[Hassam]

Q-Q Plots
Let's compare the Student t Distribution with the Cauchy Distribution. Cauchy distribution is a special case of a student t distribution when the degrees of freedom n=1. Q-Q plot is the method that we use to compare the tails of  two distributions. In a Q-Q plot we plot the quantiles of one distribution against the quantiles of the second distribution. Quantile is an important concept. Consider the 10th quantile. It means the value of the random variable for which the cumulative probability distribution is 0.1. Similarly 90th quantile will be the value of the random variable for which the cumulative probability distribution is 0.9.

First let compare a normal distribution with a Cauchy distribution. As said above, a Cauchy distribution has undefined mean and an undefined variance. On the other hand the mean and variance of a normal distribution are always well defined. Let's see how these two distributions differ:

> qnorm(c(0.8,0.85,0.9,0.95,0.99), mean=0, sd=1)
[1] 0.8416212 1.0364334 1.2815516 1.6448536 2.3263479
> qcauchy(c(0.8,0.85,0.9,0.95,0.99), location=0, scale=1)
[1]  1.376382  1.962611  3.077684  6.313752 31.820516

Above we calculated the 80th Quantile first. For the standard normal random variable value of 0.84, we have 80% of the cumulative probability on the left of it. For the Cauchy random variable to have 80% of the cumulative probability on the left, its value should be 1.37. For 99% of the cumulative probability, we find standard normal random variable to be 2.32 while the Cauchy random variable to be 31.8 which is a long way from the location =0. So you can see Cauchy has heavy tails, a lot heavier than the normal distribution.

> p <- seq(from=0,to=1, length=1200)
> plot(qt(p,1), qcauchy(p, location=0, scale=1),, type="l",
+ xlab="Student t Distribution Quantiles",
+ ylab="Cauchy Distribution Quantiles")
> plot(qt(p,2), qcauchy(p, location=0, scale=1),, type="l",
+ xlab="Student t Distribution Quantiles",
+ ylab="Cauchy Distribution Quantiles")

Let's return to the student t distribution and the Cauchy distribution once again. As said above, student t distribution with one degree of freedom is the Cauchy distribution. Q-Q plot shows a nice straight diagonal line.

As you can see above we have a straight line for the student t distribution with one degree of freedom and the Cauchy distribution meaning they are the same. Now let's check the student t distribution with two degrees of freedom with the Cauchy distribution.

Viola Cauchy distribution has very heavy tails as compared to the student t distribution as shown by the above Q-Q plot.