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Let $X_1,...,X_n$ be iid lognormally distributed variables and $X_{sum} = X_1+...+X_n$. What is the distribution of $\frac{X_k}{X_{sum}}$ for each $k$ in $1..n$? What are their density functions?

More broadly, are there any general approaches for answering this kind of question (e.g. with some other distribution than lognormal)? The usual strategy one uses when dealing with, say, finding the distribution of the minimum of a group of variables doesn't seem to yield anything here, since $X_{sum}$ depends on $X_1,...,X_n$.

Lesser Cormorant
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    The case where the $X_i$ have Gamma distributions is standard and well-known: it is the basis for all the Normal-theory ANOVA tests, for instance. – whuber Feb 04 '15 at 22:45
  • There is a strategy to compute the expectation: https://stats.stackexchange.com/a/74199/242321 – Gopi Mar 25 '19 at 07:32

2 Answers2

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I suspect that this problem does not have an algebraic solution. If I had to make inferences in a case such as this I would do a simulation exercise to determine appropriate confidence intervals etc.

John C Frain
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The density functions of convolutions of lognormal distributions do not have closed form expressions in general. As whuber mentioned, the case of the gamma is known. For gamma distributions the ratios have beta distributions.

As an example, suppose $X_1 \sim \mathrm{Gamma} \left( \alpha_1, \beta \right)$ and $X_2 \sim \mathrm{Gamma} \left( \alpha_2, \beta \right),$ with $X_1$ and $X_2$ being independent. Then $$ {{X_1} \over {X_1 + X_2}} \sim \mathrm{Beta} \left( \alpha_1, \alpha_2 \right)$$

For your case you will likely need to use simulation, although a beta model could potentially meet your needs depending on how accurate you need it to be.

soakley
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