> # HW 14
> 
> fn1<-function(theta){ prod(exp(-.5*(xvec-theta)^2))}  # Normal likelihood
> fn2<-function(theta){ prod(.5*exp(-1*abs(xvec-theta)))}  # Laplace likelihood
> fn3<-function(theta){ prod((exp(-1*(xvec-theta)))/( 1+exp(-1*(xvec-theta)) )^2 )} # Logistic
> 
> a<-rep(0,1000)   # INITIALIZE NORMAL MLE STORAGE
> b<-rep(0,1000)   # INITIALIZE LAPLACE MLE STORAGE
> c<-rep(0,1000)   # INITIALIZE LOGISTIC MLE STORAGE
> 
> for(i in 1:1000){
+ xvec<-rnorm(20)        # GENERATE NORMAL DATA n=20
+ a[i]<-optimize(fn1, c(-5,5), maximum=T)$maximum
+ b[i]<-optimize(fn2, c(-5,5), maximum=T)$maximum
+ c[i]<-optimize(fn3, c(-5,5), maximum=T)$maximum
+ }
> # COMPARE VARIANCE OF THREE MLE's (NORMAL, LAPLACE, LOGISTIC)
> # NORMAL DATA
> var(a)
[1] 0.0516323
> var(b)
[1] 0.07510302
> var(c)
[1] 0.05175143
> 
> library(smoothmest)    # FOR GENERATING LAPLACE DISTRIBUTION
> for(i in 1:1000){
+   xvec<-rdoublex(20)     # GENERATE LAPLACE DATA n=20
+   a[i]<-optimize(fn1, c(-5,5), maximum=T)$maximum
+   b[i]<-optimize(fn2, c(-5,5), maximum=T)$maximum
+   c[i]<-optimize(fn3, c(-5,5), maximum=T)$maximum
+ }
> # COMPARE VARIANCE OF THREE MLE's (NORMAL, LAPLACE, LOGISTIC)
> # LAPLACE DATA
> var(a)
[1] 0.09772875
> var(b)
[1] 0.06800851
> var(c)
[1] 0.07490839
> 
> for(i in 1:1000){
+   xvec<-rlogis(20)     # GENERATE LOGISTIC DATA n=20
+   a[i]<-optimize(fn1, c(-5,5), maximum=T)$maximum
+   b[i]<-optimize(fn2, c(-5,5), maximum=T)$maximum
+   c[i]<-optimize(fn3, c(-5,5), maximum=T)$maximum
+ }
> # COMPARE VARIANCE OF THREE MLE's (NORMAL, LAPLACE, LOGISTIC)
> # LOGISTIC DATA
> var(a)
[1] 0.1691234
> var(b)
[1] 0.204597
> var(c)
[1] 0.1566512
>