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Gamma mixture distribution

gamma_mix.Rd

Finite mixture of gamma components for positive-support bulk modeling. The scalar functions in this topic are the compiled building blocks behind the gamma bulk kernel family.

Usage

dGammaMix(x, w, shape, scale, log = 0)

pGammaMix(q, w, shape, scale, lower.tail = 1, log.p = 0)

rGammaMix(n, w, shape, scale)

qGammaMix(
  p,
  w,
  shape,
  scale,
  lower.tail = TRUE,
  log.p = FALSE,
  tol = 1e-10,
  maxiter = 200
)

Arguments

x

Numeric scalar giving the point at which the density is evaluated.

w

Numeric vector of mixture weights of length \(K\). The functions normalize w internally when needed.

shape, scale

Numeric vectors of length \(K\) giving Gamma shape and scale parameters.

log

Logical; if TRUE, return the log-density.

q

Numeric scalar giving the point at which the distribution function is evaluated.

lower.tail

Logical; if TRUE (default), probabilities are \(P(X \le q)\).

log.p

Logical; if TRUE, probabilities are returned on the log scale.

n

Integer giving the number of draws. The RNG implementation supports n = 1.

p

Numeric scalar probability in \((0,1)\) for the quantile function.

tol

Numeric scalar tolerance passed to stats::uniroot.

maxiter

Integer maximum number of iterations for stats::uniroot.

Value

Density/CDF/RNG functions return numeric scalars. qGammaMix() returns a numeric vector with the same length as p.

Details

The mixture density is $$ f(x) = \sum_{k = 1}^K \tilde{w}_k f_{\Gamma}(x \mid \alpha_k, \theta_k), \qquad x > 0, $$ with normalized weights \(\tilde{w}_k\). For vectorized R usage, use gamma_lowercase().

Under the package parameterization, each component has density \(f_\Gamma(x \mid \alpha,\theta) = x^{\alpha-1}\exp(-x/\theta) / \{\Gamma(\alpha)\theta^\alpha\}\) on \(x>0\). The mixture CDF is therefore $$ F(x) = \sum_{k=1}^K \tilde{w}_k F_\Gamma(x \mid \alpha_k,\theta_k). $$ Random generation first selects a component according to the normalized mixture weights and then draws from the corresponding gamma distribution. Since finite gamma mixtures do not have closed form quantiles, qGammaMix() obtains them numerically by inverting the mixture CDF.

The analytical mean is $$ E(X) = \sum_{k=1}^K \tilde{w}_k \alpha_k \theta_k. $$ This expression is reused in posterior predictive mean calculations for gamma-based fits.

Functions

  • dGammaMix(): Gamma mixture density

  • pGammaMix(): Gamma mixture distribution function

  • rGammaMix(): Gamma mixture random generation

  • qGammaMix(): Gamma mixture quantile function

See also

gamma_mixgpd(), gamma_gpd(), gamma_lowercase(), build_nimble_bundle(), kernel_support_table().

Other gamma kernel families: gamma_gpd, gamma_mixgpd

Examples

w <- c(0.55, 0.30, 0.15)
scale <- c(1.0, 2.5, 5.0)
shape <- c(2, 4, 6)

dGammaMix(2.0, w = w, scale = scale, shape = shape, log = 0)
#> [1] 0.1534717
pGammaMix(2.0, w = w, scale = scale, shape = shape, lower.tail = 1, log.p = 0)
#> [1] 0.3294213
qGammaMix(0.50, w = w, scale = scale, shape = shape)
#> [1] 3.623739
qGammaMix(0.95, w = w, scale = scale, shape = shape)
#> [1] 33.81667
replicate(10, rGammaMix(1, w = w, scale = scale, shape = shape))
#>  [1]  7.4874491  7.6825718 58.5380719  0.7576791 10.4751979 21.6543832
#>  [7]  0.4246413  9.1363824  1.1727450 17.8997664