Comments on Assignment 7 
			========================

1.(a) Using the default "width", for example, you found that all the 
      kernels lead to essentially the same density estimate, except 
      perhaps for the rectangular kernel which produces a rather 
      "bumpy" estimate (Why is that?).  

  (b) Findings as discussed in lab: small ==> rough estimate (low 
      bias and high variability); large ==> smooth estimate (high bias
      and low variability). Need to balance these two aspects.


2.(a) For an exponential with mean = theta:
	- var = theta^2
	- median = theta x log(2)

  (b) The normal approximations to the sampling distributions become:
	sample mean: 	mean = theta, var = theta^2/n
	sample median:  mean = theta x log(2), var = theta^2/n (also!) 
 
      For theta = 1, the normal approximations become: 
	sample mean: 	mean = 1.00, var = 0.10
	sample median:  mean = 0.69, var = 0.10

      Then you needed to simulate a bunch (500 at a minimum) of such 
      exponential samples to obtain a bunch of sample means and medians 
      (use the function you wrote for Assignment #3). Do QQplots with 
      the quantiles of the specific normals (not any normal) above.     


      #R codes

      expmean<-function(b)
                {
                  meanvec <- NULL  
                  for (i in 1:b)
                     { 
                      meanvec[i] <- mean(rexp(10,rate=1))
                     } 
                  return(meanvec)
                } 


      expmedian<-function(b)
                {
                  medianvec <- NULL  
                  for (i in 1:b)
                     { 
                      medianvec[i] <- median(rexp(10,rate=1))
                     } 
                  return(medianvec)
                } 

      meanvector<-expmean(1000)
      medianvector<-expmedian(1000)

      qqplot( meanvector, qnorm( ppoints(meanvector),mean=1,sd=sqrt(.10) ) )
      qqplot(medianvector, qnorm(ppoints(medianvector),0.69,sd=sqrt(.10) ) ) 
  
      mean(meanvector) # should be close to 1
      var(meanvector) # should be close to .10

      mean(medianvector) # should be close to .69
      var(medianvector) # should be close to .10
    
      For n = 10, these normal approximations are far from perfect, 
      but they will be adequate for rough calculations.