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The hazard function or instantaneous failure rate is very popular in survival analysis:

https://en.wikipedia.org/wiki/Failure_rate

$$\lambda(t) = \dfrac{f(t)}{1-F(t)}.$$

However, I cannot find a reference tracking back its origins. Is the history of the hazard function known? If so, who invented it or used it for the first time?

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    $\begingroup$ Actuaries have used the notion for a very long time (under a different name, force of mortality). It may be that the first occurrence of a hazard function will be in the early work on "laws of mortality", perhaps somewhere in the writings of say Gompertz, or perhaps even earlier, but it may not initially be easy to recognize. $\endgroup$ – Glen_b -Reinstate Monica Dec 4 '18 at 23:52
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    $\begingroup$ Indeed, looking at Gompertz, (1825) he discusses an individuals intensity of mortality, and this does appear to correspond to the hazard function; so the use of hazard functions is presumably at least as old as this. $\endgroup$ – Glen_b -Reinstate Monica Dec 4 '18 at 23:57
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    $\begingroup$ @Glen_b Sounds like a good reference. I could accept that as an answer (with the obvious warning that there may be older references after digging deeper, but for this question, your reference is more than enough). $\endgroup$ – Eden Michael Dec 5 '18 at 0:02
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    $\begingroup$ I'll need to reread that section more carefully first, but sure; I could offer a quote and complete reference. Also; Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics -(H) discusses the use of the term 'hazard rate', and it looks like that's from the 50s and 60s. With the references there, I think there's perhaps enough for a partial answer. $\endgroup$ – Glen_b -Reinstate Monica Dec 5 '18 at 0:21
  • $\begingroup$ @Glen_b Indeed, it is a bit tricky as it seems that paper is about what we know as the Gompertz growth curve. But certainly, the literature on tables of mortality may give some hints to earlier references to this concept. The link to Jeff Miller's website is great, thanks! $\endgroup$ – Eden Michael Dec 5 '18 at 8:23
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The term for it seems to be relatively recent but the notion is considerably older.

Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics discusses the use of the term 'hazard rate', and it looks like that's from the 50s and 60s. It reports that the term "death-hazard rate" occurs in D. J. Davis "An Analysis of Some Failure Data," Journal of the American Statistical Association, 47, (1952), 113-150, and "hazard rate" occurs in R. E. Barlow; A. W. Marshall & F. Proschan "Properties of Probability Distributions with Monotone Hazard Rate," Annals of Mathematical Statistics, 34, (1963), 375-389.

It turns out that hazard functions occur (without that specific term hazard) in the early work on "laws of mortality" - for example, the notion occurs in the writing of Gompertz. This might not be the very first use of a hazard function, but it's the earliest I have found so far. Here's Gompertz in 1825:

If the average exhaustions of a man's power to avoid death were such that at the end of equal infinitely smaIl intervals of time, he lost equal portions of his remaining power to oppose destruction which he had at the commencement of those intervals, then at the age $x$, his power to avoid death, or the intensity of his mortality might be denoted by $a\,q^x$, $a$ and $q$ being constant quantities; and if $L_x$, be the number of living at the age $x$, we shall have $a L_x \times q^x.\dot{x}$ for the fluxion of the number of deaths $= -(L_x)^{^ \bullet};\:\therefore abq^x = -\frac{\dot{L}_x}{L_x}$,

(NB I am unsure that the "$a L_x \times q^x.\dot{x}$" is accurately reproduced; Gompertz may originally have written something slightly different, but that's what it looks like in the pdf image)

Noting that $L_x$ here represents a scaled $S(x)$ ($L_x=N S(x)$, where $L_0=N$), so $-\frac{\dot{L}_x}{L_x}=\frac{f(x)}{S(x)}$, we can see that he describes the hazard function in words (an instantaneous intensity of mortality) and then at the end gives a formula that corresponds to our definition of the hazard function.

Milller, Jeff,
Earliest Known Uses of Some of the Words of Mathematics
http://jeff560.tripod.com/mathword.html

Gompertz, B. (1825),
"On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies."
Philosophical Transactions of the Royal Society of London 115: 513–583. doi:10.1098/rstl.1825.0026

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