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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.

kjetil b halvorsen
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user48855
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    One possible reference: Donald F. Morrison (1971), "The Distribution of Linear Functions of Independent F Variates," *JASA*, **66**:334 – Glen_b Oct 14 '14 at 07:03
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    Also, somewhat relevant info [here](http://stats.stackexchange.com/questions/6304/what-is-the-sum-of-squared-t-variates). – Glen_b Oct 14 '14 at 07:09
  • The characteristic function $$s\to e^{n i s} \Gamma\left(\frac{1}{2}, is\right)^n$$ is particularly simple--and arguably responds to the question, because it completely determines the distribution. – whuber Mar 15 '18 at 14:29

1 Answers1

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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
set.seed(1);
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));
plot(density(SUMS),
     main = paste0("Density of log-sum of ", n, " IID F(1,1) variables"),
     xlab = "Log-Sum", ylab = "Density");
Ben
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