Thinning a Poisson count: the recording rate is not identified — morie_thinning

Recording each event independently with probability gamma turns a Poisson(lambda) count into a Poisson(lambda * gamma) count (Research.P1.Thinning.thinned_pgf), independent of the unrecorded remainder (joint_pgf_factorises); the pair is identified only through the product (not_identified).

Usage

morie_thinning_identification(lambda, gamma, s = c(0, 0.5, 0.9, 1))

Arguments

lambda: True rate.
gamma: Recording probability in (0, 1).
s: Points at which to compare generating functions.

Value

A list: recorded_rate, alternative (a different lambda, gamma pair with the same recorded law), and pgf_check (the thinned and the Poisson generating functions agree).

References

Grimmett G, Stirzaker D (2020). Probability and Random Processes, 4th ed., Theorem 6.13.14. Oxford.

Examples

morie_thinning_identification(lambda = 5, gamma = 0.3)
#> $recorded_rate
#> [1] 1.5
#> 
#> $alternative
#> lambda  gamma 
#>    1.5    1.0 
#> 
#> $pgf_check
#> [1] TRUE
#> 
#> attr(,"theorem")
#> [1] "Research.P1.Thinning.not_identified"