Comparing the usual and model based estimates of variance for the ratio estimator

Consider a population where the ratio estimator makes sense. Assuming simple random sampling without replacement we have the usual estimate of its variance. In class I also described the model based approach to the ratio estimate and gave an estimate of variance based on the model.

For R samples the function compratiolp calculates the ratio estimate, its absolute error and the two estimates of variance. For each sample it also calculates the sample mean of the x values. The output is a R by five matrix where each row contains the ratio estimate, its absolute error, the usual estimate of variance, the model based estimate of variance and the sample mean of the x values. The matrix has been ordered so that the sample with the smallest sample mean of the x values is in the first row. The second row contains the results for the second smallest sample mean of the x values and so on.

ratiototboth<-function(smp,popy,popx) { n <- length(smp) N<-length(popx) ff<-n/N ysamp<-popy[smp] xsamp<-popx[smp] xnsamp<-popx[-smp] tx<-sum(popx) trtot<-sum(popy) rhat <- sum(ysamp)/sum(xsamp) esttot <- rhat * tx err<-abs(esttot - trtot) dum1<-(N*N*(1-ff))/(n*(n-1)) usualvartot <- dum1*sum((ysamp-rhat*xsamp)^2) usualans<-c(esttot,err,usualvartot) dum2<-sum(((ysamp -rhat*xsamp)^2/xsamp))*((mean(xnsamp) *mean(popx))/mean(xsamp)) modelvartot<-dum1*dum2 ans<-c(esttot,err,usualvartot,modelvartot) return(ans) } compratiolp<-function(popy,popx,n,R) { N<-length(popy) xbar<-rep(0,R) ans<-matrix(0,R,4) for(i in 1:R){ samp<-sample(1:N,n) xbar[i]<-mean(popx[samp]) ans[i,]<-ratiototboth(samp,popy,popx) } xbarord<-order(xbar) xbar<-xbar[xbarord] ans[1:R,]<-ans[xbarord,] ans<-cbind(ans,xbar) ans<-round(ans,digits=2) return(ans) } popx<-rgamma(100,5) + 50 popy<-rnorm(100,2.2*popx,.3*popx) out<-compratiolp(popy,popx,10,5) out

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