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Noncentral F-distribution is used frequently in communication areas. In one of the applications, I need to do a sum of two i.i.d R.V having non-central F-distribution with parameter 1 (d.o.f for numerator), $N-1$ (d.o.f for the denominator) and $\lambda$ (non-centrality) parameter of the numerator. Is there any standard result on the sum of these two R.V.? or is there any approximation result? Since applying a straightforward approach, i.e., the convolution formula, is tough or doesn't seem to have a closed-form solution.

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I think you end up having a non-standard distribution. My (tentative) approach would be the following. Let $W_{j}\sim \mathrm{F}_{1,N-1,\lambda},j=1,2$.

First, we know that $\widetilde{T}_j\sim \mathrm{t}_{N-1,\lambda},\widetilde{T}_j:=\sqrt{W_j},j=1,2,$ where $\mathrm{t}_{\nu,\mu}$ is the non-central $\mathrm{t}$ distribution with $\nu$ degrees of freedom and non-centrality parameter $\lambda$.

Second, I would consider the centralized version of the two random variables above. Namely, $$T_j = \widetilde{T}_j-\frac{\lambda}{\sqrt{V/(N-1)}},\quad V\sim \chi_{N-1}^2.$$

Finally, this paper provides a formula for the density of two independent $d$-dimensional Student-$t$ random vectors. In your case $d=1$. If you can work with $(T_1,T_2)$ instead of $(W_1,W_2)$ you are done. Otherwise, you simply transform the density for $T_1+T_2$ given in the paper into the density of $W_1+W_2$ using the relationships mentioned above.

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