Bernstein-Feller approximation of a CDF

F_K(x) = sum_k F(k/K) C(K, k) x^k (1 - x)^(K-k). The Bernstein operator applied to a CDF converges UNIFORMLY, and it maps CDFs to CDFs, which is why it turns a prior on the finite vector (F(0/K), ..., F(K/K)) into a prior on genuine distribution functions.

Usage

Ghosalbernsteinfeller(x, F = NULL, K = 30)

Arguments

x

Evaluation points; clipped to [0, 1].

F

A CDF on [0, 1]; t^2 when NULL.

K

Bernstein degree, at least 1.

Value

List with estimate (value at the middle point), F_K, sup_error, method.

Details

Formula: as displayed; the reported error is the sup over the supplied points of |F_K(x) - F(x)|.

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 2.3.4.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Ghosalbernsteinfeller(V)
#> $estimate
#> [1] 1
#> 
#> $F_K
#> [1] 1 1 1 1 1 1 1 1
#> 
#> $sup_error
#> [1] 0
#> 
#> $method
#> [1] "Bernstein-Feller CDF approximation (GvdV 2017 sec. 2.3.4)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"