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Verify Quadrature Accuracy Against Known Gamma Moments

Source: R/02_quadrature.R
verify_quadrature.Rd

Validates quadrature implementation by comparing computed expectations against known closed-form Gamma distribution moments.

Usage

verify_quadrature(a, b, M, tol = 1e-10, verbose = TRUE)

Arguments

a

Numeric; shape parameter of Gamma distribution.

b

Numeric; rate parameter of Gamma distribution.

M

Integer; number of quadrature nodes.

tol

Numeric; tolerance for verification (default: 1e-10).

verbose

Logical; if TRUE, print detailed results.

Value

Logical; TRUE if all moments match within tolerance.

Details

For \(\alpha \sim \text{Gamma}(a, b)\):

  • \(E[\alpha] = a/b\)

  • \(E[\alpha^2] = a(a+1)/b^2\)

  • \(Var(\alpha) = a/b^2\)

Examples

# Should return TRUE
verify_quadrature(2.5, 1.5, M = 80)
#> Quadrature verification (a=2.50, b=1.50, M=80):
#>   E[alpha]:   true=1.666667, quad=1.6666666667, error=4.44e-16 [PASS]
#>   E[alpha^2]:  true=3.888889, quad=3.8888888889, error=0.00e+00 [PASS]
#>   Var(alpha): true=1.111111, quad=1.1111111111, error=1.33e-15 [PASS]
#>   Overall: PASS

# More challenging case
verify_quadrature(0.5, 2.0, M = 100, verbose = TRUE)
#> Quadrature verification (a=0.50, b=2.00, M=100):
#>   E[alpha]:   true=0.250000, quad=0.2500000000, error=1.94e-16 [PASS]
#>   E[alpha^2]:  true=0.187500, quad=0.1875000000, error=6.11e-16 [PASS]
#>   Var(alpha): true=0.125000, quad=0.1250000000, error=5.13e-16 [PASS]
#>   Overall: PASS