HT.Mean.Hajek
	VE.Jk.CBS.SYG.Mean.Hajek
	VE.Jk.B.Mean.Hajek
	VE.EB.HT.Mean.Hajek
	VE.EB.SYG.Mean.Hajek
Est.Ratio:	VE.Lin.HT.Ratio
	VE.Lin.SYG.Ratio
	VE.Jk.Tukey.Ratio
	VE.Jk.CBS.HT.Ratio
	VE.Jk.CBS.SYG.Ratio
	VE.Jk.B.Ratio
	VE.EB.HT.Ratio
	VE.EB.SYG.Ratio
Est.Corr.NHT:	VE.Jk.Tukey.Corr.NHT
Est.Corr.Hajek:	VE.Jk.Tukey.Corr.Hajek
	VE.Jk.CBS.HT.Corr.Hajek
	VE.Jk.CBS.SYG.Corr.Hajek
	VE.Jk.B.Corr.Hajek
Est.RegCoI.Hajek:	VE.Jk.Tukey.RegCoI.Hajek
	VE.Jk.CBS.HT.RegCoI.Hajek
	VE.Jk.CBS.SYG.RegCoI.Hajek
	VE.Jk.B.RegCoI.Hajek
Est.RegCo.Hajek:	VE.Jk.Tukey.RegCo.Hajek
	VE.Jk.CBS.HT.RegCo.Hajek
	VE.Jk.CBS.SYG.RegCo.Hajek
	VE.Jk.B.RegCo.Hajek

For these point estimators:	The available variance estimators for self-weighted two-stage samples are:
Est.Total.Hajek:	VE.Jk.EB.SW2.Total.Hajek
Est.Mean.Hajek:	VE.Jk.EB.SW2.Mean.Hajek
Est.Ratio:	VE.Jk.EB.SW2.Ratio
Est.Corr.Hajek:	VE.Jk.EB.SW2.Corr.Hajek
Est.RegCoI.Hajek:	VE.Jk.EB.SW2.RegCoI.Hajek
Est.RegCo.Hajek:	VE.Jk.EB.SW2.RegCo.Hajek

For the inclusion probabilities:	The available functions are:
1st order inclusion probabilities:	Pk.PropNorm.U
2nd order (joint) inclusion probabilities:	Pkl.Hajek.s
	Pkl.Hajek.U

datasets

oaxaca

Details

To return to this description type:
help(samplingVarEst)
or type:
?samplingVarEst
To cite, use:
citation("samplingVarEst")

---

Estimator of a correlation coefficient using the Hajek point estimator

Description

Estimates a population correlation coefficient of two variables using the Hajek (1971) point estimator.

Usage

Est.Corr.Hajek(VecY.s, VecX.s, VecPk.s)

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.

Details

For the population correlation coefficient of two variables $y$ and $x$:

$C = \frac{\sum_{k\in U} (y_k - \bar{y})(x_k - \bar{x})}{\sqrt{\sum_{k\in U} (y_k - \bar{y})^2}\sqrt{\sum_{k\in U} (x_k - \bar{x})^2}}$

the point estimator of $C$, assuming that $N$ is unknown (see Sarndal et al., 1992, Sec. 5.9) (implemented by the current function), is:

$\hat{C}_{Hajek} = \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sqrt{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})^2}\sqrt{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2}}$

where $\hat{\bar{y}}_{Hajek}$ is the Hajek (1971) point estimator of the population mean $\bar{y} = N^{-1} \sum_{k\in U} y_k$,

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

and $w_k=1/\pi_k$ with $\pi_k$ denoting the inclusion probability of the $k$-th element in the sample $s$.

Value

The function returns a value for the correlation coefficient point estimator.

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.

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

See Also

Est.Corr.NHT
VE.Jk.Tukey.Corr.Hajek
VE.Jk.CBS.HT.Corr.Hajek
VE.Jk.CBS.SYG.Corr.Hajek
VE.Jk.B.Corr.Hajek
VE.Jk.EB.SW2.Corr.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 correlation coefficient estimator for y1 and x
Est.Corr.Hajek(y1[s==1], x[s==1], pik.U[s==1])
#Computes the correlation coefficient estimator for y2 and x
Est.Corr.Hajek(y2[s==1], x[s==1], pik.U[s==1])

---

Estimator of a correlation coefficient using the Narain-Horvitz-Thompson point estimator

Description

Estimates a population correlation coefficient of two variables using the Narain (1951); Horvitz-Thompson (1952) point estimator.

Usage

Est.Corr.NHT(VecY.s, VecX.s, VecPk.s, N)

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.

Details

For the population correlation coefficient of two variables $y$ and $x$:

$C = \frac{\sum_{k\in U} (y_k - \bar{y})(x_k - \bar{x})}{\sqrt{\sum_{k\in U} (y_k - \bar{y})^2}\sqrt{\sum_{k\in U} (x_k - \bar{x})^2}}$

the point estimator of $C$ (implemented by the current function) is given by:

$\hat{C} = \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{NHT})(x_k - \hat{\bar{x}}_{NHT})}{\sqrt{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{NHT})^2}\sqrt{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{NHT})^2}}$

where $\hat{\bar{y}}_{NHT}$ is the Narain (1951); Horvitz-Thompson (1952) estimator for the population mean $\bar{y} = N^{-1} \sum_{k\in U} y_k$,

$\hat{\bar{y}}_{NHT} = \frac{1}{N}\sum_{k\in s} w_k y_k$

and $w_k=1/\pi_k$ with $\pi_k$ denoting the inclusion probability of the $k$-th element in the sample $s$.

Value

The function returns a value for the correlation coefficient point estimator.

Author(s)

Emilio Lopez Escobar.

References

Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169–175.

See Also

Est.Corr.Hajek
VE.Jk.Tukey.Corr.NHT

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
x     <- oaxaca$HOMES10                     #Defines the variable of interest x
#Computes the correlation coefficient estimator for y1 and x
Est.Corr.NHT(y1[s==1], x[s==1], pik.U[s==1], N)
#Computes the correlation coefficient estimator for y2 and x
Est.Corr.NHT(y2[s==1], x[s==1], pik.U[s==1], N)

---

The Hajek estimator for the empirical cumulative distribution function

Description

Computes the Hajek (1971) estimator for the empirical cumulative distribution function (ECDF).

Usage

Est.EmpDistFunc.Hajek(VecY.s, VecPk.s, t)

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.
t	value to be evaluated for the empirical cumulative distribution function. It must be an integer or a double-precision scalar.

Details

For the population empirical cumulative distribution function (ECDF) of the variable $y$ at the value $t$:

$Fn(t) = \frac{\#(k\in U:y_k \leq t)}{N} = \frac{1}{N} \sum_{k\in U} I(y_k \leq t)$

the approximately unbiased Hajek (1971) estimator of $Fn(t)$ (implemented by the current function) is given by:

$\hat{F}n_{Hajek}(t) = \frac{\sum_{k\in s} w_k I(y_k \leq t)}{\sum_{k\in s} w_k}$

where $I(y_k \leq t)$ denotes the indicator function that takes the value $1$ if $y_k \leq t$ and the value $0$ otherwise, and where $w_k=1/\pi_k$ and $\pi_k$ denotes the inclusion probability of the $k$-th element in the sample $s$.

Value

The function returns a value for the empirical cumulative distribution function evaluated at $t$.

Author(s)

Emilio Lopez Escobar [aut, cre], Juan Francisco Munoz Rosas [ctb].

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.

See Also

Est.EmpDistFunc.NHT

Examples

data(oaxaca)                                      #Loads Oaxaca municipalities dataset
pik.U <- Pk.PropNorm.U(373, oaxaca$HOMES00)       #Reconstructs the inclusion probs.
s     <- oaxaca$sHOMES00                          #Defines the sample to be used
y1    <- oaxaca$POP10                             #Defines the variable of interest y1
Est.EmpDistFunc.Hajek(y1[s==1], pik.U[s==1], 950) #Hajek est. of ECDF for y1 at t=950

---

The Narain-Horvitz-Thompson estimator for the empirical cumulative distribution function

Description

Computes the Narain (1951); Horvitz-Thompson (1952) estimator for the empirical cumulative distribution function (ECDF).

Usage

Est.EmpDistFunc.NHT(VecY.s, VecPk.s, N, t)

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.
t	value to be evaluated for the empirical cumulative distribution function. It must be an integer or a double-precision scalar.

Details

For the population empirical cumulative distribution function (ECDF) of the variable $y$ at the value $t$:

$Fn(t) = \frac{\#(k\in U:y_k \leq t)}{N} = \frac{1}{N} \sum_{k\in U} I(y_k \leq t)$

the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of $Fn(t)$ (implemented by the current function) is given by:

$\hat{F}n_{NHT}(t) = \frac{1}{N} \sum_{k\in s} \frac{I(y_k \leq t)}{\pi_k}$

where $I(y_k \leq t)$ denotes the indicator function that takes the value $1$ if $y_k \leq t$ and the value $0$ otherwise, and where $\pi_k$ denotes the inclusion probability of the $k$-th element in the sample $s$.

Value

The function returns a value for the empirical cumulative distribution function evaluated at $t$.

Author(s)

Emilio Lopez Escobar [aut, cre], Juan Francisco Munoz Rosas [ctb].

References

Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169–175.

See Also

Est.EmpDistFunc.Hajek

Examples

data(oaxaca)                                       #Loads Oaxaca municipalities dataset
pik.U <- Pk.PropNorm.U(373, oaxaca$HOMES00)        #Reconstructs the inclusion probs.
s     <- oaxaca$sHOMES00                           #Defines the sample to be used
N     <- dim(oaxaca)[1]                            #Defines the population size
y1    <- oaxaca$POP10                              #Defines the variable of interest y1
Est.EmpDistFunc.NHT(y1[s==1], pik.U[s==1], N, 950) #NHT est. of ECDF for y1 at t=950

---

The Hajek estimator for a mean

Description

Computes the Hajek (1971) estimator for a population mean.

Usage

Est.Mean.Hajek(VecY.s, VecPk.s)

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.

Details

For the population mean of the variable $y$:

$\bar{y} = \frac{1}{N} \sum_{k\in U} y_k$

the approximately unbiased Hajek (1971) estimator of $\bar{y}$ (implemented by the current function) is given by:

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

where $w_k=1/\pi_k$ and $\pi_k$ denotes the inclusion probability of the $k$-th element in the sample $s$.

Value

The function returns a value for the mean point estimator.

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.

See Also

Est.Mean.NHT
VE.Jk.Tukey.Mean.Hajek
VE.Jk.CBS.HT.Mean.Hajek
VE.Jk.CBS.SYG.Mean.Hajek
VE.Jk.B.Mean.Hajek
VE.Jk.EB.SW2.Mean.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$POP10                       #Defines the variable of interest y1
y2    <- oaxaca$HOMES10                     #Defines the variable of interest y2
Est.Mean.Hajek(y1[s==1], pik.U[s==1])       #Computes the Hajek est. for y1
Est.Mean.Hajek(y2[s==1], pik.U[s==1])       #Computes the Hajek est. for y2

---

The Narain-Horvitz-Thompson estimator for a mean

Description

Computes the Narain (1951); Horvitz-Thompson (1952) estimator for a population mean.

Usage

Est.Mean.NHT(VecY.s, VecPk.s, N)

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.

Details

For the population mean of the variable $y$:

$\bar{y} = \frac{1}{N} \sum_{k\in U} y_k$

the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of $\bar{y}$ (implemented by the current function) is given by:

$\hat{\bar{y}}_{NHT} = \frac{1}{N} \sum_{k\in s} \frac{y_k}{\pi_k}$

where $\pi_k$ denotes the inclusion probability of the $k$-th element in the sample $s$.

Value

The function returns a value for the mean point estimator.

Author(s)

Emilio Lopez Escobar.

References

Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169–175.

See Also

Est.Mean.Hajek
VE.HT.Mean.NHT
VE.SYG.Mean.NHT
VE.Hajek.Mean.NHT

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$POP10                       #Defines the variable of interest y1
y2    <- oaxaca$HOMES10                     #Defines the variable of interest y2
Est.Mean.NHT(y1[s==1], pik.U[s==1], N)      #The NHT estimator for y1
Est.Mean.NHT(y2[s==1], pik.U[s==1], N)      #The NHT estimator for y2

---

Estimator of a ratio

Description

Estimates a population ratio of two totals/means.

Usage

Est.Ratio(VecY.s, VecX.s, VecPk.s)

Arguments

VecY.s	vector of the numerator variable of interest; 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 denominator variable of interest; 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. All values of VecX.s should be greater than zero. A warning is displayed if this does not hold, and computations continue if mathematical expressions allow this kind of values for the denominator variable.
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.

Details

For the population ratio of two totals/means of the variables $y$ and $x$:

$R = \frac{\sum_{k\in U} y_k/N}{\sum_{k\in U} x_k/N} = \frac{\sum_{k\in U} y_k}{\sum_{k\in U} x_k}$

the ratio estimator of $R$ (implemented by the current function) is given by:

$\hat{R} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k x_k}$

where $w_k=1/\pi_k$ and $\pi_k$ denotes the inclusion probability of the $k$-th element in the sample $s$.

Value

The function returns a value for the ratio point estimator.

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.

Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169–175.

See Also

VE.Jk.Tukey.Ratio
VE.Jk.CBS.HT.Ratio
VE.Jk.CBS.SYG.Ratio
VE.Jk.B.Ratio
VE.Jk.EB.SW2.Ratio

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$POP10                       #Defines the numerator variable y1
y2    <- oaxaca$POPMAL10                    #Defines the numerator variable y2
x     <- oaxaca$HOMES10                     #Defines the denominator variable x
Est.Ratio(y1[s==1], x[s==1], pik.U[s==1])   #Ratio estimator for y1 and x
Est.Ratio(y2[s==1], x[s==1], pik.U[s==1])   #Ratio estimator for y2 and x

---

Estimator of the regression coefficient using the Hajek point estimator

Description

Estimates the population regression coefficient using the Hajek (1971) point estimator.

Usage

Est.RegCo.Hajek(VecY.s, VecX.s, VecPk.s)

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.

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 $s$.

Value

The function returns a value for the regression coefficient point estimator.

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.

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

See Also

Est.RegCoI.Hajek
VE.Jk.Tukey.RegCo.Hajek
VE.Jk.CBS.HT.RegCo.Hajek
VE.Jk.CBS.SYG.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
#Computes the regression coefficient estimator for y1 and x
Est.RegCo.Hajek(y1[s==1], x[s==1], pik.U[s==1])
#Computes the regression coefficient estimator for y2 and x
Est.RegCo.Hajek(y2[s==1], x[s==1], pik.U[s==1])

---

Estimator of the intercept regression coefficient using the Hajek point estimator

Description

Estimates the population intercept regression coefficient using the Hajek (1971) point estimator.

Usage

Est.RegCoI.Hajek(VecY.s, VecX.s, VecPk.s)

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.

Details

From Linear Regression Analysis, for an imposed population model

$y=\alpha + \beta x$

the population intercept regression coefficient $\alpha$, assuming that the population size $N$ is unknown (see Sarndal et al., 1992, Sec. 5.10), can be estimated by:

$\hat{\alpha}_{Hajek} = \hat{\bar{y}}_{Hajek} - \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2} \hat{\bar{x}}_{Hajek}$

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 $s$.

Value

The function returns a value for the intercept regression coefficient point estimator.

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.

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

See Also

Est.RegCo.Hajek
VE.Jk.Tukey.RegCoI.Hajek
VE.Jk.CBS.HT.RegCoI.Hajek
VE.Jk.CBS.SYG.RegCoI.Hajek
VE.Jk.B.RegCoI.Hajek
VE.Jk.EB.SW2.RegCoI.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 intercept regression coefficient estimator for y1 and x
Est.RegCoI.Hajek(y1[s==1], x[s==1], pik.U[s==1])
#Computes the intercept regression coefficient estimator for y2 and x
Est.RegCoI.Hajek(y2[s==1], x[s==1], pik.U[s==1])

---

The Hajek estimator for a total

Description

Computes the Hajek (1971) estimator for a population total.

Usage

Est.Total.Hajek(VecY.s, VecPk.s, N)

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.

Details

For the population total of the variable $y$:

$t = \sum_{k\in U} y_k$

the approximately unbiased Hajek (1971) estimator of $t$ (implemented by the current function) 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 $s$.

Value

The function returns a value for the total point estimator.

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.

See Also

Est.Total.NHT
VE.Jk.Tukey.Total.Hajek
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$POP10                       #Defines the variable y1
y2    <- oaxaca$HOMES10                     #Defines the variable y2
Est.Total.Hajek(y1[s==1], pik.U[s==1], N)   #The Hajek estimator for y1
Est.Total.Hajek(y2[s==1], pik.U[s==1], N)   #The Hajek estimator for y2

---

The Narain-Horvitz-Thompson estimator for a total

Description

Computes the Narain (1951); Horvitz-Thompson (1952) estimator for a population total.

Usage

Est.Total.NHT(VecY.s, VecPk.s)

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.

Details

For the population total of the variable $y$:

$t = \sum_{k\in U} y_k$

the unbiased Narain (1951); Horvitz-Thompson (1952) estimator of $t$ (implemented by the current function) is given by:

$\hat{t}_{NHT} = \sum_{k\in s} \frac{y_k}{\pi_k}$

where $\pi_k$ denotes the inclusion probability of the $k$-th element in the sample $s$.

Value

The function returns a value for the total point estimator.

Author(s)

Emilio Lopez Escobar.

References

Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169–175.

See Also

Est.Total.Hajek
VE.HT.Total.NHT
VE.SYG.Total.NHT
VE.Hajek.Total.NHT

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$POP10                       #Defines the variable of interest y1
y2    <- oaxaca$HOMES10                     #Defines the variable of interest y2
Est.Total.NHT(y1[s==1], pik.U[s==1])        #Computes the NHT estimator for y1
Est.Total.NHT(y2[s==1], pik.U[s==1])        #Computes the NHT estimator for y2

---

Municipalities of the state of Oaxaca in Mexico

Description

Dataset with information about the free and sovereign state of Oaxaca, which is located in the southern part of Mexico. The dataset contains information on population, surface, indigenous language, agriculture, and income from years ranging from 2000 to 2010. The information was originally collected and processed by Mexico's National Institute of Statistics and Geography (INEGI by its name in Spanish, ‘Instituto Nacional de Estadistica y Geografia’, http://www.inegi.org.mx/).

Usage

data(oaxaca)

Format

A data frame with 570 observations on the following 41 variables:

IDREGION

region INEGI code.

LBREGION

region name (without accents and Spanish language characters).

IDDISTRI

district INEGI code.

LBDISTRI

district name (without accents and Spanish language characters).

IDMUNICI

municipality INEGI code.

LBMUNICI

municipality name (without accents and Spanish language characters).

SURFAC05

surface in squared kilometres 2005.

POP00

population 2000.

POP10

population 2010.

HOMES00

number of homes 2000.

HOMES10

number of homes 2010.

POPMAL00

male population 2000.

POPMAL10

male population 2010.

POPFEM00

female population 2000.

POPFEM10

female population 2010.

INLANG00

5 or more years old population which speaks indigenous language 2000.

INLANG10

5 or more years old population which speaks indigenous language 2010.

INCOME00

gross income in thousands of Mexican pesos 2000.

INCOME01

gross income in thousands of Mexican pesos 2001.

INCOME02

gross income in thousands of Mexican pesos 2002.

INCOME03

gross income in thousands of Mexican pesos 2003.

PTREES00

planted trees 2000.

PTREES01

planted trees 2001.

PTREES02

planted trees 2002.

PTREES03

planted trees 2003.

MARRIA07

marriages 2007.

MARRIA08

marriages 2008.

MARRIA09

marriages 2009.

HARVBE07

harvested bean surface in hectares 2007.

HARVBE08

harvested bean surface in hectares 2008.

HARVBE09

harvested bean surface in hectares 2009.

VALUBE07

value of bean production in thousands of Mexican pesos 2007.

VALUBE08

value of bean production in thousands of Mexican pesos 2008.

VALUBE09

value of bean production in thousands of Mexican pesos 2009.

VOLUBE07

volume of bean production in tons 2007.

VOLUBE08

volume of bean production in tons 2008.

VOLUBE09

volume of bean production in tons 2009.

sHOMES00

a sample (column vector of ones and zeros; 1 = selected, 0 = otherwise) of 373 municipalities drawn using the Hajek (1964) maximum-entropy sampling design with inclusion probabilities proportional to the variable HOMES00.

sSURFAC

a sample (column vector of ones and zeros; 1 = selected, 0 = otherwise) of 373 municipalities drawn using the Hajek (1964) maximum-entropy sampling design with inclusion probabilities proportional to the variable SURFAC05.

SIZEDIST

the size of the district, i.e., the number of municipalities in each district.

sSW_10_3

a sample (column vector of ones and zeros; 1 = selected, 0 = otherwise) of 30 municipalities drawn using a self-weighted two-stage sampling design. The first stage draws 10 districts using the Hajek (1964) maximum-entropy sampling design with clusters' inclusion probabilities proportional to the size of the clusters (variable SIZEDIST). The second stage draws 3 municipalities within the selected districts at the first stage, using equal-probability without-replacement sampling.

Source

Mexico's National Institute of Statistics and Geography (INEGI), ‘Instituto Nacional de Estadistica y Geografia’ http://www.inegi.org.mx/

Examples

data(oaxaca)                         #Loads the Oaxaca municipalities dataset
mean(oaxaca$INCOME00, na.rm= TRUE)   #Computes INCOME00 mean (note it has NA's)
median(oaxaca$INCOME00, na.rm= TRUE) #Computes INCOME00 median (note it has NA's)

---

Inclusion probabilities proportional to a specified variable.

Description

Creates and normalises the 1st order inclusion probabilities proportional to a specified variable. In the current context, normalisation means that the inclusion probabilities are less than or equal to 1. Ideally, they should sum up to $n$, the sample size.

Usage

Pk.PropNorm.U(n, VecMOS.U)

Arguments

n	the sample size. It must be an integer or a double-precision scalar with zero-valued fractional part.
VecMOS.U	vector of the variable called measure of size (MOS) to which the first-order inclusion probabilities are to be proportional; its length is equal to the population size. Values in VecMOS.U should be greater than zero (a warning message appears if this does not hold). There must not be missing values.

Details

Although the normalisation procedure is well-known in the survey sampling literature, we follow the procedure described in Chao (1982, p. 654). Hence, we obtain a unique set of inclusion probabilities that are proportional to the MOS variable.

Value

The function returns a vector of length $n$ with the inclusion probabilities.

Author(s)

Emilio Lopez Escobar.

References

Chao, M. T. (1982) A general purpose unequal probability sampling plan. Biometrika 69, 653–656.

See Also

Pkl.Hajek.s
Pkl.Hajek.U

Examples

data(oaxaca) #Loads the Oaxaca municipalities dataset
             #Creates the normalised 1st order incl. probs. proportional
             #to the variable oaxaca$HOMES00 and with sample size 373
pik.U <- Pk.PropNorm.U(373, oaxaca$HOMES00)
sum(pik.U)   #Shows the sum is equal to the sample size 373
any(pik.U>1) #Shows there isn't any probability greater than 1
any(pik.U<0) #Shows there isn't any probability less than 0

---

The Hajek approximation for the 2nd order (joint) inclusion probabilities (sample based)

Description

Computes the Hajek (1964) approximation for the 2nd order (joint) inclusion probabilities utilising only sample-based quantities.

Usage

Pkl.Hajek.s(VecPk.s)

Arguments

VecPk.s

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

Let $\pi_k$ denote the inclusion probability of the $k$-th element in the sample $s$, and let $\pi_{kl}$ denote the joint-inclusion probabilities of the $k$-th and $l$-th elements in the sample $s$. If the joint-inclusion probabilities $\pi_{kl}$ are not available, the Hajek (1964) approximation can be used. Note that this approximation is 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).

The sample-based version of the Hajek (1964) approximation for the joint-inclusion probabilities $\pi_{kl}$ (implemented by the current function) is:

$\pi_{kl} \doteq \pi_k \pi_l \{1 - \hat{d}^{-1}(1-\pi_k)(1-\pi_l)\}$

where $\hat{d} =\sum_{k\in s}(1-\pi_k)$.

The approximation was originally developed for $d\rightarrow\infty$, under the maximum-entropy sampling design (see Hajek 1981, Theorem 3.3, Ch. 3 and 6), the Rejective Sampling design. It requires that the utilised sampling design is of large entropy. An overview can be found in Berger and Tille (2009). An account of different sampling designs, $\pi_{kl}$ approximations, and approximate variances under large-entropy designs can be found in Tille (2006), Brewer and Donadio (2003), and Haziza, Mecatti, and Rao (2008). Recently, Berger (2011) gave sufficient conditions under which Hajek's results still hold for large-entropy sampling designs that are not the maximum-entropy one.

Value

The function returns a ($n$ by $n$) square matrix with the estimated joint inclusion probabilities, where $n$ is the sample size.

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. (2011) Asymptotic consistency under large entropy sampling designs with unequal probabilities. Pakistan Journal of Statististics, 27, 407–426.

Berger, Y. G. and Tille, Y. (2009) Sampling with unequal probabilities. In Sample Surveys: Design, Methods and Applications (eds. D. Pfeffermann and C. R. Rao), 39–54. Elsevier, Amsterdam.

Brewer, K. R. W. and Donadio, M. E. (2003) The large entropy variance of the Horvitz-Thompson estimator. Survey Methodology 29, 189–196.

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. (1981) Sampling From a Finite Population. Dekker, New York.

Haziza, D., Mecatti, F. and Rao, J. N. K. (2008) Evaluation of some approximate variance estimators under the Rao-Sampford unequal probability sampling design. Metron, LXVI, 91–108.

Tille, Y. (2006) Sampling Algorithms. Springer, New York.

See Also

Pkl.Hajek.U
Pk.PropNorm.U

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
#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
#First 5 rows/cols of (sample-based) 2nd order incl. probs. matrix
pikl.s[1:5,1:5]

---

The Hajek approximation for the 2nd order (joint) inclusion probabilities (population based)

Description

Computes the Hajek (1964) approximation for the 2nd order (joint) inclusion probabilities utilising population-based quantities.

Usage

Pkl.Hajek.U(VecPk.U)

Arguments

VecPk.U

vector of the first-order inclusion probabilities; its length is equal to the population size. Values in VecPk.U must be greater than zero and less than or equal to one. There must not be missing values.

Details

Let $\pi_k$ denote the inclusion probability of the $k$-th element in the sample $s$, and let $\pi_{kl}$ denote the joint-inclusion probabilities of the $k$-th and $l$-th elements in the sample $s$. If the joint-inclusion probabilities $\pi_{kl}$ are not available, the Hajek (1964) approximation can be used. Note that this approximation is 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).

The population-based version of the Hajek (1964) approximation for the joint-inclusion probabilities $\pi_{kl}$ (implemented by the current function) is:

$\pi_{kl} \doteq \pi_k \pi_l \{1 - d^{-1}(1-\pi_k)(1-\pi_l)\}$

where $d =\sum_{k\in U}\pi_k(1-\pi_k)$.

The approximation was originally developed for $d\rightarrow\infty$, under the maximum-entropy sampling design (see Hajek 1981, Theorem 3.3, Ch. 3 and 6), the Rejective Sampling design. It requires that the utilised sampling design is of large entropy. An overview can be found in Berger and Tille (2009). An account of different sampling designs, $\pi_{kl}$ approximations, and approximate variances under large-entropy designs can be found in Tille (2006), Brewer and Donadio (2003), and Haziza, Mecatti, and Rao (2008). Recently, Berger (2011) gave sufficient conditions under which Hajek's results still hold for large-entropy sampling designs that are not the maximum-entropy one.

Value

The function returns a ($N$ by $N$) square matrix with the estimated joint inclusion probabilities, where $N$ is the population size.

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. (2011) Asymptotic consistency under large entropy sampling designs with unequal probabilities. Pakistan Journal of Statististics, 27, 407–426.

Berger, Y. G. and Tille, Y. (2009) Sampling with unequal probabilities. In Sample Surveys: Design, Methods and Applications (eds. D. Pfeffermann and C. R. Rao), 39–54. Elsevier, Amsterdam.

Brewer, K. R. W. and Donadio, M. E. (2003) The large entropy variance of the Horvitz-Thompson estimator. Survey Methodology 29, 189–196.

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. (1981) Sampling From a Finite Population. Dekker, New York.

Haziza, D., Mecatti, F. and Rao, J. N. K. (2008) Evaluation of some approximate variance estimators under the Rao-Sampford unequal probability sampling design. Metron, LXVI, 91–108.

Tille, Y. (2006) Sampling Algorithms. Springer, New York.

See Also

Pkl.Hajek.s
Pk.PropNorm.U

Examples

data(oaxaca)                                 #Loads the Oaxaca municipalities dataset
pik.U  <- Pk.PropNorm.U(373, oaxaca$HOMES00) #Reconstructs the 1st order incl. probs.
#(This approximation is only suitable for large-entropy sampling designs)
pikl.U <- Pkl.Hajek.U(pik.U)                 #Approximates 2nd order incl. probs. from U
#First 5 rows/cols of (population-based) 2nd order incl. probs. matrix
pikl.U[1:5,1:5]

---

The Escobar-Berger unequal probability replicate variance estimator for the Hajek (1971) estimator of a mean (Horvitz-Thompson form)

Description

Computes the Escobar-Berger (2013) unequal probability replicate variance estimator for the Hajek estimator of a mean. It uses the Horvitz-Thompson (1952) variance form.

Usage

VE.EB.HT.Mean.Hajek(VecY.s, VecPk.s, MatPkl.s,
                    VecAlpha.s = rep.int(1, length(VecPk.s)))

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.
MatPkl.s	matrix of the second-order inclusion probabilities; its number of rows and columns equals $n$, 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.
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 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 $\alpha_k>1$ approximates the Demnati-Rao (2004) linearisation variance estimators. There must not be missing values.

Details

For the population mean of the variable $y$:

$\bar{y} = \frac{1}{N} \sum_{k\in U} y_k$

the approximately unbiased Hajek (1971) estimator of $\bar{y}$ is given by:

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

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 $\hat{\bar{y}}_{Hajek}$ can be estimated by the Escobar-Berger (2013) unequal probability replicate variance estimator (implemented by the current function):

$\hat{V}(\hat{\bar{y}}_{Hajek}) = \sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} \breve{\nu}_k \breve{\nu}_l$

where

$\breve{\nu}_k = w_k^{\alpha_k} \left(\hat{\bar{y}}_{Hajek}-\hat{\bar{y}}_{Hajek,k}^{*}\right)$

for some $\alpha_k\geq0$ (suggested to be 1, see below comments) and with

$\hat{\bar{y}}_{Hajek,k}^{*} = \frac{\sum_{l\in s} w_l y_l - w_k^{1-\alpha_k} y_k}{\sum_{l\in s} w_l - w_k^{1-\alpha_k}}$

Regarding the value of $\alpha_k$, Escobar-Berger (2013) show that $\hat{V}(\hat{\bar{y}}_{Hajek})$ 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 $\alpha_k=1$, $\hat{V}(\hat{\bar{y}}_{Hajek})$ reduces to the Escobar-Berger (2011) jackknife. Using $\alpha_k>1$ approximates 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.

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

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.

Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

See Also

VE.Jk.Tukey.Mean.Hajek
VE.Jk.CBS.SYG.Mean.Hajek
VE.Jk.B.Mean.Hajek
VE.Jk.EB.SW2.Mean.Hajek
VE.EB.SYG.Mean.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$POP10                       #Defines the variable y1
y2      <- oaxaca$POPMAL10                    #Defines the variable 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 mean point estimator using y1
VE.EB.HT.Mean.Hajek(y1[s==1], pik.U[s==1], pikl.s)
#Computes the var. est. of the Hajek mean point estimator using y2
VE.EB.HT.Mean.Hajek(y2[s==1], pik.U[s==1], pikl.s, Alpha.s)

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The Escobar-Berger unequal probability replicate variance estimator for the estimator of a ratio (Horvitz-Thompson form)

Description

Computes the Escobar-Berger (2013) unequal probability replicate variance estimator for the estimator of a ratio of two totals/means. It uses the Horvitz-Thompson (1952) variance form.

Usage

VE.EB.HT.Ratio(VecY.s, VecX.s, VecPk.s, MatPkl.s,
               VecAlpha.s = rep.int(1, length(VecPk.s)))

Arguments

VecY.s	vector of the numerator variable of interest; 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 denominator variable of interest; 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. All values of VecX.s should be greater than zero. A warning is displayed if this does not hold, and computations continue if mathematical expressions allow this kind of values for the denominator variable.
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 equals $n$, 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.
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 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 $\alpha_k>1$ approximates the Demnati-Rao (2004) linearisation variance estimators. There must not be missing values.

Details

For the population ratio of two totals/means of the variables $y$ and $x$:

$R = \frac{\sum_{k\in U} y_k/N}{\sum_{k\in U} x_k/N} = \frac{\sum_{k\in U} y_k}{\sum_{k\in U} x_k}$

the ratio estimator of $R$ is given by:

$\hat{R} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k x_k}$

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 $\hat{R}$ can be estimated by the Escobar-Berger (2013) unequal probability replicate variance estimator (implemented by the current function):

$\hat{V}(\hat{R}) = \sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} \breve{\nu}_k \breve{\nu}_l$

where

$\breve{\nu}_k = w_k^{\alpha_k} \left(\hat{R}-\hat{R}_k^{*}\right)$

for some $\alpha_k\geq0$ (suggested to be 1, see below comments) and with

$\hat{R}_k^{*} = \frac{\left(\sum_{l\in s} w_l y_l - w_k^{1-\alpha_k} y_k\right)/\left(\sum_{l\in s} w_l - w_k^{1-\alpha_k} \right)}{\left(\sum_{l\in s} w_l x_l - w_k^{1-\alpha_k} x_k\right)/\left(\sum_{l\in s} w_l - w_k^{1-\alpha_k} \right)} = \frac{\sum_{l\in s} w_l y_l - w_k^{1-\alpha_k} y_k}{\sum_{l\in s} w_l x_l - w_k^{1-\alpha_k} x_k}$

Regarding the value of $\alpha_k$, Escobar-Berger (2013) show that $\hat{V}(\hat{R})$ 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 $\alpha_k=1$, $\hat{V}(\hat{R})$ reduces to the Escobar-Berger (2011) jackknife. Using $\alpha_k>1$ approximates 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.

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

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.

Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

See Also

VE.Lin.HT.Ratio
VE.Lin.SYG.Ratio
VE.Jk.Tukey.Ratio
VE.Jk.CBS.HT.Ratio
VE.Jk.CBS.SYG.Ratio
VE.Jk.B.Ratio
VE.Jk.EB.SW2.Ratio
VE.EB.SYG.Ratio

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$POP10                       #Defines the numerator variable y1
y2      <- oaxaca$POPMAL10                    #Defines the numerator variable y2
x       <- oaxaca$HOMES10                     #Defines the denominator variable x
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 ratio point estimator using y1
VE.EB.HT.Ratio(y1[s==1], x[s==1], pik.U[s==1], pikl.s) #Using default VecAlpha.s
#Computes the var. est. of the ratio point estimator using y2
VE.EB.HT.Ratio(y2[s==1], x[s==1], pik.U[s==1], pikl.s, Alpha.s)

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The Escobar-Berger unequal probability replicate variance estimator for the Hajek (1971) estimator of a total (Horvitz-Thompson form)

Description

Computes the Escobar-Berger (2013) unequal probability replicate variance estimator for the Hajek estimator of a total. It uses the Horvitz-Thompson (1952) variance form.

Usage

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

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.
MatPkl.s	matrix of the second-order inclusion probabilities; its number of rows and columns equals $n$, 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 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 $\alpha_k>1$ approximates the Demnati-Rao (2004) linearisation variance estimators. There must not be missing values.

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):

$\hat{V}(\hat{t}_{Hajek}) = \sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} \breve{\nu}_k \breve{\nu}_l$

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 $\alpha_k>1$ approximates 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.

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

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.

Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.

See Also

VE.Jk.Tukey.Total.Hajek
VE.Jk.CBS.SYG.Total.Hajek
VE.Jk.B.Total.Hajek
VE.Jk.EB.SW2.Total.Hajek
VE.EB.SYG.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
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.HT.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.HT.Total.Hajek(y2[s==1], pik.U[s==1], pikl.s, N, Alpha.s)

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The Escobar-Berger unequal probability replicate variance estimator for the Hajek (1971) estimator of a mean (Sen-Yates-Grundy form)

Description

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

Usage

VE.EB.SYG.Mean.Hajek(VecY.s, VecPk.s, MatPkl.s,
                     VecAlpha.s = rep.int(1, length(VecPk.s)))

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.
MatPkl.s	matrix of the second-order inclusion probabilities; its number of rows and columns equals $n$, 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.
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 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 $\alpha_k>1$ approximates the Demnati-Rao (2004) linearisation variance estimators. There must not be missing values.

Details

For the population mean of the variable $y$:

$\bar{y} = \frac{1}{N} \sum_{k\in U} y_k$

the approximately unbiased Hajek (1971) estimator of $\bar{y}$ is given by:

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

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 $\hat{\bar{y}}_{Hajek}$ can be estimated by the Escobar-Berger (2013) unequal probability replicate variance estimator (implemented by the current function):

$\hat{V}(\hat{\bar{y}}_{Hajek}) = \frac{-1}{2}\sum_{k\in s}\sum_{l\in s} \frac{\pi_{kl}-\pi_k\pi_l}{\pi_{kl}} (\breve{\nu}_k - \breve{\nu}_l)^{2}$

where

$\breve{\nu}_k = w_k^{\alpha_k} \left(\hat{\bar{y}}_{Hajek}-\hat{\bar{y}}_{Hajek,k}^{*}\right)$

for some $\alpha_k\geq0$ (suggested to be 1, see below comments) and with

$\hat{\bar{y}}_{Hajek,k}^{*} = \frac{\sum_{l\in s} w_l y_l - w_k^{1-\alpha_k} y_k}{\sum_{l\in s} w_l - w_k^{1-\alpha_k}}$

Regarding the value of $\alpha_k$, Escobar-Berger (2013) show that $\hat{V}(\hat{\bar{y}}_{Hajek})$ 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 $\alpha_k=1$, $\hat{V}(\hat{\bar{y}}_{Hajek})$ reduces to the Escobar-Berger (2011) jackknife. Using $\alpha_k>1$ approximates 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.

Value

The function returns a value for the estimated variance.

Author(s)

Emilio Lopez Escobar.

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.

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.

See Also

VE.Jk.Tukey.Mean.Hajek
VE.Jk.CBS.HT.Mean.Hajek
VE.Jk.B.Mean.Hajek
VE.Jk.EB.SW2.Mean.Hajek
VE.EB.HT.Mean.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
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 mean point estimator using y1
VE.EB.SYG.Mean.Hajek(y1[s==1], pik.U[s==1], pikl.s)
#Computes the var. est. of the Hajek mean point estimator using y2
VE.EB.SYG.Mean.Hajek(y2[s==1], pik.U[s==1], pikl.s, Alpha.s)

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