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Amoroso mixture distribution

amoroso_mix.Rd

Finite mixture of Amoroso components for flexible positive-support bulk modeling.

Usage

dAmorosoMix(x, w, loc, scale, shape1, shape2, log = 0)

pAmorosoMix(q, w, loc, scale, shape1, shape2, lower.tail = 1, log.p = 0)

rAmorosoMix(n, w, loc, scale, shape1, shape2)

qAmorosoMix(
  p,
  w,
  loc,
  scale,
  shape1,
  shape2,
  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 treat the weights as non-negative and normalize them internally when needed.

loc

Numeric vector of length \(K\) giving component locations.

scale

Numeric vector of length \(K\) giving component scales. Negative values flip support for the corresponding component.

shape1

Numeric vector of length \(K\) giving the first Amoroso shape parameter for each component.

shape2

Numeric vector of length \(K\) giving the second Amoroso shape parameter for each component.

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

Mixture density/CDF/RNG functions return numeric scalars. qAmorosoMix() returns a numeric vector with the same length as p.

Details

The mixture density is $$ f(x) = \sum_{k = 1}^K \tilde{w}_k f_{Amoroso}(x \mid a_k, \theta_k, \alpha_k, \beta_k), $$ with normalized weights \(\tilde{w}_k\). These scalar functions are NIMBLE-compatible; for vectorized R usage, use amoroso_lowercase().

The Amoroso family is especially useful for positive-support data because it can reproduce a wide range of skewed and heavy-right-tail shapes while remaining analytically tractable through its gamma transformation. The mixture CDF is $$ F(x) = \sum_{k=1}^K \tilde{w}_k F_{Amoroso}(x \mid a_k,\theta_k,\alpha_k,\beta_k), $$ and random generation proceeds by selecting a component and sampling from that component.

Closed-form mixture quantiles are not available, so qAmorosoMix() inverts the mixture CDF numerically. The analytical mixture mean is the weighted average of the component means, \(a_k + \theta_k \Gamma(\alpha_k + 1/\beta_k) / \Gamma(\alpha_k)\), whenever those component moments exist.

Functions

  • dAmorosoMix(): Density Function of Amoroso Mixture Distribution

  • pAmorosoMix(): Cumulative Distribution Function of Amoroso Mixture Distribution

  • rAmorosoMix(): Random Generation for Amoroso Mixture Distribution

  • qAmorosoMix(): Quantile Function of Amoroso Mixture Distribution

See also

amoroso_mixgpd(), amoroso_gpd(), amoroso_lowercase(), build_nimble_bundle(), kernel_support_table().

Other amoroso kernel families: amoroso_gpd, amoroso_mixgpd

Examples

w <- c(0.60, 0.25, 0.15)
loc <- c(0, 1, 2)
scale <- c(1.0, 1.2, 1.6)
shape1 <- c(2, 4, 6)
shape2 <- c(1.0, 1.2, 1.5)

dAmorosoMix(2.0, w, loc, scale, shape1, shape2, log = 0)
#> [1] 0.1717337
pAmorosoMix(2.0, w, loc, scale, shape1, shape2, lower.tail = 1, log.p = 0)
#> [1] 0.3587001
qAmorosoMix(0.50, w, loc, scale, shape1, shape2)
#> [1] 2.929829
qAmorosoMix(0.95, w, loc, scale, shape1, shape2)
#> [1] 8.017293
replicate(10, rAmorosoMix(1, w, loc, scale, shape1, shape2))
#>  [1] 6.6966960 1.1390523 1.7812916 2.8622365 2.5033891 1.2467692 3.6989854
#>  [8] 1.1954265 0.8008628 7.2399839