Question

The water quality of a river was investigated by taking 15 water samples and recording the counts of a particular bacteria for each sample, where ???? is the number of bacteria in sample ?? and the ????’s are assumed to be independent and follow a Poisson ( ?oi(?) ) distribution. The data are as follows: {27, 24, 25, 21, 30, 22, 20, 22, 29, 19, 19, 22, 23, 36, 19}

1. Find a point estimate for the population mean number of bacteria in the water using maximum likelihood estimation.

2. Use R to plot the log-likelihood function with the maximum likelihood estimate highlighted.

3. Show that the maximum likelihood estimator of ? is a consistent estimator.

4. Show that the maximum likelihood estimator of ? achieves the Cramer-Rao lower bound.

5. Derive the asymptotic distribution of ? ^, the MLE of ?.

6. Find an approximate 95% confidence interval for ?? using the Wald method

Answer 1

b.

R code:

x=c(27, 24, 25, 21, 30, 22, 20, 22, 29, 19, 19, 22, 23, 36, 19)
mean(x)
theta=seq(1,100,by=0.01)
M=1:length(theta)*0
for(i in 1:length(theta))
M[i]=-15*theta[i]+sum(x)*log(theta[i])-sum(log(factorial(x)))
plot(theta,M,type="l",xlab=expression(theta),ylab=expression(log(theta)))
abline(v=23.86667,col=2)
text(35,-700,expression(hat(theta)==23.6667))