Robbins' nonparametric empirical Bayes

Formula: E[theta | y = k] = (k + 1) f(k + 1) / f(k)

Usage

Empbnp(y, prior_family = "poisson")

Arguments

y

Observed counts (poisson) or values (gaussian).

prior_family

"poisson" for Robbins' rule, "gaussian" for Tweedie's.

Value

List with estimate, theta_hat, support, counts, n, method.

Details

For Poisson observations the posterior mean of the rate needs no parametric prior at all: it is recovered from the marginal counts alone. The gaussian branch is Tweedie's formula E[theta | y] = y + d/dy log f(y), estimated from a kernel-smoothed marginal.

References

Robbins (1956), Proc. 3rd Berkeley Symp. 1:157-163; Efron (2011), JASA 106(496):1602-1614.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Empbnp(V)
#> $estimate
#> [1] 4.375
#> 
#> $theta_hat
#> [1] NaN   2   3   4   5   6   7   8   0
#> 
#> $support
#> [1] 0 1 2 3 4 5 6 7 8
#> 
#> $counts
#> [1] 0 1 1 1 1 1 1 1 1
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Robbins nonparametric empirical Bayes (Poisson)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"