CDF of K Given Alpha

Computes the cumulative distribution function \(P(K_J \leq k \mid \alpha)\) for \(k = 0, 1, \ldots, J\).

Usage

cdf_K_given_alpha(J, alpha, logS)

Arguments

J

Integer; sample size.

alpha

Numeric; DP concentration parameter.

logS

Matrix; pre-computed log-Stirling matrix.

Value

Numeric vector of length \(J+1\) containing \(P(K_J \leq k \mid \alpha)\) for \(k = 0, 1, \ldots, J\).

Details

The CDF satisfies:

• \(F(0) = 0\) (since \(P(K_J = 0) = 0\))

• \(F(J) = 1\)

• \(F(k)\) is non-decreasing in \(k\)

Examples

logS <- compute_log_stirling(50)
cdf <- cdf_K_given_alpha(50, 2.0, logS)

# Verify CDF ends at 1
cdf[51]  # Should be 1
#> [1] 1

# P(K <= 5)
cdf[6]
#> [1] 0.2418684