optimStrat (version 2.0)

expmsepips: Anticipated Mean Squared Error of a PIps design

Description

Compute the anticipated Mean Squared Error of a PIps design.

Usage

expmsepips(x, pik, n, Beta11, Beta12, Beta21, Beta22, Delta12, Rfy, ak = 1)

Arguments

x

a matrix or data.frame giving the values of the auxiliary variables.

pik

a numeric vector giving the variable with respect to which the inclusion probabilities are to be obtained.

n

a positive integer indicating the desired sample size.

Beta11

a numeric vector of length equal to the number of variables in x giving the coefficients of the trend term in the true superpopulation model (see ‘Details’).

Beta12

a numeric vector of length equal to the number of variables in x giving the exponents of the trend term in the true superpopulation model (see ‘Details’).

Beta21

a numeric vector of length equal to the number of variables in x giving the coefficients of the spread term in the true superpopulation model (see ‘Details’).

Beta22

a numeric vector of length equal to the number of variables in x giving the exponents of the spread term in the true superpopulation model (see ‘Details’).

Delta12

a numeric vector of length equal to the number of variables in x giving the exponents of the trend term in the assumed superpopulation model (see ‘Details’).

Rfy

a number giving the square root of the coefficient of determination between the auxiliary variables and the study varible.

ak

a vector of weights.

Value

A numeric value giving the anticipated Mean Squared Error.

Details

The Anticipated Mean Squared Error of the strategy that couples a PIps design with the general regression estimator of a sample of size n is computed.

It is assumed that the underlying superpopulation model is of the form $$Y_{k} = \sum_{j=1}^{J}\delta_{1,j}x_{jk}^{\delta_{1,J+j}} + \epsilon_{k}$$ with \(E\epsilon_{k}=0\), \(V\epsilon_{k}=\sigma^{2}\sum_{j=1}^{J}\delta_{2,j}x_{jk}^{\delta_{2,J+j}}\) and \(Cov(\epsilon_{k},\epsilon_{l})=0\).

But the true generating model is of the form $$Y_{k} = \sum_{j=1}^{J}\beta_{1,j}x_{jk}^{\beta_{1,J+j}} + \epsilon_{k}$$ with \(E\epsilon_{k}=0\), \(V\epsilon_{k}=\sigma^{2}\sum_{j=1}^{J}\beta_{2,j}x_{jk}^{\beta_{2,J+j}}\) and \(Cov(\epsilon_{k},\epsilon_{l})=0\).

The coefficients \(\beta_{1,j}\) (\(j=1,\cdots,J\)) are given by Beta11. The exponents \(\beta_{1,j}\) (\(j=J+1,\cdots,2J\)) are given by Beta12. The coefficients \(\beta_{2,j}\) (\(j=1,\cdots,J\)) are given by Beta21. The exponents \(\beta_{2,j}\) (\(j=J+1,\cdots,2J\)) are given by Beta22.

The exponents \(\delta_{1,j}\) (\(j=J+1,\cdots,2J\)) are given by Delta12.

The inclusion probabilities are calculated as \(n\times x_{k}/t_{x}\) and corrected, if necessary, to ensure that they are smaller or equal than one.

References

Bueno, E. (2018). A Comparison of Stratified Simple Random Sampling and Probability Proporional-to-size Sampling. Research Report, Department of Statistics, Stockholm University 2018:6. http://gauss.stat.su.se/rr/RR2018_6.pdf.

Examples

# NOT RUN {
x1<- 1 + sort( rgamma(5000, shape=4/9, scale=108) )
x2<- 1 + sort( rgamma(5000, shape=4/9, scale=108) )
x3<- 1 + sort( rgamma(5000, shape=4/9, scale=108) )
x<- cbind(x1,x2,x3)
expmsepips(x,pik=x3,n=150,Beta11=c(1,-1,0),Beta12=c(1,1,0),Beta21=c(0,0,1),
   Beta22=c(0,0,0.5),Delta12=c(1,1,0),Rfy=0.8)
expmsepips(x,pik=x3,n=150,Beta11=c(1,-1,0),Beta12=c(1,1,0),Beta21=c(0,0,1),
   Beta22=c(0,0,0.5),Delta12=c(1,1,1),Rfy=0.8)
# }