Multigam {joker}R Documentation

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 signature 'Multigam,numeric'
d(distr, x, log = FALSE)

## S4 method for signature 'Multigam,matrix'
d(distr, x, log = FALSE)

## S4 method for signature 'Multigam,numeric'
r(distr, n)

## S4 method for signature 'Multigam'
mean(x)

## S4 method for signature 'Multigam'
var(x)

## S4 method for signature 'Multigam'
finf(x)

llmultigam(x, shape, scale)

## S4 method for signature 'Multigam,matrix'
ll(distr, x)

emultigam(x, type = "mle", ...)

## S4 method for signature 'Multigam,matrix'
mle(
  distr,
  x,
  par0 = "same",
  method = "L-BFGS-B",
  lower = 1e-05,
  upper = Inf,
  na.rm = FALSE
)

## S4 method for signature 'Multigam,matrix'
me(distr, x, na.rm = FALSE)

## S4 method for signature 'Multigam,matrix'
same(distr, x, na.rm = FALSE)

vmultigam(shape, scale, type = "mle")

## S4 method for signature 'Multigam'
avar_mle(distr)

## S4 method for signature 'Multigam'
avar_me(distr)

## S4 method for signature 'Multigam'
avar_same(distr)

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).

Value

Each type of function returns a different type of object:

References

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")

[Package joker version 0.14.2 Index]