Add invariance principle visualization

ch08.r

@@ -55,7 +55,9 @@ lambda = as.data.frame(load.image("cameraman.tif"))
 lambda = xtabs(value ~ x+y, data=lambda)
 T = 100
 x = c()
-for (i in 1:T) x = append(x, rpois(length(lambda), lambda))
+for (i in 1:T) { 
+    x = append(x, rpois(length(lambda), lambda))
+}
 x = array(x, c(256, 256, 100))
 y = (x>=1)
 mu = apply(y, c(1,2), mean)
@@ -70,4 +72,29 @@ image(flip_matrix(fig1), col=gray.colors(255))
 image(flip_matrix(lambdahat), col=gray.colors(255))
 
 #############
+# Chapter 8.2 Properties of the ML estimation
+
+## Visualizing the invariance principle
+
+# R code
+N = 50
+S = 20
+theta = seq(0.1, 0.9, (0.1+0.9)/1000)
+L = S * log(theta) + (N-S) * log(1-theta)
+plot(theta, L, type="n", xlab=expression(theta), ylab=expression(paste("Log L(", theta, "|S = 20)")))
+title("Bernoulli")
+lines(theta, L, lwd=6, col="#8080BF")
+grid()
+
+h_theta = -log(1-theta)
+plot(theta, h_theta, type="n", xlab=expression(theta), ylab=expression(paste(eta, " = h(", theta, ")")))
+lines(theta, h_theta, lwd=6)
+grid()
+
+theta = seq(0.1, 2.5, (0.1+2.5)/1000)
+L = S * log(1-exp(-theta)) - theta * (N-S)
+plot(theta, L, type="n")
+title("Truncated Poisson")
+lines(theta, L, lwd=6, col="#0000BF")
+grid()