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Inverse Gaussian mixture distribution

InvGauss_mix.Rd

Finite mixture of inverse Gaussian components for positive-support bulk modeling. Each component is parameterized by mean[j] and shape[j].

Usage

dInvGaussMix(x, w, mean, shape, log = 0)

pInvGaussMix(q, w, mean, shape, lower.tail = 1, log.p = 0)

rInvGaussMix(n, w, mean, shape)

qInvGaussMix(
  p,
  w,
  mean,
  shape,
  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.

mean, shape

Numeric vectors of length \(K\) giving component means and shapes.

log

Logical; if TRUE, return the log-density (integer flag 0/1 in NIMBLE).

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. For portability inside NIMBLE, 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 in quantile inversion.

maxiter

Integer maximum number of iterations for stats::uniroot.

Value

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

Details

The scalar functions in this topic are the compiled building blocks for inverse-Gaussian bulk kernels. For vectorized R usage, use invgauss_lowercase().

The mixture distribution is $$ F(x) = \sum_{k=1}^K \tilde{w}_k F_{IG}(x \mid \mu_k,\lambda_k), $$ where each inverse Gaussian component has mean \(\mu_k\) and variance \(\mu_k^3/\lambda_k\). Random generation selects a component using the normalized weights and then generates from the corresponding inverse Gaussian law. Quantiles are computed numerically because the finite-mixture inverse CDF is not available in closed form.

The analytical mixture mean is $$ E(X) = \sum_{k=1}^K \tilde{w}_k \mu_k. $$ That expression is used by the package whenever inverse-Gaussian mixtures contribute to posterior predictive means.

Functions

  • dInvGaussMix(): Inverse Gaussian mixture density

  • pInvGaussMix(): Inverse Gaussian mixture distribution function

  • rInvGaussMix(): Inverse Gaussian mixture random generation

  • qInvGaussMix(): Inverse Gaussian mixture quantile function

See also

InvGauss_mixgpd(), InvGauss_gpd(), invgauss_lowercase(), build_nimble_bundle(), kernel_support_table().

Other inverse-gaussian kernel families: InvGauss_gpd, InvGauss_mixgpd

Examples

w <- c(0.55, 0.30, 0.15)
mean <- c(1.0, 2.5, 5.0)
shape <- c(2, 4, 8)

dInvGaussMix(2.0, w = w, mean = mean, shape = shape, log = 0)
#> [1] 0.17698
pInvGaussMix(2.0, w = w, mean = mean, shape = shape,
            lower.tail = 1, log.p = 0)
#> [1] 0.6866789
qInvGaussMix(0.50, w = w, mean = mean, shape = shape)
#> [1] 1.251694
qInvGaussMix(0.95, w = w, mean = mean, shape = shape)
#> [1] 6.489781
replicate(10, rInvGaussMix(1, w = w, mean = mean, shape = shape))
#>  [1] 0.9962017 1.5444369 0.9499274 0.3434187 1.5752745 4.7585974 1.5624384
#>  [8] 0.4135278 0.7628506 4.1941022