Quantile of K Given Alpha

Computes the \(p\)-th quantile of \(K_J \mid \alpha\), defined as the smallest \(k\) such that \(P(K_J \leq k \mid \alpha) \geq p\).

Usage

quantile_K_given_alpha(p, J, alpha, logS)

Arguments

p

Numeric; probability level(s) in \([0, 1]\). Can be scalar or vector.

J

Integer; sample size.

alpha

Numeric; DP concentration parameter.

logS

Matrix; pre-computed log-Stirling matrix.

Value

Integer (or integer vector); the \(p\)-th quantile(s) of \(K_J \mid \alpha\).

Details

For \(p = 0.5\), this gives the median. Note that for discrete distributions, the quantile is defined as the smallest value where the CDF meets or exceeds \(p\).

Edge cases:

• \(p = 0\): returns 0 (the first k where CDF >= 0)

• \(p = 1\): returns J (the maximum possible K)

Examples

logS <- compute_log_stirling(50)

# Single quantile (median)
quantile_K_given_alpha(0.5, 50, 2.0, logS)
#> [1] 7

# Multiple quantiles
quantile_K_given_alpha(c(0.25, 0.5, 0.75), 50, 2.0, logS)
#> [1] 6 7 8

# Edge cases
quantile_K_given_alpha(0, 50, 2.0, logS)  # Returns 0
#> [1] 0
quantile_K_given_alpha(1, 50, 2.0, logS)  # Returns 50
#> [1] 50