# Uses the function varbeta0hat to calculate the variance of the y-intercept in a 
# straight line regression with 10 values of x.  
#

#Set up the constraints on the x-values
diag10 <- diag(1,10)
cmat <- rbind(diag10,-diag10)
uvec <- rep(c(0,-1),each=10)

varbeta0hat <- function(x)
{
 a <- sum(x^2)
 v <- a/(10*a - (sum(x))^2)
 v
}


#Then the R function constrOptim is used to minimise varbeta0hat.
start <- c(0.0026,0.0076,0.0088,0.0266,0.0450,0.0523,0.0564,0.0631,
  0.1301,0.9665)
out <- constrOptim(start,varbeta0hat,NULL,cmat,uvec,method="Nelder-Mead")
out


# The function gradvar gives the partial derivatives of the variance with repect to each of each x-value.
gradvar <- function(x)
{
 a <- sum(x^2)
 b <- sum(x)
 num <- 2*b*(a - b*x)
 denom <- (10*a - b^2)^2
 gradvector <- num/denom
 gradvector
}

#Now use constrOptim with gradvar to find the optimum solution
start <- c(0.0026,0.0076,0.0088,0.0266,0.0450,0.0523,0.0564,0.0631,
  0.1301,0.9665)
out <- constrOptim(start,varbeta0hat,gradvar,cmat,uvec)
out