hesela.devMean residual life
Mean residual life is the expected additional lifetime of a unit conditional on its having survived beyond a specified age.
For a nonnegative lifetime T and survival S(t) = P(T > t), m(t) = E(T - t | T > t) = integral from t to infinity of S(u) du, divided by S(t). This requires S(t) > 0 and a finite tail integral for a finite mean; the units of time are not probability or inverse time. [1]
MRL concerns survivors, so it is not generally the original mean minus age. Exponential survival with rate lambda > 0 gives m(t) = 1/lambda: a model property, not evidence that drives avoid aging. [2]
A fictional population split equally between lifetimes of 2 and 10 years has initial mean 6, but remaining mean 7 at age 3. Replacing 10 by 100 changes that remaining mean to 97. Five-year follow-up cannot distinguish them: both record failure at year 2 and a survivor censored at year 5. These are constructed populations, not measured drive lifetimes.
Reciprocal hazard is not generally MRL. For an absolutely continuous lifetime with finite differentiable m, m'(t) = h(t)m(t) - 1. Uniform lifetimes on 0 to 10 years give m(3) = 3.5 years but 1/h(3) = 7 years. The smooth identity does not apply at discrete jumps. [1]
Follow-up ending with positive survival leaves an unobserved tail. Complete MRL needs tail information or assumptions; a limited future horizon answers a different question. A conditional population mean is not a predicted failure date for one disk. Define whether the event is device failure or operational replacement. [3]
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Citations
- Gupta and Bradley: On Representing the Mean Residual Life in Terms of the Failure Rate, section 2, equations 2.1 and 2.4 (2003; author manuscript)
- NIST e-Handbook 8.1.6.1: Exponential distribution, survival and mean
- Alvarez-Iglesias et al.: Summarising censored survival data using the mean residual life function (2015), abstract