Bracketing entropy integral and the Donsker conditions

\(J_{[]}(\delta) = \int_0^\delta \sqrt{\log N_{[]}(\epsilon)} d\epsilon\). The square root is what makes polynomial bracketing growth integrable and exponential growth divergent – that IS the Donsker boundary. Also reports the two envelope conditions, which differ between GC (needs P\*F) and Donsker (needs P\*F^2). Mirrors morie.fn.ksr035–ksr038.

Usage

morie_entropy_integral(
  N_bracket,
  delta = 1,
  envelope_mean = NULL,
  envelope_sq_mean = NULL
)

Arguments

N_bracket

function mapping eps to a bracketing number.

delta

upper limit.

envelope_mean

optional P*F, for the GC condition.

envelope_sq_mean

optional P*F^2, for the Donsker condition.

Value

list: J, finite, gc_conditions_met, donsker_conditions_met.

References

Kosorok (2008), Ch. 2.

Examples

morie_entropy_integral(function(e) (1 / e)^3)$J
#> [1] 1.53499