I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$.

1) Does this quantity $f(X,t)$ have a name? As the title question suggests, it is a tail of the integral that is cut out when computing the trimmed mean.

2) Are there any bounds on $f(X,t)\ / \ f(X,0)$ in terms of $t$ and properties only of $X$ (e.g. moments or quantiles of $X$)? Note $f(X,0)=E[X]$.

Chebyshev's inequality has the right form for a bound on a different quantity, $$\int_t^\infty\negthinspace p(x)\ dx \le \frac{\text{Var}(X)}{(t - E[X])^2}.$$ I am looking for both upper and lower bounds on $f(X,t)$ and Chebyshev's inequality doesn't seem to provide either.

I am especially interested in the case where $X=|Y_1-Y_2|$ where the $Y$'s are i.i.d. variables. Results for the general case might be better-known, and I would appreciate either.


I think results from extreme value theory will be helpful here. The standard condition to put on $p(x)$ in these kinds of situations is that $p$ is regularly varying in $x$, and that there exists a constant $\gamma > 0$ such that $$ \lim_{x \rightarrow \infty} \frac{1 - F(tx)}{1 - F(x)} = t^{-1/\gamma} \text{ for all } t > 0, $$ where $F$ is the CDF of $p$. The constant $\gamma$ is called the tail index. (There also exists a more general theory with $\gamma \leq 0$ that applies to thin-tailed distributions like the Gaussian.)

Once you assume that your density is regularly varying, the quantity $f(X, \, t)$ becomes easier to analyze. For example, you can show that $f(X, \, t)$ scales with $t$.

A good reference for getting started is Chapter 4.3 of Coles (2001) "An Introduction to Statistical Modeling of Extreme Values." For a more theoretical approach, the textbook by de Haan and Ferreira (2006) is excellent.

  • $\begingroup$ An example of the type of result you can get is the following. If $P(X>x) \sim C x^{-\alpha}$ with $\alpha > 1$ then it isn't very hard to show that $$f(X,t) \sim \frac{C\alpha}{\alpha-1} t^{1-\alpha}.$$ It may be that similar asymptotics are true under weaker assumptions on the tail like the regular varying assumption that eisit mentioned above. $\endgroup$ – Jon Peterson Apr 16 '14 at 13:18

The paper "Bounds on Conditional Moments of Weibull and Other Monotone Failure Rate Families" by Patel and Read (1975) gives some useful bounds for distributions with monotone failure rates (either increasing or decreasing), where the failure rate is given by $h(t) = p(t) / (1 - P(t) )$.

Let $H(t) = \int_0^t h(x) dx$, and assume that $h(t)$ is increasing. Among many other results, they show that the conditional expectation is bounded by $$ E[X | X > t] = \frac{f(X,t)} {1-P(t)} \le H^{-1}(1 + H(t)) \le t + \frac 1 {h(t)}\,. $$ I hope that you find this helpful.


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