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compatible.Rmd
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---
title: "Compatible model selection"
author: "Jean-Michel Marin"
date: "4/02/2022"
output: pdf_document
fontsize: 11pt
numbersections: true
editor_options:
markdown:
wrap: 72
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
# Gaussian example
We consider an $n$-sample from $F=\mathcal{N}(0,1)$.
## Known variance - True model belongs to the parametric family
We consider the parametric family $Q_\theta=\mathcal{N}(\theta,1)$ where
$\theta\in\mathbb{R}$.
In such a case, we have
$\displaystyle \beta=\max_{\theta\in\mathbb{R}}\int\log(q(y|\theta))f(y)\text{d}y=-\log(2\pi)/2-1/2$.
For the $n$-sample we have $\beta_n=-n/2\log(2\pi)-n/2$.
```{r gaussian1, cache=TRUE}
beta <- -1/2*log(2*pi)-1/2
N <- 1000 ; n <- 500 ; B <- 500 ; alpha <- 0.1
statboot <- rep(0,B) ; cover <- 0
for (i in 1:N)
{
x <- rnorm(n)
sn2 <- var(x)*(n-1)/n
for (j in 1:B)
{
xstar <- sample(x,n,rep=TRUE)
sn2star <- var(xstar)*(n-1)/n
statboot[j] <- sn2/2-sn2star/2
}
interva <- -1/2*log(2*pi)-sn2/2-
quantile(statboot,c(1-alpha/2,alpha/2),names=FALSE)
cover <- cover + (beta >= interva[1] & beta <= interva[2])
}
cover/N
```
## Unknown variance - True model belongs to the parametric family
We consider the parametric family
$Q_{(\mu,\sigma^2)}=\mathcal{N}(\mu,\sigma^2)$ where
$(\mu,\sigma^2)\in\mathbb{R}\times\mathbb{R}_+^*$.
In such a case, we have
$\displaystyle \beta=\max_{(\mu,\sigma^2)\in\in\mathbb{R}\times\mathbb{R}_+^*}\int\log(q(y|(\mu,\sigma^2)))f(y)\text{d}y=-\log(2\pi)/2-1/2$.
For the $n$-sample, we have $\beta_n=-n/2\log(2\pi)-n/2$.
```{r gaussian2, cache=TRUE}
beta <- -1/2*log(2*pi)-1/2
N <- 1000 ; n <- 500 ; B <- 500 ; alpha <- 0.1
statboot <- rep(0,B) ; cover <- 0
for (i in 1:N)
{
x <- rnorm(n)
sn2 <- var(x)*(n-1)/n
for (j in 1:B)
{
xstar <- sample(x,n,rep=TRUE)
sn2star <- var(xstar)*(n-1)/n
statboot[j] <- log(sn2)/2-log(sn2star)/2
}
interva <- -1/2*(log(2*pi)+1)-log(sn2)/2-
quantile(statboot,c(1-alpha/2,alpha/2),names=FALSE)
cover <- cover + (beta >= interva[1] & beta <= interva[2])
}
cover/N
```
# Regression example
We consider an $n$-sample from the Gaussian regression model $$
F=[y|x_1,x_2]=\mathcal{N}(1+x_1+x_2,1)\,.
$$
```{r regression-design-matrix, eval=FALSE}
n <- 500
X <- cbind(rep(1,n),matrix(rnorm(n*2),n,2))
Xm <- solve(t(X)%*%X)
p <- dim(X)[2]
betan <- -n/2*log(2*pi)-n/2
```
## Known variance - True model belongs to the parametric family
We consider the parametric family
$Q_{(\theta_0,\theta_1,\theta_2)}=\mathcal{N}(\theta_0+\theta_1x_1+\theta_2x_2,1)$.
In such a case, for the $n$-sample, we have
$\beta_n=-n/2\log(2\pi)-n/2$.
```{r regression1, eval=FALSE}
N <- 1000 ; B <- 500 ; alpha <- 0.1
statboot <- rep(0,B) ; moy <- ampli <- cover <- 0
# tp <- txtProgressBar(min = 1, max = N, style = 3, char = "*")
for (i in 1:N)
{
y <- 1*X[,1]+1*X[,2]+1*X[,3]+rnorm(n,0,1)
yhat <- X%*%Xm%*%t(X)%*%y
r <- y-yhat
radj <- r/sqrt(1-p/n)
for (j in 1:B)
{
radjstar <- sample(radj,n,rep=TRUE)
ystar <- yhat+radjstar
yhatstar <- X%*%Xm%*%t(X)%*%ystar
statboot[j] <- sum(r^2)/2-sum((ystar-yhatstar)^2)/2
}
interva <- -n/2*log(2*pi)-sum(r^2)/2-
quantile(statboot,c(1-alpha/2,alpha/2),names=FALSE)
moy <- moy+sum(interva)/2
ampli <- ampli+diff(interva)
cover <- cover + (betan >= interva[1] & betan <= interva[2])
# setTxtProgressBar(tp,i)
}
c(betan,moy/N,ampli/N,cover/N)
```
## Unknown variance - True model belongs to the parametric family
We consider the parametric family
$Q_{(\theta_0,\theta_1,\theta_2,\theta_3)}=\mathcal{N}(\theta_0+\theta_1x_1+\theta_2x_2,\theta_3)$
where $\theta_3>0$. In such a case, for the $n$-sample, we have $$
\beta_n=-n/2\log(2\pi)-n/2
$$
```{r regression2, eval=FALSE}
N <- 1000 ; B <- 500 ; alpha <- 0.1
statboot <- rep(0,B) ; moy <- ampli <- cover <- 0
# tp <- txtProgressBar(min = 1, max = N, style = 3, char = "*")
for (i in 1:N)
{
y <- 1*X[,1]+1*X[,2]+1*X[,3]+rnorm(n,0,1)
yhat <- X%*%Xm%*%t(X)%*%y
r <- y-yhat
radj <- r/sqrt(1-p/n)
for (j in 1:B)
{
radjstar <- sample(radj,n,rep=TRUE)
ystar <- yhat+radjstar
yhatstar <- X%*%Xm%*%t(X)%*%ystar
statboot[j] <- n*log(mean(r^2))/2-n*log(mean((ystar-yhatstar)^2))/2
}
interva <- -n/2*(log(2*pi)+1)-n*log(mean(r^2))/2-
quantile(statboot,c(1-alpha/2,alpha/2),names=FALSE)
moy <- moy+sum(interva)/2
ampli <- ampli+diff(interva)
cover <- cover + (betan >= interva[1] & betan <= interva[2])
# setTxtProgressBar(tp,i)
}
c(betan,moy/N,ampli/N,cover/N)
```
## Unknown variance - True model bigger than the quasi true model
We consider the parametric family $$
Q_{(\theta_0,\theta_1)}=\mathcal{N}(\theta_0+\theta_1x_1,\theta_3)
$$
```{r regression3, eval=FALSE}
N <- 1000 ; B <- 500 ; alpha <- 0.1
statboot <- rep(0,B) ; moy <- ampli <- cover <- 0
# tp <- txtProgressBar(min = 1, max = N, style = 3, char = "*")
for (i in 1:N)
{
y <- 1*X[,1]+1*X[,2]+1*X[,3]+rnorm(n,0,1)
yhat <- X%*%Xm%*%t(X)%*%y
r <- y-yhat
radj <- r/sqrt(1-p/n)
for (j in 1:B)
{
radjstar <- sample(radj,n,rep=TRUE)
ystar <- yhat+radjstar
yhatstar <- X%*%Xm%*%t(X)%*%ystar
statboot[j] <- sum(r^2)/2-sum((ystar-yhatstar)^2)/2
}
interva <- -n/2*log(2*pi)-sum(r^2)/2-
quantile(statboot,c(1-alpha/2,alpha/2),names=FALSE)
moy <- moy+sum(interva)/2
ampli <- ampli+diff(interva)
cover <- cover + (betan >= interva[1] & betan <= interva[2])
# setTxtProgressBar(tp,i)
}
c(betan,moy/N,ampli/N,cover/N)
```