joker (version 0.14.2)

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 \(\alpha_i > 0, i\in\{1, \dots, k\}\) and the same scale \(\beta > 0\).

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: $$ f(x; \alpha, \beta) = \frac{\beta^{-\alpha_0}}{\prod_{i=1}^k\Gamma(\alpha_i)}, e^{-x_k/\beta} x_1^{\alpha_1-1}\prod_{i=1}^k (x_i - x_{i-1})^{(\alpha_i-1)} \quad x > 0. $$

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 \(\alpha, \beta\), the optimization can be performed on the shape parameter sum \(\alpha_0:=\sum_{i=1}^k\alpha \in(0,+\infty)^k\). 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

# -----------------------------------------------------
# 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")