{
  "slug": "mean-residual-life",
  "term": "Mean residual life",
  "aliases": [
    "MRL",
    "Expected remaining lifetime"
  ],
  "definition": "Mean residual life is the expected additional lifetime of a unit conditional on its having survived beyond a specified age.",
  "note": "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. [gupta-bradley-mrl]\n\nMRL 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. [nist-exponential-mrl]\n\nA 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.\n\nReciprocal 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. [gupta-bradley-mrl]\n\nFollow-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. [alvarez-censored-mrl]",
  "status": "shipping",
  "advantages": [],
  "limitations": [],
  "related": [
    "weibull-distribution",
    "survival-function",
    "hazard-rate",
    "right-censoring",
    "mtbf"
  ],
  "supersedes": [],
  "supersededBy": [],
  "firstSeen": null,
  "lastSeen": null,
  "citations": [
    {
      "id": "gupta-bradley-mrl",
      "title": "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)",
      "url": "https://arxiv.org/html/math/0411297v1",
      "accessed": "2026-11-05"
    },
    {
      "id": "nist-exponential-mrl",
      "title": "NIST e-Handbook 8.1.6.1: Exponential distribution, survival and mean",
      "url": "https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm",
      "accessed": "2026-11-05"
    },
    {
      "id": "alvarez-censored-mrl",
      "title": "Alvarez-Iglesias et al.: Summarising censored survival data using the mean residual life function (2015), abstract",
      "url": "https://onlinelibrary.wiley.com/doi/10.1002/sim.6431",
      "accessed": "2026-11-05"
    }
  ],
  "updated": "2026-11-05"
}