Estimating Max Value of a Population

This is an experiment to verify a technique for estimating the maximum value of a sample given a gaussian population with sigma sigma and mean mu.

sigma <- 150
mu <- 500

The expected value is the value for which there is a 50% probability of occurring at that value or greater.

Also calculate a 95% confidence interval for the max value.

est.max <- function(sample_size) {
    round(qnorm(c(`Mean`=1 - (0.5/sample_size),
                  `2.5%`=1 - (0.975/sample_size),
                  `97.5%`=1 - (0.025/sample_size)),
                mean=mu, sd=sigma))
}

Define a function to grab a random sample from a population with the given parameters.

Run it 1000 times.

simmax <- function(sample_size) {
    round(max(rnorm(sample_size, mean=mu, sd=sigma)))
}

Run the simulation 10,000 times using a sample size of 500. Accumulate the max values over all the simulations.

N <- 500
sample_maxes <- sapply(1:10000, function(v) { simmax(N) })

How does the average value compare to the expected value?

How does the 95 percent confidence interval of the expected max compare to the actual 95% range of the simulated values?

t <- data.frame(Expected=est.max(N), 
                Simulated=round(quantile(probs=c(0.5, 0.025, 0.975), sample_maxes)))
t$Difference <- t$Expected - t$Simulated
rownames(t) <- c(' Mean', ' 2.5%', '97.5%')
t

This seems to work pretty well. The expected value is close to the simulated value and the confidence intervals line up pretty well.