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demo-asus

This vignette provides a quick demo of the functionalities of the asus package.

We will consider a two sample estimation problem where the goal is to estimate an \(n\) dimensional parameter \(\boldsymbol{\theta} =\boldsymbol{\theta}_x-\boldsymbol{\theta}_y\) based on observations \(X_i\sim N(\theta_{x,i},1)\) and \(Y_i\sim N(\theta_{y,i},1)\) where \(i=1,\ldots,n\).

Example 1 - SureShrink estimator and SURE estimate of risk

# Parameter of interest
n<-500
set.seed(42)
theta.x<- c(matrix(0,1,400),runif(n-400,2,6))
set.seed(42)
theta.y<- c(matrix(0,1,450),runif(n-450,2,3))
theta<-theta.x-theta.y

#observations
v.x<- rep(1,n)
v.y<- rep(1,n)
set.seed(42)
x<-rnorm(n,theta.x,v.x)
set.seed(84)
y<-rnorm(n,theta.y,v.y)

# SureShrink estimator of theta
library("asus")
u<- x-y
v.u<- v.x+v.y
theta.ss<-sureshrink(u,v.u)$est

plot(1:n,theta,pch=19,ylab ="theta and theta.ss in red")
points(1:n,theta.ss,col="red",pch=19)

# SURE estimate of risk of theta.hat and choice of threshold
out<- sureshrink.mse(u,v.u)
mse.ss<- out$sure.est
t.ss<- out$t

Example 2 - Extended James-Stein estimator

# EJS estimator of theta
theta.ejs<- ejs(u,v.u)

plot(1:n,theta,pch=19,ylab ="theta and theta.ejs in green")
points(1:n,theta.ejs,col="green",pch=19)

Example 3 - ASUS and SURE estimate of risk of ASUS

# side information on theta
s<- abs(x+y) 
out<- asus(u,v.u,s)

# ASUS estimator
theta.asus<- out$est

plot(1:n,theta,pch=19,ylab ="theta and theta.asus in cyan")
points(1:n,theta.asus,col="cyan",pch=19)

# SURE estimate of risk of ASUS
mse.asus<-out$mse

# Grouping and thresholding parameters
tau<- out$tau
t.asus<- out$t

# Group sizes
n.asus<- out$size

You will notice that mse.asus = 187.4638917 < 816.9164054 = mse.ss due to the efficient incorporation of side information \(\boldsymbol{s}\) into the shrinkage estimation procedure.

Finally in the plot above, the green and the red dots represent the two groups constructed by ASUS using the side information in \(\boldsymbol{s}\).

i1<- (s<=tau)
i2<-(s>tau)
idx<-1:n
plot(idx[i1],u[i1],xlab="1:n",ylab="u",col="red",pch=19)#group 1
points(idx[i2],u[i2],col="green",pch=19)#group 2

Notice how the two groups differ with respect to their signal sparsity with group 1 (red) being more sparse than group 2 (green). Indeed, t.asus[1] = 3.4535719 > 0.4808622 = t.asus[2] and compare this with t.ss = 3.5255094.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.