# F79MA 2018-19: Assessed Project 1 Report

## Introduction and Summary

This project is aim to look at parameter estimation for an inverse gamma model, which can be considered as a survival distribution. Also, we will show that computer simulation can be a very useful tool to characterize the performance of statistical estimators.

## Analysis

The probability density function of inverse distribution is where is a shape parameter, is a scale parameter, and is the gamma function.

Let be a random sample from an inverse gamma distribution with shape parameter and unknown scale parameter .

As is i.i.d, the joint density function is Then Differentiating with respect to , we get the likelihood equation Differentiating again, we get Which is negative for all , and any n.

Solving equation (1.3) for , we get   sol-F79MA.docx

r code：

```rm(list=ls(all=TRUE))
library('invgamma')

alpha<-10
beta<-1   #true value

ntime<-1000
nmax<-50
betahat<-rep(0,ntime)
vac<-rep(0,nmax)
MSE<-rep(0,nmax)

for (n in 1:nmax){
# repeat times
for (t in 1:ntime){
Y<-rinvgamma(n,alpha,rate=1)
betahat[t]<-alpha/mean(1/Y)
}
vac[n]<-var(betahat)
MSE[n]<-1/ntime*sum(betahat-beta)^2
}

n<-c(1:50)
plot(n,MSE,main='MSE vs n')

crbound<-beta^2/alpha/n^2
plot(MSE-crbound)
abline(h=0,col='red')

betas<-c(0.3,0.5,0.8,1,1.2)
MSEMME<-matrix(0,nrow=5,ncol=3)
MSEMLE<-matrix(0,nrow=5,ncol=3)
ns<-c(20,40,60)

for (i in 1:5){
beta<-betas[i]
for (j in 1:3){
n<-ns[j]
mme<-rep(0,100)
mle<-rep(0,100)
for (t in 1:100){
sp<- rinvgamma(n,alpha,beta)
mme[t]<-mean(sp)*((mean(sp))^2/var(sp)+1)
mle[t]<-alpha/mean(1/sp)
}
MSEMME[i,j]<-1/100*sum((mme-beta)^2)
MSEMLE[i,j]<-1/100*sum((mle-beta)^2)
}
}

round(MSEMME,4)
round(MSEMLE,4)

betas<-c(0.3,0.5,0.8,1,1.2)
AVsMLE<-matrix(0,nrow=5,ncol=3)
ns<-c(20,40,60)

for (i in 1:5){
beta<-betas[i]
for (j in 1:3){
n<-ns[j]
mle<-rep(0,100)
for (t in 1:100){
sp<- rinvgamma(n,alpha,beta)
mle[t]<-alpha/mean(1/sp)
}
AVsMLE[i,j]<-sd(mle)
}
}
round(AVsMLE,4)``` 