Fitting Models

Fitting Models

R andom sample from the normal distribution, estimate the parameters

simulated sample

mu <- 43; sigma <- 5.97   # parameters
n <- 300

x <- rnorm(n, mu, sigma)

hist(x, probability = TRUE)

Estimate the parameters with x.bar and sd

mu.hat <- mean(x); mu.hat

[1] 42.54465

sigma.hat <- sd(x); sigma.hat

[1] 6.137992

Plot the fitted model on the histogram

hist(x, probability = TRUE)
curve(dnorm(x, mu.hat, sigma.hat), add=T)

Random sample from the exponential distribution, estimate the parameters

simulated sample

lambda <- 11.34   # parameter
n <- 30

x <- rexp(n, rate = lambda)

hist(x, probability = TRUE)

Estimate the parameters with x.bar

lambda.hat = 1/mean(x); lambda.hat

[1] 9.726872

Plot the fitted model on the histogram

hist(x, probability = TRUE)
curve(dexp(x, rate = lambda.hat), add=T)

Now suppose we want to fit the gamma distribution to a random sample how can we do this?

Can we use x.bar and the sd?

The big question is, how do we estimate parameters in models in general?