remaininglife

Remaining life, or remaining lifetime, refers to the amount of time that remains before a specified event occurs, such as death in humans or failure of a component, given current age or state. In statistics and reliability theory, remaining life is treated as a conditional random variable describing future lifetime beyond a given time.

Let T be the total lifetime with survival function S(t) = P(T > t). The remaining lifetime at age x is T − x conditional on T > x. Its expected value, called the mean residual life at age x, is m(x) = E[T − x | T > x] = (1 / S(x)) ∫_x^∞ S(t) dt. It is related to the hazard rate h(t) = f(t)/S(t) and can be used to summarize expected future lifetime given survival to age x. In practice, remaining life is estimated from survival data and the chosen statistical model.

In human mortality studies, remaining life features in survival analysis to inform prognosis. In reliability engineering, remaining useful life (RUL) describes the time to system failure given current condition, guiding maintenance and replacement decisions. Estimation methods include parametric models such as Weibull or Gompertz, nonparametric approaches like the Kaplan-Meier estimator, and machine learning techniques for complex data.

Applications include actuarial calculations for life insurance and pensions, medical prognosis, maintenance planning, and warranty optimization. Limitations include dependence on model assumptions, data censoring, and limited predictive power for individuals. See also life expectancy, survival analysis, hazard rate, and remaining useful life.