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Given a random variable $X$ with density function $f_X$, we can define the function $H$ as:

$$H(a) \equiv \mathbb{P}(f_X(X) \geqslant a) \quad \quad \text{for all } a \geqslant 0.$$

This function measures the probability of an outcome with density at least as large as a stipulated minimum cut-off level. The function is useful for the examination of highest density regions (HDRs). Although this function is useful in that context, I have not run across it before in the statistics work, and so I'm not sure if this one has a name. I have been unable to find references to this function in my own searches, so I am coming to the learned CV.SE community for help. As a placeholder name, I am calling it the "intensity function", on the basis that if the density is more concentrated ("intense") then the function values will tend to be higher.


Question: Does this function have an existing name in the statistical literature (or in the literature of any related field)? If not, can anyone suggest a good name/notation for this function? (Please feel free to offer any suggestions that are better names than the one I am presently using.)

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    $\begingroup$ if $f_x(\cdot)$ were the identity function this would be the survival function (provided you omit the equality part of the $\geq$, so perhaps the survival function with respect to $f$? Consider that $Y=f_x(X)$ is itself a random variable so alternately, a name could be w.r.t. $Y$ if the redefinition makes sense in the application you are considering. $\endgroup$ – Lucas Roberts Oct 18 '19 at 0:46
  • $\begingroup$ @LucasRoberts: So taking $Y = -f_X(X)$ then $H(a) = F_Y(-a)$ is a reversal of the CDF of $Y$. This might be a useful way to look at it (thanks!) but it doesn't suggest an obvious name to me. $\endgroup$ – Ben Oct 18 '19 at 1:16
  • $\begingroup$ Are you looking for a single word name? What I was suggesting the name "survival function w.r.t 𝑓" as the name. Alternately this could be considered a "tail weight" (either in the right tail, per your OP, or the left tail, per your interpretation of my comment). If a terseness if a criterion, I'm struggling w/conjuring a single word. $\endgroup$ – Lucas Roberts Oct 18 '19 at 1:57
  • $\begingroup$ Yeah, I think terseness/simplicity/interpretability are all definitely criteria. $\endgroup$ – Ben Oct 18 '19 at 2:07
  • $\begingroup$ I don't see why survival function wouldn't work, unless there is some specific context where that might be confusing. In probability and statistics, as well as reliability and survival analysis fields, this naming is very standard. If you went with some other name and the audience from the probability and statistics community read your work, they would likely be confused. If the equality part of $\Pr(f_x(X)\geq a)$ is really needed then it is different. However, in that case then $f(\cdot)$ wouldn't be what is usually called a density. $\endgroup$ – Lucas Roberts Oct 18 '19 at 2:18
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Markov's inequality states that, if $X$ is a non-negative random variable and $a > 0$, then $P(X \ge a) \le E(X)/a.$ So the inequality provides a bound on the specified tail probability.

The phrase 'tail probability' is commonly used in discussing Markov and Chebyshev inequalities and in statistical testing. So it seems OK to refer to $P(X \ge a)$ as a tail probability, and people would know what you mean. This would work especially well if $a$ is relatively large and $P(X > a)$ is relatively small. However, making a formal definition of a commonly-used phrase would seem strange.

Also, as mentioned in a comment $P(X > t),$ is often called a survival function $S_x(t)$ or a reliability function $R_X(t) = \bar F_X(t) = 1-F_X(t).$ But unless you are discussing survival analysis or reliability theory, it would seem strange (perhaps distracting) to use that terminology.

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