Cesaro-average Hellinger discrepancy of the predictive densities

The martingale approach needs no tests: what is checked is the conclusion n^-1 sum d_H^2(phat_i, p_0) -> 0, plus the Lemma 6.52 summability condition.

Usage

Martcons(dh2, variances = NULL)

Arguments

dh2

Squared Hellinger distances, each in [0, 1].

variances

Optional variances of the martingale differences.

Value

List with cesaro, final, tail_mean, lemma652_sum, summable, n.

Details

Formula: n^-1 sum_{i=1}^{n} d_H^2(phat_i, p_0) -> 0 a.s.; Lemma 6.52 needs sum_{n>=1} n^-2 var_0(...) < Inf

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, Section 6.8.4, equations (6.17) and (6.18), and Lemma 6.52. The approach is due to Walker (2003, 2004) as the book's historical notes record. Read from the copy of the book held in the corpus.

Examples

Martcons(dh2 = c(0.1, 0.2, 0.15))
#> $cesaro
#> [1] 0.10 0.15 0.15
#> 
#> $final
#> [1] 0.15
#> 
#> $tail_mean
#> [1] 0.175
#> 
#> $lemma652_sum
#> [1] NaN
#> 
#> $summable
#> [1] NaN
#> 
#> $n
#> [1] 3
#> 
#> $method
#> [1] "Martingale consistency check, Ghosal Section 6.8.4"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"