hesela.devLength-biased sampling
Length-biased sampling selects units with probability proportional to the duration being measured.
A cross-sectional sample of operating devices can overrepresent long lifetimes because those devices remain available for selection longer. Under stationary Poisson entry, a stable lifetime distribution and uniform selection among operating devices, the sampled full-lifetime density is t f(t) / E(T), for positive T with finite mean. This is a specific sampling model, not a property of every fleet snapshot. [1]
The mean full lifetime in that sample is E(T^2) / E(T) when the second moment is finite. The difference from the entrant mean is Var(T) / E(T), zero for identical lifetimes. It includes time before and after selection: it is neither current age nor remaining life. [2]
Selection after survival to entry creates left-truncation; length bias is a special case. Right-censoring after selection is a separate observation limit and can complicate inference. Fleet growth, changing model mix and early retirement require a justified sampling model, not an automatic duration-weight correction. [1]
Relations
Citations
- Chan, Chen and Di (2012): Proportional mean residual life model for right-censored length-biased data, sections 1-2
- Wrigley-Field and Feehan (2022): In a stationary population, the average lifespan of the living is a length-biased life expectancy, equations 5-6
Further reading
Sources & caveats
- Statistical sources establish a sampling mechanism, not an empirical estimate of bias in a storage fleet.
- The linked worked example uses fictional stationary arrivals and fixed lifetimes; it is not Backblaze data.