VE.Jk.Tukey.Corr.NHT {samplingVarEst} | R Documentation |
The Tukey (1958) jackknife variance estimator for the estimator of a correlation coefficient using the Narain-Horvitz-Thompson point estimator
Description
Computes the Quenouille(1956); Tukey (1958) jackknife variance estimator for the estimator of a correlation coefficient of two variables using the Narain (1951); Horvitz-Thompson (1952) point estimator.
Usage
VE.Jk.Tukey.Corr.NHT(VecY.s, VecX.s, VecPk.s, N, FPC= TRUE)
Arguments
VecY.s |
vector of the variable of interest Y; its length is equal to |
VecX.s |
vector of the variable of interest X; its length is equal to |
VecPk.s |
vector of the first-order inclusion probabilities; its length is equal to |
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 |
logical value. If an ad hoc finite population correction |
Details
For the population correlation coefficient of two variables and
:
the point estimator of is given by:
where is the Narain (1951); Horvitz-Thompson (1952) estimator for the population mean
,
and with
denoting the inclusion probability of the
-th element in the sample
. The variance of
can be estimated by the Quenouille(1956); Tukey (1958) jackknife variance estimator (implemented by the current function):
where has the same functional form as
but omitting the
-th element from the sample
.
We are implementing the Tukey (1958) jackknife variance estimator using the ‘ad hoc’ finite population correction
(see Shao and Tu, 1995; Wolter, 2007). If
FPC=FALSE
, then the term is omitted from the above formula.
Value
The function returns a value for the estimated variance.
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.
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.
See Also
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 var. est. of the corr. coeff. point estimator using y1
VE.Jk.Tukey.Corr.NHT(y1[s==1], x[s==1], pik.U[s==1], N)
#Computes the var. est. of the corr. coeff. point estimator using y2
VE.Jk.Tukey.Corr.NHT(y2[s==1], x[s==1], pik.U[s==1], N, FPC= FALSE)