Efficient influence function for F(t)

psi-tilde(x) = 1{x <= t} - F0(t) has mean zero under P0 and variance F0(t)(1 - F0(t)), which IS the semiparametric efficiency bound for estimating F(t). The influence function is the whole content of the efficiency calculation: once it is identified, the bound is just its second moment.

Usage

Ghosaleffinflfn(data, t)

Arguments

data

Numeric vector of observations, non-empty.

t

Point at which the CDF functional is taken.

Value

List with estimate (the variance), mean_zero_gap, matches_bernoulli_var, method.

Details

Formula: var = (1/n) sum_i (1{x_i <= t} - F_n(t))^2, which equals F_n(t)(1 - F_n(t)) identically.

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 12.3.1.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Ghosaleffinflfn(V, V)
#> $estimate
#> [1] 0
#> 
#> $mean_zero_gap
#> [1] 0
#> 
#> $matches_bernoulli_var
#> [1] TRUE
#> 
#> $method
#> [1] "efficient influence function (GvdV 2017 sec. 12.3.1)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"