Distribution of the ratio of dependent chi-square random variables

Assume that X=X1+X2++Xn where XiN(0,σ2) are independent.

My question is, what distribution does

Z=X2X21+X22++X2n

follow? I know from here that the ratio of two chi-squared random variables expressed as WW+Y follows a Beta distribution. I think that this assumes independence between W and Y. In my case though, the denominator of Z contains the components of X squared.

I think Z must also follow a variation of the Beta distribution but I am not sure. And if this assumption is correct, I don’t know how to prove it.

Answer

This post elaborates on the answers in the comments to the question.


Let X=(X1,X2,,Xn). Fix any e1Rn of unit length. Such a vector may always be completed to an orthonormal basis (e1,e2,,en) (by means of the Gram-Schmidt process, for instance). This change of basis (from the usual one) is orthogonal: it does not change lengths. Thus the distribution of

(e1X)2||X||2=(e1X)2X21+X22++X2n

does not depend on e1. Taking e1=(1,0,0,,0) shows this has the same distribution as

X21X21+X22++X2n.

Since the Xi are iid Normal, they may be written as σ times iid standard Normal variables Y1,,Yn and their squares are σ2 times Γ(1/2) distributions. Since the sum of n1 independent Γ(1/2) distributions is Γ((n1)/2), we have determined that the distribution of (1) is that of

σ2Uσ2U+σ2V=UU+V

where U=X21/σ2Γ(1/2) and V=(X22++X2n)/σ2Γ((n1)/2) are independent. It is well known that this ratio has a Beta(1/2,(n1)/2) distribution. (Also see the closely related thread at Distribution of XY if X Beta(1,K1) and Y chi-squared with 2K degrees.)

Since X1++Xn=(1,1,,1)(X1,X2,,Xn)=ne1X

for the unit vector e1=(1,1,,1)/n, we conclude that Z is (n)2=n times a Beta(1/2,(n1)/2) variate. For n2 it therefore has density function

fZ(z)=n1n/2B(12,n12)(nz)n3z

on the interval (0,n) (and otherwise is zero).


As a check, I simulated 100,000 independent realizations of Z for σ=1 and n=2,3,10, plotted their histograms, and superimposed the graph of the corresponding Beta density (in red). The agreements are excellent.

Figure

Here is the R code. It carries out the simulation by means of the formula sum(x)^2 / sum(x^2) for Z, where x is a vector of length n generated by rnorm. The rest is just looping (for, apply) and plotting (hist, curve).

for (n in c(2, 3, 10)) {
  z <- apply(matrix(rnorm(n*1e5), nrow=n), 2, function(x) sum(x)^2 / sum(x^2))
  hist(z, freq=FALSE, breaks=seq(0, n, length.out=50), main=paste("n =", n), xlab="Z")
  curve(dbeta(x/n, 1/2, (n-1)/2)/n, add=TRUE, col="Red", lwd=2)
}

Attribution
Source : Link , Question Author : x0dros , Answer Author : Community

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