Normalized inverse-Gaussian process

The NIG process normalizes a completely random measure whose Levy intensity is the tilted stable-1/2 density rho(u) = (2 pi)^-1/2 u^-3/2 exp(-tau^2 u / 2), so int u rho(u) du = 1 / tau exactly. Its tails are heavier than the Dirichlet process's, which is why it is used in its place. The Python arm delegates the quadrature to gh_c14_14, which has no R mirror; the midpoint rule is five lines and is written here.

Usage

Nrmprc(y, alpha = 1, tau = 1, u_max = 10, n_grid = 6000)

Arguments

y

Observed values; only the count and the number of distinct values enter the summary.

alpha

Total mass of the base measure, positive.

tau

Exponential tilting parameter, positive.

u_max

Upper quadrature limit.

n_grid

Midpoint-rule grid size.

Value

List with estimate, theory, gap, alpha, tau, n, n_distinct.

References

Lijoi, A., Mena, R. H. & Prunster, I. (2005). Hierarchical mixture modeling with normalized inverse-Gaussian priors. Journal of the American Statistical Association, 100(472), 1278-1291. Ghosal, S. & van der Vaart, A. (2017). Fundamentals of Nonparametric Bayesian Inference, CUP, section 14.6.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Nrmprc(V)
#> $estimate
#> [1] 0.9885819
#> 
#> $theory
#> [1] 1
#> 
#> $gap
#> [1] 0.01141806
#> 
#> $alpha
#> [1] 1
#> 
#> $tau
#> [1] 1
#> 
#> $n
#> [1] 8
#> 
#> $n_distinct
#> [1] 8
#> 
#> $method
#> [1] "Normalized inverse-Gaussian process (Lijoi-Mena-Prunster 2005)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"