Mean functional of a normalized random measure

An NRMI is a completely random measure divided by its own total mass. Regazzini, Lijoi & Prunster asked what the linear functional M = int x P~(dx) then looks like; for the gamma CRM, whose normalization is the Dirichlet process, the first two moments are closed form: E[M] = mu0 and \(Var[M] = sigma0^2 / (alpha + 1)\). After observing y_1, ..., y_n the posterior is again a Dirichlet process with mass alpha + n and base measure (alpha P0 + sum delta_yi) / (alpha + n), so E[M | y] = (alpha mu0 + sum y_i) / (alpha + n) and Var[M | y] = s2_post / (alpha + n + 1). None of these depend on tau.

Usage

Nrgmwd(y, alpha = 1, tau = 1, mu0 = 0, sigma0 = 1)

Arguments

y

Observed values.

alpha

Total mass of the CRM, positive.

tau

Scale of the CRM, positive; carried through only to the unnormalized total mass.

mu0

Mean of the base measure.

sigma0

Standard deviation of the base measure, non-negative.

Value

List with estimate, prior_mean, prior_var, post_mean, post_var, post_base_var, post_mass, total_mass, alpha, tau, n.

References

Regazzini, E., Lijoi, A. & Prunster, I. (2003). Distributional results for means of normalized random measures with independent increments. Annals of Statistics, 31(2), 560-585. doi:10.1214/aos/1051027881

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Nrgmwd(V)
#> $estimate
#> [1] 4
#> 
#> $prior_mean
#> [1] 0
#> 
#> $prior_var
#> [1] 0.5
#> 
#> $post_mean
#> [1] 4
#> 
#> $post_var
#> [1] 0.6777778
#> 
#> $post_base_var
#> [1] 6.777778
#> 
#> $post_mass
#> [1] 9
#> 
#> $total_mass
#> [1] 1
#> 
#> $alpha
#> [1] 1
#> 
#> $tau
#> [1] 1
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Mean functional of a normalized random measure"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"