递增失效率分布证明问询：a<b时，(F(b)-F(a))/(f(b)-f(a))的单调性

Great question—this ties into some key properties of increasing failure rate (IFR) distributions, which are foundational in reliability and survival analysis. Let’s break this down step by step, with definitions, intuition, and concrete examples.

First, let’s recap the core definition to avoid confusion:
For a distribution with PDF $f(x)$ and CDF $F(x)$, the failure rate (hazard function) is:
$$h(x) = \frac{f(x)}{1-F(x)}$$
An IFR distribution means $h(x)$ is non-decreasing in $x$. We’ll also use the survival function $S(x) = 1-F(x)$, which for IFR distributions can be written as $S(x) = \exp\left(-\int_0^x h(t)dt\right)$ (assuming the distribution’s support starts at 0).

We’re analyzing the ratio:
$$Q(a,b) = \frac{F(b)-F(a)}{f(b)-f(a)}$$
for $a < b$, focusing on its monotonicity in $a$ (fixed $b$) and $b$ (fixed $a$).

1. Monotonicity in $a$ (Fixed $b$) #

For IFR distributions, $Q(a,b)$ is non-decreasing in $a$—and strictly increasing if $h(x)$ is strictly increasing. Here’s why:

• Rewrite $Q(a,b)$ using the survival function:
  $$Q(a,b) = \frac{S(a)-S(b)}{f(b)-f(a)} = \frac{S(a)-S(b)}{-S'(b)+S'(a)}$$
• Since $h(x)$ is non-decreasing, $S(x)$ is a log-concave function: the logarithm of $S(x)$ is concave (its second derivative is $-h'(x) \leq 0$). A key property of log-concave functions is that the ratio $\frac{g(x)-g(y)}{g'(x)-g'(y)}$ (which matches our $Q(a,b)$ when $g(x)=S(x)$) is non-decreasing in $x$ for fixed $y > x$.

Concrete Examples: #

• Exponential distribution (constant hazard rate, trivially IFR):
  $F(x)=1-e^{-\lambda x}$, $f(x)=\lambda e^{-\lambda x}$. Calculating $Q(a,b)$ gives a constant value:
  $$Q(a,b) = \frac{(1-e^{-\lambda b})-(1-e^{-\lambda a})}{\lambda e^{-\lambda b}-\lambda e^{-\lambda a}} = -\frac{1}{\lambda}$$
  It doesn’t change as $a$ increases, so it’s trivially non-decreasing.
• Weibull distribution ($\alpha=2$, $\beta=1$) (strictly increasing hazard rate):
  Fixed $b=2$, compute $Q(a,2)$ for $a=1$ and $a=1.5$:
  • $a=1$: $Q≈-0.48$
  • $a=1.5$: $Q≈-0.28$
    As $a$ increases, $Q(a,b)$ increases (becomes less negative), confirming strict monotonicity.

2. Monotonicity in $b$ (Fixed $a$) #

The claim that $Q(a,b)$ increases as $b$ decreases is not true in general. We can easily find counterexamples with strictly IFR distributions:

• Rayleigh distribution ($h(x)=x$, strictly increasing hazard rate):
  Fixed $a=1$, compute $Q(1,b)$ for $b=3$ and $b=2$:
  • $b=3$: $Q≈-1.04$
  • $b=2$: $Q≈-1.40$
    As $b$ decreases from 3 to 2, $Q(a,b)$ becomes more negative (i.e., decreases), directly contradicting the "increasing as $b$ decreases" claim.
• Even in the exponential case, $Q(a,b)$ is constant—it doesn’t increase as $b$ decreases, it stays the same.

Key Takeaways #

• For IFR distributions, $\frac{F(b)-F(a)}{f(b)-f(a)}$ is non-decreasing in $a$ (fixed $b$), with strict monotonicity if the hazard rate is strictly increasing.
• The ratio does not generally increase as $b$ decreases (fixed $a$); it can even decrease for strictly IFR distributions.

内容的提问来源于stack exchange，提问作者Zhiyuan Wang