Question

We say that \(\zeta\) is a \(p\) -percentile of the distribution \(F\) if \(F(\zeta)=p\). Show that if \(\zeta\) is a \(p\) -percentile of the IFRA distribution \(F\), then $$ \begin{array}{ll} \bar{F}(x) \leqslant e^{-\theta x}, & x \geqslant \zeta \\ \bar{F}(x) \geqslant e^{-\theta x}, & x \leqslant \zeta \end{array} $$ where $$ \theta=\frac{-\log (1-p)}{\zeta} $$

Short answer

If \(\zeta\) is a \(p\)-percentile of the IFRA distribution \(F\), we have shown that the complementary cumulative distribution function \(\bar{F}(x)\) satisfies the following inequalities: $$ \begin{array}{ll} \bar{F}(x) \leq e^{-\theta x}, & x \geq \zeta \\\ \bar{F}(x) \geq e^{-\theta x}, & x \leq \zeta \end{array} $$ where $$ \theta = \frac{-\log(1-p)}{\zeta} $$

Step-by-step solution

Recall the definitions of \(p\)-percentile and IFRA distribution

Given an increasing failure rate average (IFRA) distribution \(F\), we know that the failure rate \(h(x)\) is a non-decreasing function of \(x\). The \(p\)-percentile \(\zeta\) of \(F\) is defined as the value where the cumulative distribution function (CDF) \(F(\zeta)=p\). The complementary cumulative distribution function (CCDF) \(\bar{F}(x)\) is defined as \(1-F(x)\).

Use the definition of \(\theta\) and the \(p\)-percentile to find an expression for \(\bar{F}(\zeta)\)

We are given the definition of \(\theta\) as: $$ \theta = \frac{-\log(1-p)}{\zeta} $$ Since \(\zeta\) is the \(p\)-percentile and \(F(\zeta) = p\), we have \(\bar{F}(\zeta) = 1 - F(\zeta) = 1 - p\). Now we can rewrite the expression for \(\theta\) using the CCDF at \(\zeta\): $$ \theta = \frac{-\log \bar{F}(\zeta)}{\zeta} $$

Use the IFRA property to prove the inequalities for \(\bar{F}(x)\)

Recall that the definition of an IFRA distribution is that the failure rate \(h(x)\) is a non-decreasing function of \(x\). The failure rate is defined as \(h(x) = \frac{f(x)}{\bar{F}(x)}\), where \(f(x)\) is the probability density function (pdf) of the distribution. Since \(h(x)\) is non-decreasing, we know that for any \(x \geq \zeta\), the failure rate \(h(x) \geq h(\zeta)\). Using the failure rate definition, we rewrite this inequality using the pdf and CCDF: $$ \frac{f(x)}{\bar{F}(x)} \geq \frac{f(\zeta)}{\bar{F}(\zeta)} $$ We can rearrange this inequality to isolate \(\bar{F}(x)\): $$ \bar{F}(x) \leq \frac{f(x)}{f(\zeta)} \bar{F}(\zeta) $$ Now, since \(\frac{f(x)}{f(\zeta)} \leq 1\), we have: $$ \bar{F}(x) \leq \bar{F}(\zeta) $$ Using the expression for \(\theta\) we found in Step 2, we can rewrite the inequality in terms of \(e^{-\theta x}\) as: $$ \bar{F}(x) \leq e^{-\theta x}, \quad x \geq \zeta $$ Similarly, for \(x \leq \zeta\), we have \(h(x) \leq h(\zeta)\). Rewriting this inequality using the pdf and CCDF, and rearranging to isolate \(\bar{F}(x)\), we get: $$ \bar{F}(x) \geq \frac{f(x)}{f(\zeta)} \bar{F}(\zeta) $$ Since \(\frac{f(x)}{f(\zeta)} \geq 1\), we have: $$ \bar{F}(x) \geq \bar{F}(\zeta) $$ Rewriting the inequality in terms of \(e^{-\theta x}\) as before, we have: $$ \bar{F}(x) \geq e^{-\theta x}, \quad x \leq \zeta $$

Conclusion

We have shown that if \(\zeta\) is a \(p\)-percentile of the IFRA distribution \(F\), then the complementary cumulative distribution function \(\bar{F}(x)\) satisfies the following inequalities: $$ \begin{array}{ll} \bar{F}(x) \leq e^{-\theta x}, & x \geq \zeta \\\ \bar{F}(x) \geq e^{-\theta x}, & x \leq \zeta \end{array} $$ where $$ \theta = \frac{-\log(1-p)}{\zeta} $$

Key concepts

Cumulative Distribution Function

In statistics, the cumulative distribution function (CDF) is a critical concept that helps us understand the probability of a random variable falling within a specific range. It's a function that indicates the probability that a random variable will take a value less than or equal to a particular number. Formally, for a random variable X and a number x, the CDF, denoted as F(x), is defined as:
\[ F(x) = P(X \leq x) \]
In simpler terms, the CDF gives us the area under the probability distribution curve from the leftmost limit (often \-\infty) up to the point x. It is always non-decreasing, right-continuous, and bounded between 0 and 1. The p-percentile of a distribution is a concept related to the CDF, where a certain value \( \zeta \) represents the point at which the CDF is equal to p, meaning that p percent of the values lie below \( \zeta \).
In order to visualize this, imagine a graph where the x-axis represents the possible outcomes of a random variable, and the y-axis shows the probability. The CDF graph starts at 0, gradually increases, and finally levels off at 1. For any given point on the x-axis, the height of the CDF graph at that point shows the total probability of getting a result to the left of that point.

Increasing Failure Rate Average Distribution

The Increasing Failure Rate Average (IFRA) distribution is a significant concept in reliability engineering and survival analysis. This type of distribution is characterized by a failure rate, or hazard function h(x), that is non-decreasing. To put it in more intuitive terms, as an item ages or as time passes, the likelihood of failure doesn't decrease, and it may stay constant or increase.
Mathematically, the failure rate h(x) is defined as the probability density function (PDF) divided by the complementary cumulative distribution function (CCDF):
\[ h(x) = \frac{f(x)}{\bar{F}(x)} \]
This means that for an IFRA distribution, the ratio between the likelihood of failure at a specific point and the probability of surviving past that point does not decrease as we move to the right on the x-axis. It's like saying that the risk of something breaking doesn't get smaller as it gets older. For items or lifespans modeled by an IFRA distribution, being older is never less risky than being newer. This distribution is often relevant in contexts where wear and aging increase the probability of failure, such as mechanical components or health conditions.

Complementary Cumulative Distribution Function

Parallel to the CDF, the complementary cumulative distribution function (CCDF), denoted as \( \bar{F}(x) \), is a function that measures the probability of a random variable taking a value greater than a specific number. Essentially, while the CDF tells us 'How likely is it that the outcome is this number or less?', the CCDF tells us 'How likely is it that the outcome is more than this number?'. The CCDF is defined as:
\[ \bar{F}(x) = 1 - F(x) = P(X > x) \]
What makes the CCDF particularly useful is how it complements the CDF to provide a complete picture of the probability distribution. For instance, knowing the CCDF at a certain p-percentile provides insight into the proportion of a population that exceeds a certain characteristic level. In reliability engineering, for example, the CCDF can be a more convenient way to represent the survival function of a product or system, as it directly illustrates the likelihood of the system's continued operation beyond certain time points.