Variance of the number of clusters under the Gnedin prior

Compute the closed-form variance of \(K_n\) under the Gnedin finite-type model (\(\sigma=-1\)).

Usage

gnedin_K_var(n, gamma)

Arguments

n

Integer vector of sample sizes (each \(\ge 1\)).

gamma

Numeric vector of Gnedin parameters, each in \((0,1)\).

Value

A numeric vector with \(\mathrm{Var}(K_n)\) for each pair (n, gamma).

Details

Using the factorial–ordinary moment relation \(\mathbb{E}[K_n^2]=\mathbb{E}[K_n(K_n-1)]+\mathbb{E}[K_n]\), the variance is $$ \mathrm{Var}(K_n) \;=\; \mathbb{E}[K_n(K_n-1)] + \mathbb{E}[K_n] - \{\mathbb{E}[K_n]\}^2, $$ where $$ \mathbb{E}[K_n(K_n-1)] \;=\; n(n-1)(1-\gamma)\,\frac{\Gamma(n)\,\Gamma(1+\gamma)}{\Gamma(n+\gamma)}. $$ The implementation uses lgamma for numerical stability and is vectorized over n and gamma (with standard R recycling rules).

References

Gnedin, A. (2010). A species sampling model with finitely many types. Electronic Communications in Probability, 15, 79–88.

Favaro, S., Lijoi, A., & Prünster, I. (2013). Extending the class of Gibbs-type priors: theoretical properties and new examples. Annals of Applied Probability, 23(4), 1729–1754.

Pitman, J. (2006). Combinatorial Stochastic Processes. Springer.

See also

gnedin_K_mean()

Examples

# Scalar inputs
gnedin_K_var(105, 0.8)
#> [1] 45.95132

# Consistency check: variance is nonnegative
all(gnedin_K_var(20, c(0.3, 0.5, 0.8)) >= 0)
#> [1] TRUE