Multinom: Multinomial Distribution

In joker: Probability Distributions and Parameter Estimation

Multinomial Distribution

Description

The multinomial distribution is a discrete probability distribution which models the probability of having x successes in n independent categorical trials with success probability vector p.

Usage

Multinom(size = 1, prob = c(0.5, 0.5))

## S4 method for signature 'Multinom,numeric'
d(distr, x, log = FALSE)

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

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

## S4 method for signature 'Multinom'
mode(x)

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

## S4 method for signature 'Multinom'
entro(x)

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

llmultinom(x, size, prob)

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

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

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

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

vmultinom(size, prob, type = "mle")

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

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

Arguments

size	number of trials (zero or more).
prob	numeric. Probability of success on each trial.
distr	an object of class Multinom.
x	For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Multinom. 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.
type	character, case ignored. The estimator type (mle or me).
...	extra arguments.
na.rm	logical. Should the NA values be removed?

Details

The probability mass function (PMF) of the Multinomial distribution is:

P(X_1 = x_1, ..., X_k = x_k) = \frac{n!}{x_1! x_2! ... x_k!} \prod_{i=1}^k p_i^{x_i},

subject to \sum_{i=1}^{k} x_i = n .

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.

See Also

Functions from the stats package: dmultinom(), rmultinom()

Examples

# -----------------------------------------------------
# Multinomial Distribution Example
# -----------------------------------------------------

# Create the distribution
N <- 10 ; p <- c(0.1, 0.2, 0.7)
D <- Multinom(N, p)

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

d(D, c(2, 3, 5)) # density function

# alternative way to use the function
df <- d(D) ; df(c(2, 3, 5)) # df is a function itself

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

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

mean(D) # Expectation
mode(D) # Mode
var(D) # Variance
entro(D) # Entropy
finf(D) # Fisher Information Matrix

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

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

ll(D, x)
llmultinom(x, N, p)

emultinom(x, type = "mle")
emultinom(x, type = "me")

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

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

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

vmultinom(N, p, type = "mle")
vmultinom(N, p, type = "me")

avar_mle(D)
avar_me(D)

v(D, type = "mle")

joker documentation built on June 8, 2025, 12:12 p.m.