If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T03:50:27Z http://mathoverflow.net/feeds/question/95191 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95191/if-mean-residual-lifetime-is-approximately-constant-residual-lifetime-is-approxi If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense Stephan 2012-04-25T20:10:57Z 2012-07-04T14:20:36Z <p>Suppose the "mean residual lifetime," $\mathbb{E}[X-x|X≥x]$ is approximately constant for large $x$. Then, I believe that the conditional tail distribution is approximately exponential, in the sense of being stochastically dominated by an exponential and dominating a similar exponential. Formally:</p> <p><strong>Conjecture</strong> Given any random variable $X$ with support on $[0,∞)$. If, for some $\lambda \in(0,∞)$, $$\lim_{x→∞}\mathbb{E}[X-x|X≥x]= \lambda ,$$</p> <p>then, for all $ε>0$ and for all $\Delta>0$, there is some $c$ such that $x≥c$ implies $$e^{-\frac{t}{λ-ε}}\leq \mathbb{P}[X≥x+t|X≥x] \leq e^{-\frac{t}{λ+ε}} \qquad ∀t≥\Delta.$$</p> <p>I posted this question on <a href="http://math.stackexchange.com/questions/136489/sufficient-condition-for-asymptotically-exponential-tail-corrected-but-still-un" rel="nofollow">StackExchange</a>. <a href="http://mathoverflow.net/users/13650/robert-israel" rel="nofollow">Robert Israel</a> provided a counterexample to an earlier conjecture, which was wrong.</p> <p><strong>Update</strong> The approximation result is stronger than weak convergence. Let $Y$ be distributed exponentially with parameter $\lambda$. The conclusion of the conjecture implies that</p> <p>$$\lim_{x→∞}\mathbb{E}[f(X-x)|X≥x]=\mathbb{E}[f(Y)]$$</p> <p>for all nondecreasing functions for which $\mathbb{E}[f(Y)]$ exists. In particular, $f$ is allowed to be <em>unbounded</em>. The first great response by <a href="http://mathoverflow.net/users/1061/ori-gurel-gurevich" rel="nofollow">Ori Gurel-Gurevich</a> implies a slightly weaker approximation result.</p> http://mathoverflow.net/questions/95191/if-mean-residual-lifetime-is-approximately-constant-residual-lifetime-is-approxi/95230#95230 Answer by Ori Gurel-Gurevich for If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense Ori Gurel-Gurevich 2012-04-26T05:04:00Z 2012-04-26T05:04:00Z <p>$\newcommand{\eps}{\varepsilon} \newcommand{\E}{\mathbb{E}} \renewcommand{\P}{\mathbb{P}}$ Fix $\eps>0$ and let $a=\E[X]$ and $b=\E[X-\eps \mid X>\eps]$. Let $p=\P(X > \eps)$. Then $$(1-p)(b+\eps)\le a \le (1-p)(b+\eps) + p \eps \ .$$</p> <p>Solving, we get $$1-\frac{a+\eps}{b+\eps}+\frac{\eps}{b+\eps} &lt; p &lt; 1-\frac{a}{b}+\frac{\eps}{b}$$</p> <p>In other words, $$\frac{\eps}{\lambda+\eps} -\delta &lt; p &lt; \frac{\eps}{\lambda} +\delta$$</p> <p>Where $\delta=\delta(a,b)$ goes to 0 as $a$ and $b$ approach $\lambda$.</p> <p>Compounding this we get that when $a &lt; \E[X-s \mid X > s] &lt; b$ for all $s>0$ and we take $a$ and $b$ to $\lambda$, we have for any $t>0$</p> <p>$$(1-\frac{\eps}{\lambda+\eps})^{\frac{t}{\eps}} - \delta_1 &lt; \P(X > t) &lt; (1-\frac{\eps}{\lambda})^{t/\eps} + \delta_1$$</p> <p>where $\delta_1 \to 0$ when $a,b \to \lambda$.</p> <p>Taking now $\eps$ small enough yields the desired result.</p> http://mathoverflow.net/questions/95191/if-mean-residual-lifetime-is-approximately-constant-residual-lifetime-is-approxi/101310#101310 Answer by miladydesummer for If Mean Residual Lifetime is approximately constant, Residual Lifetime is Approximately Exponential in a Strong Sense miladydesummer 2012-07-04T14:20:36Z 2012-07-04T14:20:36Z <p>Here is a work which gives general conditions under which your conjecture is true.</p> <p>"Limiting Properties of the Mean Residual Lifetime Function" by Isaac Meilijson in Ann. Math. Statist. Volume 43, Number 1 (1972), 354-357.</p> <p><a href="http://projecteuclid.org/euclid.aoms/1177692731" rel="nofollow">http://projecteuclid.org/euclid.aoms/1177692731</a></p>