Stock Portfolio example: 
* alpha represents the fraction of funds invested in X 
  (and thus 1-alpha invested in Y). 
* The variance of the total investment, var(alpha*X+(1-alpha)*Y), is minimized at
    alpha = [var(Y)-cov(X,Y)]/[var(X)+var(Y)-2*cov(X,Y)]
* Estimating alpha is simple, but an estimate of SE(alpha) is difficult 
  without a resampling procedure such as the bootstrap. 

#=================================================
#excerpt from ISLR_ALL-Labs.txt Chapter 5 (with additions)
#The Bootstrap
 library(ISLR)
 library(boot)

#SET RandomNumberGenerator TYPE TO PRE-3.6.0 DEFAULT IN ORDER TO MATCH TEXT RESULTS 
 RNGversion('3.5.3')

   alpha.fn=function(data,index){
     X=data$X[index]
     Y=data$Y[index]
     return((var(Y)-cov(X,Y))/(var(X)+var(Y)-2*cov(X,Y)))
   }
 dim(Portfolio)
 alpha.fn(Portfolio,1:100) #0.576
 set.seed(1)
 alpha.fn(Portfolio,sample(100,100,replace=T)) #0.596
 boot(Portfolio,alpha.fn,R=1000)

  #-------------------------------------------------
   set.seed(2)
   temp = boot(Portfolio,alpha.fn,R=1000)
   print(temp)
   attributes(temp)
   alpha.fn(Portfolio,1:100) 
   temp$t0                #mean of original data
   hist(temp$t)           #histogram of Bootstrap replicates           
   mean(temp$t)           #Bootstrap estimate (mean of Bootstrap samples)
   mean(temp$t) - temp$t0 #Bias
   hist(temp$t)           
   text(temp$t0,0,"X")    #place an "X" at the original mean
  #-------------------------------------------------

# Estimating the Accuracy of a Linear Regression Model

 data(Auto)
 boot.fn=function(data,index)
   return(coef(lm(mpg~horsepower,data=data,subset=index)))
 boot.fn(Auto,1:392)
 set.seed(1)
 boot.fn(Auto,sample(392,392,replace=T))
 boot.fn(Auto,sample(392,392,replace=T))
 boot(Auto,boot.fn,1000)
 summary(lm(mpg~horsepower,data=Auto))$coef
 boot(Auto,boot.fn,1000)
 boot(Auto,boot.fn,1000)

 boot.fn=function(data,index)
   coefficients(lm(mpg~horsepower+I(horsepower^2),data=data,subset=index))
 set.seed(1)
 boot(Auto,boot.fn,1000)
 mod=lm(mpg~horsepower+I(horsepower^2),data=Auto)
 summary(mod)$coef
#=================================================