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## Learning from the extremes

Here is an interesting problem with “selected data”.  Suppose you are measuring the speeds of cars driving on an interstate.  You assume the speeds are normally distributed with mean $\mu$ and standard deviation $\sigma$.  You see 10 cars pass by and you only record the minimum and maximum speeds.  What have you learned about the normal parameters?

First we’ll describe the construction of the likelihood function.  We’ll combine the likelihood with the standard noninformative prior for a mean and standard deviation.   Then we’ll illustrate the use of a normal approximation to learn about the parameters.

Here we focus on the construction of the likelihood.  Given values of the normal parameters, what is the probability of observing minimum = x and the maximum = y in a sample of size n?

Essentially we’re looking for the joint density of two order statistics which is a standard result.  Let f and F denote the density and cdf of a normal density with mean $\mu$ and standard deviation $\sigma$.  Then the joint density of (x, y) is given by

$f(x, y | \mu, \sigma) \propto f(x) f(y) [F(y) - F(x)]^{n-2}, x < y$

After we observe data, the likelihood is this sampling density viewed as function of the parameters.  Suppose we take a sample of size 10 and we observe x = 52, y = 84.  Then the likelihood is given by

$L(\mu, \sigma) \propto f(52) f(84) [F(84) - F(52)]^{8}$

In the next blog posting, I’ll describe how to summarize this posterior by a normal approximation in LearnBayes.