VE.Jk.Tukey.Corr.NHT | R Documentation |
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.
VE.Jk.Tukey.Corr.NHT(VecY.s, VecX.s, VecPk.s, N, FPC= TRUE)
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 |
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 |
vector of the first-order inclusion probabilities; its length is equal to n, the sample size. Values in |
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 FPC=1-n/N is to be used. The default is TRUE. |
For the population correlation coefficient of two variables y and x:
C = \frac{∑_{k\in U} (y_k - \bar{y})(x_k - \bar{x})}{√{∑_{k\in U} (y_k - \bar{y})^2}√{∑_{k\in U} (x_k - \bar{x})^2}}
the point estimator of C is given by:
\hat{C} = \frac{∑_{k\in s} w_k (y_k - \hat{\bar{y}}_{NHT})(x_k - \hat{\bar{x}}_{NHT})}{√{∑_{k\in s} w_k (y_k - \hat{\bar{y}}_{NHT})^2}√{∑_{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} ∑_{k\in U} y_k,
\hat{\bar{y}}_{NHT} = \frac{1}{N}∑_{k\in s} w_k y_k
and w_k=1/π_k with π_k denoting the inclusion probability of the k-th element in the sample s. The variance of \hat{C} can be estimated by the Quenouille(1956); Tukey (1958) jackknife variance estimator (implemented by the current function):
\hat{V}(\hat{C}) = ≤ft(1-\frac{n}{N}\right)\frac{n-1}{n}∑_{k\in s} ≤ft( \hat{C}_{(k)}-\hat{C} \right)^2
where \hat{C}_{(k)} has the same functional form as \hat{C} 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.
The function returns a value for the estimated variance.
Emilio Lopez Escobar.
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.
Est.Corr.Hajek
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)
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