Chernoff bound on an upper tail

Formula: P(X >= a) <= min_{s>0} exp(-s a) E[exp(s X)]

Usage

Chernbnd(mgf, a, s_grid = NULL)

Arguments

mgf

See Usage.

a

See Usage.

s_grid

See Usage.

Value

List with bound, s, log_bound, at_boundary.

References

Chernoff (1952), A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Annals of Mathematical Statistics 23:493-507. The exponential Markov bound is stated in this exact form in every standard reference.

Examples

Chernbnd(mgf = function(s) exp(0.5 * s^2), a = 2)
#> $bound
#> [1] 0.1354437
#> 
#> $s
#> [1] 2.040016
#> 
#> $log_bound
#> [1] -1.999199
#> 
#> $at_boundary
#> [1] FALSE
#> 
#> $method
#> [1] "Chernoff bound"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"