Compute quantile of sum of distributions from particular quantiles

Question

Let's assume $N$ independent random variables $X_1, ..., X_N$ for which the quantiles at some specific level $\alpha$ are known through estimation from data: $\alpha = P(X_1 < q_1)$, ..., $\alpha = P(X_N < q_N)$. Now let's define the random variable $Z$ as the sum $Z = \sum_{i=1}^N X_i$. Is there a way to compute the value of the quantile of the sum at level $\alpha$, that is, $q_z$ in $\alpha = P(Z < q_Z)$?

I think that in particular cases, such as if $X_i$ follows a Gaussian distribution $\forall i$ this is easy, but I'm not so sure for the case where the distribution of the $X_i$ is unknown. Any ideas?

Answer 1

$q_Z$ could be anything.

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To understand this situation, let us make a preliminary simplification. By working with $Y_i = X_i - q_i$ we obtain a more uniform characterization

$$\alpha = \Pr(X_i \le q_i) = \Pr(Y_i \le 0).$$

That is, each $Y_i$ has the same probability of being negative. Because

$$W = \sum_i Y_i = \sum_i X_i - \sum_i q_i = Z - \sum_i q_i,$$

the defining equation for $q_Z$ is equivalent to

$$\alpha = \Pr(Z \le q_Z) = \Pr(Z - \sum_i q_i \le q_Z - \sum_i q_i) = \Pr(W \le q_W)$$

with $q_Z = q_W + \sum_i q_i$.

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What are the possible values of $q_W$? Consider the case where the $Y_i$ all have the same distribution with all probability on two values, one of them negative ($y_{-}$) and the other one positive ($y_{+}$). The possible values of the sum $W$ are limited to $ky_{-} + (n-k)y_{+}$ for $k=0, 1, \ldots, n$. Each of these occurs with probability

$${\Pr}_W(ky_{-} + (n-k)y_{+}) = \binom{n}{k}\alpha^k(1-\alpha)^{n-k}.$$

The extremes can be found by

1. Choosing $y_{-}$ and $y_{+}$ so that $y_{-} + (n-1)y_{+} \lt 0$; $y_{-}=-n$ and $y_{+}=1$ will accomplish this. This guarantees that $W$ will be negative except when all the $Y_i$ are positive. This chance equals $1 - (1-\alpha)^n$. It exceeds $\alpha$ when $n\gt 1$, implying the $\alpha$ quantile of $W$ must be strictly negative.

2. Choosing $y_{-}$ and $y_{+}$ so that $(n-1) y_{-} + y_{+} \gt 0$; $y_{-}=-1$ and $y_{+}=n$ will accomplish this. This guarantees that $W$ will be negative only when all the $Y_i$ are negative. This chance equals $\alpha^n$. It is less than $\alpha$ when $n\gt 1$, implying the $\alpha$ quantile of $W$ must be strictly positive.

This shows that the $\alpha$ quantile of $W$ could be either negative or positive, but is not zero. What could its size be? It has to equal some integral linear combination of $y_{-}$ and $y_{+}$. Making both these values integers assures all the possible values of $W$ are integral. Upon scaling $y_{\pm}$ by an arbitrary positive number $s$, we can guarantee that all integral linear combinations of $y_{-}$ and $y_{+}$ are integral multiples of $s$. Since $q_W \ne 0$, it must be at least $s$ in size. Consequently, the possible values of $q_W$ (and whence of $q_Z$) are unlimited, no matter what $n\gt 1$ may equal.

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The only way to derive any information about $q_Z$ would be to make specific and strong constraints on the distributions of the $X_i$, in order to prevent and limit the kind of unbalanced distributions used to derive this negative result.

Answer 2

(version 3)

Since you say you have sample data, you could use the following numerical method:

a) fit pdfs to the data for each $X_i$ variable...maybe using kernel densities

b) take the DFT (discrete Fourier transform) of each kernel density

c) multiply the DFTs together

d) take the inverse DFT

That would give you an estimate of the pdf of $Z$.

It's a standard technique for finding the distribution of the sum of independent random variables, covered by many authors, and not too hard to derive. It's used in the insurance industry for combining distributions of possible insurance claims. For the sake of providing a citation I googled it, and found this one: doi.org/10.1016/S0167-4730(96)00032-X (although I have only read the abstract).