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title author date output
IP2
Henry Jones
2023-07-26
html_document
keep_md
true
 
library(rootSolve)
times <- read.csv("/Users/henryjones/Desktop/Math_CC/MA_417/times.csv")
times = times$x
newxbar = mean(times)
sighat = var(times)

The method of moments

ahat = newxbar*((newxbar *(1-newxbar))/sighat -1)
betahat = (1-newxbar)*((newxbar *(1-newxbar))/sighat -1)
print('method of moments estimates')
## [1] "method of moments estimates"

print(ahat)
## [1] 3.554313

print(betahat)
## [1] 4.441891

The maximum likelihood estimator

set.seed(8)
n <- length(times)
fnew <- function(y){
  c(n*digamma(y[1]+y[2]) - n* digamma(y[1]) + sum(log(times)), n*digamma(y[1]+y[2]) - n * digamma(y[2]) + sum(log(1-times)))
}
newexes <- seq(0,1,by = 0.001)
newsol <- multiroot(fnew,start = c(2,2))
hist(times,breaks=10)
lines(newexes,dbeta(newexes,newsol$root[1], newsol$root[2]),col = 'red')
lines(newexes,dbeta(newexes,ahat,betahat), col = 'blue')
legend("topright", legend = c("MLE",'Method of moments'), pch = "|", col = c('red','blue','green'))

print('new MLE estimates')
## [1] "new MLE estimates"

trueahat <- newsol$root[1]
truebhat <- newsol$root[2]
print(trueahat)
## [1] 3.988659

print(truebhat)
## [1] 4.988091

Non-parametric standard error estimates

f1 <- function(y,newsamp){
  c(n*digamma(y[1]+y[2]) - n* digamma(y[1]) + sum(log(newsamp)), n*digamma(y[1]+y[2]) - n * digamma(y[2]) + sum(log(1-newsamp)))
}

#MLE standard error
avect <- numeric(n)
bvect <- numeric(n)
for(i in 1:1000){
  samp <- sample(times, n, replace=T)
  f1 <- function(y){
    c(n*digamma(y[1]+y[2]) - n* digamma(y[1]) + sum(log(samp)), n*digamma(y[1]+y[2]) - n * digamma(y[2]) + sum(log(1-samp)))
  }
  sol <- multiroot(f1,start = c(1,1))
  avect[i] <- sol$root[1]
  bvect[i] <- sol$root[2]
}

print('standard error for MLE paramters')
## [1] "standard error for MLE paramters"

print(sqrt(var(avect)))
## [1] 1.154412

print(sqrt(var(bvect)))
## [1] 1.479494

95% confidence intervals for MLE parameters

print('CI for MLE alpha')
## [1] "CI for MLE alpha"

print(quantile(avect,0.025))
##     2.5% 
## 2.922868

print(quantile(avect,0.9725)) 
##   97.25% 
## 7.388597

print('CI for MLE beta')
## [1] "CI for MLE beta"

print(quantile(bvect,0.025))
##    2.5% 
## 3.65733

print(quantile(bvect,0.9725))
##   97.25% 
## 9.006351

The method of moments estimates

avectmm <- numeric(n)
bvectmm <- numeric(n)
for(i in 1:1000){
  samp <- sample(times, n, replace=T)
  xbar <- mean(samp)
  vhat <- var(samp)
  ahat = xbar*((xbar *(1-xbar))/vhat -1)
  betahat = (1-xbar)*((xbar *(1-xbar))/vhat -1)
  avectmm[i] <- ahat
  bvectmm[i] <- betahat
}
print('method of moments estimates')
## [1] "method of moments estimates"

print(sqrt(var(avectmm)))
## [1] 1.088669

print(sqrt(var(bvectmm)))
## [1] 1.282205

95% confidence intervals for method of momemts parameters

print('CI for MM alpha')
## [1] "CI for MM alpha"

print(quantile(avectmm,0.025))
##     2.5% 
## 2.435558

print(quantile(avectmm,0.9725))
##   97.25% 
## 6.401336

print('CI for MM beta')
## [1] "CI for MM beta"

print(quantile(bvectmm,0.025))
##     2.5% 
## 3.075294

print(quantile(bvectmm,0.9725))
##   97.25% 
## 7.916625

Prametric Boostrap error estimates and quantiles

MLE standard error

avect <- numeric(n)
bvect <- numeric(n)
for(i in 1:1000){
  samp <- rbeta(n,trueahat,truebhat)
  f1 <- function(y){
    c(n*digamma(y[1]+y[2]) - n* digamma(y[1]) + sum(log(samp)), n*digamma(y[1]+y[2]) - n * digamma(y[2]) + sum(log(1-samp)))
  }
  sol <- multiroot(f1,start = c(1,1))
  avect[i] <- sol$root[1]
  bvect[i] <- sol$root[2]
}

print('standard error for MLE paramters param')
## [1] "standard error for MLE paramters param"

print(sqrt(var(avect)))
## [1] 1.610049

print(sqrt(var(bvect)))
## [1] 2.029556

95% confidence intervals for MLE parameters

print('CI for MLE alpha param')
## [1] "CI for MLE alpha param"

print(quantile(avect,0.025))
##     2.5% 
## 2.476336

print(quantile(avect,0.9725)) 
##   97.25% 
## 8.559181

print('CI for MLE beta param')
## [1] "CI for MLE beta param"

print(quantile(bvect,0.025))
##     2.5% 
## 3.056187

print(quantile(bvect,0.9725))
##   97.25% 
## 10.87503

Method of moments estimates

avectmm <- numeric(n)
bvectmm <- numeric(n)
for(i in 1:1000){
  samp <- rbeta(n,trueahat,truebhat)
  xbar <- mean(samp)
  vhat <- var(samp)
  ahat = xbar*((xbar *(1-xbar))/vhat -1)
  betahat = (1-xbar)*((xbar *(1-xbar))/vhat -1)
  avectmm[i] <- ahat
  bvectmm[i] <- betahat
}

Parametric standard error for method of moments estimates

print('method of moments standar error param')
## [1] "method of moments standar error param"

print(sqrt(var(avectmm)))
## [1] 1.641155

print(sqrt(var(bvectmm)))
## [1] 2.072645

95% confidence intervals for method of moments parameters

print('CI for MM alpha param')
## [1] "CI for MM alpha param"

print(quantile(avectmm,0.025))
##     2.5% 
## 2.317754

print(quantile(avectmm,0.9725))
##   97.25% 
## 8.381866

print('CI for MM beta param')
## [1] "CI for MM beta param"

print(quantile(bvectmm,0.025))
##     2.5% 
## 2.855589

print(quantile(bvectmm,0.9725))
##  97.25% 
## 10.6101