Solved – sum of $n$ independent F distributed random variables


I need a help:

What will be the distribution of sum of $n$ independent F distributed random variables with parameters 1 and 1 (i.e., $F(x;1,1)$?

Great if you can suggest some references too.

Best Answer

There is no closed-form distribution for this quantity, but it can be simulated in a number of ways. One way to simulate it is to use the relationship between the $F(1,1)$ distribution and the uniform distribution. If $U_1, ..., U_n \sim \text{IID U}(0,1)$ are uniform then we can form corresponding random variables $F_1, ..., F_n \sim \text{IID F}(1,1)$ using the transformation:

$$F_i \equiv \frac{1-U_i}{U_i} = \frac{1}{U_i}-1.$$

So the sum of interest can be written as:

$$S \equiv \sum_{i=1}^n F_i = \sum_{i=1}^n \Bigg( \frac{1}{U_i}-1 \Bigg) = \sum_{i=1}^n \frac{1}{U_i} -n .$$

The distribution of this value is strongly positively skewed, so it is easiest to visualise its distribution on a log-scale. This is easy to simulate in base-R using the following code:

#Set seed and other variables
n <- 100;
m <- 10^5;

#Define function to sum IID F(1,1) values
SumF <- function(n, m) { SUMS <- rep(0, m);
                         for (i in 1:m) { SUMS[i] <- sum(1/runif(n)) - n }
                         SUMS }

#Plot kernel density of IID F(1,1) values
SUMS <- log(SumF(n,m));
     main = paste0("Density of log-sum of ", n, " IID F(1,1) variables"),
     xlab = "Log-Sum", ylab = "Density");