optimStrat (version 2.0)

expmsestsi: Anticipated Mean Squared Error of a STSI design

Description

Compute the anticipated Mean Squared Error of a Stratified Simple Random Sampling design.

Usage

expmsestsi(x, stratum, nh, Beta11, Beta12, Beta21, Beta22, Delta12, Rfy, ak = 1)

Arguments

x

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

stratum

a vector indicating the stratum to which each element belongs.

nh

a vector indicating the sample size of the stratum to which each element belongs.

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 STSI design with the general regression estimator 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.

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)
stratum1<- optiallo(n=150,x=x3,H=6)
expmsestsi(x,stratum1$stratum,stratum1$nh,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)
expmsestsi(x,stratum1$stratum,stratum1$nh,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)
# }