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I'm reporting on methods that use discrete distribution data, for example:

d <- density(rnorm(10) * 3 + 10, n=201, from=0, to=20)  # don't focus on this line!
x <- d[["x"]]   # these are the discrete points 0.0, 0.1, ..., 19.9, 20.0
fx <- d[["y"]]  # 0.000977 0.001116 0.001271 0.001445 0.001639 ...

In this example, the only data I have are two vectors of (e.g.) 201 points: the discrete points $x$ and their frequency intensities $f(x)$.

My question relates to an R function to approximate the point $x_\beta$ along $x$ where the cumulative frequency exceeds $\beta$, e.g. 0.2:

beta <- 0.2
x_beta <- approx(cumsum(fx) / sum(fx), x, beta)$y  # 6.524198

Or shown graphically as a cumulative distribution with cumsum(fx) / sum(fx):

image

How should the determination of $x_\beta$ be reported? Is it possible to describe this with typeset equations? For example, I'm sure there are some $\sum$ symbols, but I've never seen a typeset formula for cumsum, so I'm lost on where to start. Or should this method be described just as words?

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  • $\begingroup$ This is an empirical quantile, no? $\endgroup$
    – Xi'an
    Mar 13, 2016 at 14:22

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As the comment suggested, you're talking about an Empirical Quantile Function (EQF).

cumsum(fx) / sum(fx) can be written as:

For $x \in \{X_{1},X_{2},..,X_{n}\}$, define $$ \bar{g}(x) = \frac{\sum_{i=1}^{n} f(X_{i}) \mathbf{1}(X_{i}\leq x)}{\sum_{i=1}^{n} f(X_{i})} $$ Where $\mathbf{1}(.)$ is the indicator function.

Then let $g(x)$ be the linear interpolation. See https://en.wikipedia.org/wiki/Linear_interpolation for various ways to write this up. To be rigorous with the definition, you'll need to ensure that $ 0 \leq g(x) \leq 1$ for all $x$.

Finally, $x_{\beta} = g^{-1}(\beta)$, where again, you'll have to be a little careful with the inverse so that it's well defined.

I think spending a little time reading about Empirical Density Functions and Empirical Quantile Functions should clear things up for you.

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