Failure Rate Function

A Failure Rate Function, [math]\displaystyle{ F(t) }[/math], is a temporal probability function that produces a failure probability (the probability that a case will fail in time interval [math]\displaystyle{ t }[/math]).

• AKA: Hazard Function, Survival Function, Failure Rate.
• Context:
  • It can be a part of a Failure Model.
  • It can be a member of a Survival Model Family.
  • It can be the output of a Failure Rate Modeling Task.
  • It can be estimated by dividing the number of failures per time units in the respective interval, divided by the average number of surviving cases at the mid-point of the interval.
• Example(s):
  • a Stroke Rate Function.
  • an Exponential Failure Rate Function.
• See: Continuous Function, Explanatory Variable, Time Function.

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References

2013

• http://en.wikipedia.org/wiki/Survival_function
  • The survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. It captures the probability that the system will survive beyond a specified time.

    The term reliability function is common in engineering while the term survival function is used in a broader range of applications, including human mortality. Another name for the survival function is the complementary cumulative distribution function.

• http://en.wikipedia.org/wiki/Survival_function#Definition
  • Let T be a continuous random variable with cumulative distribution function F(t) on the interval [0,∞). Its survival function or reliability function is: :[math]\displaystyle{ R(t) = P(\{T \gt t\}) = \int_t^{\infty} f(u)\,du = 1-F(t). }[/math]

• http://en.wikipedia.org/wiki/Failure_rate#Failure_rate_in_the_continuous_sense
  • Calculating the failure rate for ever smaller intervals of time, results in the Template:Visible anchor (loosely and incorrectly called hazard rate), [math]\displaystyle{ h(t) }[/math]. This becomes the instantaneous failure rate as [math]\displaystyle{ \scriptstyle\Delta t }[/math] tends to zero: :[math]\displaystyle{ h(t)=\lim_{\triangle t \to 0} \frac{R(t)-R(t+\triangle t)}{\triangle t \cdot R(t)}. }[/math] A continuous failure rate depends on the existence of a failure distribution, [math]\displaystyle{ \scriptstyle F(t) }[/math], which is a cumulative distribution function that describes the probability of failure (at least) up to and including time t, :[math]\displaystyle{ \operatorname{Pr}(T\le t)=F(t)=1-R(t),\quad t\ge 0. \! }[/math] where [math]\displaystyle{ {T} }[/math] is the failure time.

    The failure distribution function is the integral of the failure density function, f(t), :[math]\displaystyle{ F(t)=\int_{0}^{t} f(\tau)\, d\tau. \! }[/math] The hazard function can be defined now as :[math]\displaystyle{ h(t)=\frac{f(t)}{R(t)}. \! }[/math] Many probability distributions can be used to model the failure distribution (see List of important probability distributions). A common model is the exponential failure distribution, :[math]\displaystyle{ F(t)=\int_{0}^{t} \lambda e^{-\lambda \tau}\, d\tau = 1 - e^{-\lambda t}, \! }[/math] which is based on the exponential density function. The hazard rate function for this is: :[math]\displaystyle{ h(t) = \frac{f(t)}{R(t)} = \frac{\lambda e^{-\lambda t}}{e^{-\lambda t}} = \lambda . }[/math]

2003

• (Davison, 2003) ⇒ Anthony C. Davison. (2003). “Statistical Models." Cambridge University Press. ISBN:0521773393

2001

• (Gage et al., 2001) ⇒ Brian F Gage, Amy D Waterman, William Shannon, Michael Boechler, Michael W Rich, and Martha J Radford. (2001). “Validation of Clinical Classification Schemes for Predicting Stroke.” In: JAMA: the journal of the American Medical Association. doi:10.1001/jama.285.22.2864
  • QUOTE: The stroke rate was lowest among the 120 patients in the NRAF cohort who had a CHADS2 score of 0, a crude stroke rate of 1.2, and an adjusted rate of 1.9 per 100 patient-years without antithrombotic therapy (Table 2). The stroke rate increased by a factor of 1.5 (95% CI, 1.3-1.7) for each 1-point increase in the CHADS2 score (P<.001). Aspirin was associated with a hazard rate of 0.80 (95% CI, 0.5-1.3), corresponding to a nonsignificant 20% RR reduction in the rate of stroke (P = .27). File:2001 ValidationofClinicalClassificat.joc01974t2.png

1972

• (Cox, 1972) ⇒ David R Cox. (1972). “Regression Models and Life-tables.” In: Journal of the Royal Statistical Society. Series B (Methodological).
  • QUOTE: The hazard function (age-specific failure rate) is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time.

1963

• (Barlow et al., 1963) ⇒ Richard E. Barlow, Albert W. Marshall, and Frank Proschan. (1963. “Properties of probability distributions with monotone hazard rate.” In: The Annals of Mathematical Statistics, 34(2).
  • ABSTRACT: In this paper, we relate properties of a distribution function [math]\displaystyle{ F }[/math] (or its density [math]\displaystyle{ f }[/math]) to properties of the corresponding hazard rate [math]\displaystyle{ q }[/math] defined for [math]\displaystyle{ F(x)\lt 1 }[/math] by [math]\displaystyle{ q(x)=f(x)/[1−F(x)] }[/math]. It is shown, e.g., that the class of distributions for which [math]\displaystyle{ q }[/math] is increasing is closed under convolution, and the class of distributions for which [math]\displaystyle{ q }[/math] is decreasing is closed under convex combinations. Using the fact that [math]\displaystyle{ q }[/math] is increasing if and only if [math]\displaystyle{ 1-F }[/math] is a Polya frequency function of order two, inequalities for the moments of [math]\displaystyle{ F }[/math] are obtained, and some consequences of monotone [math]\displaystyle{ q }[/math] for renewal processes are given. Finally, the finiteness of moments and moment generating function is related to limiting properties of [math]\displaystyle{ q }[/math].

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