Multigam: Multivariate Gamma Distribution

Description

The multivariate gamma distribution is a multivariate absolute continuous probability distribution, defined as the cumulative sum of independent gamma random variables with possibly different shape parameters $α_{i} > 0, i \in {1, \dots, k}$ and the same scale $β > 0$ .

Usage

Multigam(shape = 1, scale = 1)
dmultigam(x, shape, scale, log = FALSE)
rmultigam(n, shape, scale)
# S4 method for Multigam,numeric
d(distr, x, log = FALSE)
# S4 method for Multigam,matrix
d(distr, x, log = FALSE)
# S4 method for Multigam,numeric
r(distr, n)
# S4 method for Multigam
mean(x)
# S4 method for Multigam
var(x)
# S4 method for Multigam
finf(x)
llmultigam(x, shape, scale)
# S4 method for Multigam,matrix
ll(distr, x)
emultigam(x, type = "mle", ...)
# S4 method for Multigam,matrix
mle(
  distr,
  x,
  par0 = "same",
  method = "L-BFGS-B",
  lower = 1e-05,
  upper = Inf,
  na.rm = FALSE
)
# S4 method for Multigam,matrix
me(distr, x, na.rm = FALSE)
# S4 method for Multigam,matrix
same(distr, x, na.rm = FALSE)
vmultigam(shape, scale, type = "mle")
# S4 method for Multigam
avar_mle(distr)
# S4 method for Multigam
avar_me(distr)
# S4 method for Multigam
avar_same(distr)

Value

Each type of function returns a different type of object:

Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.
Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.
Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.
Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

Arguments

shape, scale: numeric. The non-negative distribution parameters.
x: For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Multigam. For the log-likelihood and the estimation functions, x is the sample of observations.
log: logical. Should the logarithm of the probability be returned?
n: number of observations. If length(n) > 1, the length is taken to be the number required.
distr: an object of class Multigam.
type: character, case ignored. The estimator type (mle, me, or same).
...: extra arguments.
par0, method, lower, upper: arguments passed to optim for the mle optimization. See Details.
na.rm: logical. Should the NA values be removed?

Details

The probability density function (PDF) of the multivariate gamma distribution is given by: $f (x; α, β) = \frac{β^{- α_{0}}}{\prod_{i = 1}^{k} Γ (α_{i})}, e^{- x_{k} / β} x_{1}^{α_{1} - 1} \prod_{i = 1}^{k} (x_{i} - x_{i - 1})^{(α_{i} - 1)} x > 0.$

The MLE of the multigamma distribution parameters is not available in closed form and has to be approximated numerically. This is done with optim(). Specifically, instead of solving a $(k + 1)$ optimization problem w.r.t $α, β$ , the optimization can be performed on the shape parameter sum $α_{0} := \sum_{i = 1}^{k} α \in (0, + \infty)^{k}$ . The default method used is the L-BFGS-B method with lower bound 1e-5 and upper bound Inf. The par0 argument can either be a numeric (satisfying lower <= par0 <= upper) or a character specifying the closed-form estimator to be used as initialization for the algorithm ("me" or "same" - the default value).

References

Mathal, A. M., & Moschopoulos, P. G. (1992). A form of multivariate gamma distribution. Annals of the Institute of Statistical Mathematics, 44, 97-106.
Oikonomidis, I. & Trevezas, S. (2025), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

Examples

Run this code

# -----------------------------------------------------
# Multivariate Gamma Distribution Example
# -----------------------------------------------------

# Create the distribution
a <- c(0.5, 3, 5) ; b <- 5
D <- Multigam(a, b)

# ------------------
# dpqr Functions
# ------------------

d(D, c(0.3, 2, 10)) # density function

# alternative way to use the function
df <- d(D) ; df(c(0.3, 2, 10)) # df is a function itself

x <- r(D, 100) # random generator function

# ------------------
# Moments
# ------------------

mean(D) # Expectation
var(D) # Variance
finf(D) # Fisher Information Matrix

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llmultigam(x, a, b)

emultigam(x, type = "mle")
emultigam(x, type = "me")
emultigam(x, type = "same")

mle(D, x)
me(D, x)
same(D, x)
e(D, x, type = "mle")

mle("multigam", x) # the distr argument can be a character

# ------------------
# Estimator Variance
# ------------------

vmultigam(a, b, type = "mle")
vmultigam(a, b, type = "me")
vmultigam(a, b, type = "same")

avar_mle(D)
avar_me(D)
avar_same(D)

v(D, type = "mle")

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