Who invented the hazard function? 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?
 A: 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
https://mathshistory.st-andrews.ac.uk/Miller/mathword/
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
