Mean time in custody from completed and current stays

The mean length of stay of the people released (or of the cases disposed) leaves out everyone still inside. Their elapsed time is a lower bound on their stay, so the mean over everyone admitted is at least lower, the mean of the completed stays and the elapsed times together; the released-only mean falls short of it by m / (n + m) times the excess of the mean elapsed time over the released mean, and understates the truth for certain when that excess is positive. No upper bound exists without a cap on how long the current stays will last. See proof("losbnd").

Usage

losbnd(done, open)

Arguments

done: Completed stays: non-negative numbers.
open: Elapsed time of the stays still open: non-negative numbers.

Value

A named numeric vector: mean (of the completed stays), age (the mean elapsed time of the open ones), lower (the lower bound on the mean over all) and bias (lower - mean, the least by which the completed mean understates when it is positive).

References

Manski, C. F. (2003). Partial Identification of Probability Distributions. Springer. doi:10.1007/b97478

Examples

losbnd(done = c(30, 45, 60, 90, 120), open = c(100, 150, 200))
#>    mean     age   lower    bias 
#>  69.000 150.000  99.375  30.375