Mean run length of a restarted detector with false-alarm probability alpha

With \(P(\tau < \infty) \le \alpha\) for each restarted copy, \(P(N \le n) \le (n+1)\alpha\) (Research.P10.Cusum.stop_prob_le) and \(E N \ge \sum_{n<J}(1-n\alpha) \ge J/2\) for any \(J\alpha \le 1\) (mean_run_length_ge).

Usage

morie_run_length_bound(alpha, J = floor(1/alpha))

Arguments

alpha: False-alarm probability in (0, 1].
J: Horizon; default \(\lfloor 1/\alpha \rfloor\).

Value

A list: J, partial_sum, bound (J / 2).

References

Lai TL (1995). Sequential changepoint detection in quality control and dynamical systems. JRSS-B 57(4).

Examples

morie_run_length_bound(alpha = 0.01)
#> $J
#> [1] 100
#> 
#> $partial_sum
#> [1] 50.5
#> 
#> $bound
#> [1] 50
#> 
#> attr(,"theorem")
#> [1] "Research.P10.Cusum.mean_run_length_ge"