Multinomial Distribution

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 class 'Multinom,numeric'
d(distr, x, log = FALSE)

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

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

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

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

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

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

llmultinom(x, size, prob)

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

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

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

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

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

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

# S4 method for class '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?

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.

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 $.

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
#> [1] 0.03388291

# alternative way to use the function
df <- d(D) ; df(c(2, 3, 5)) # df is a function itself
#> [1] 0.03388291

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

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

mean(D) # Expectation
#> [1] 1 2 7
mode(D) # Mode
#> [1] 3
var(D) # Variance
#>      [,1] [,2] [,3]
#> [1,]  0.9 -0.2 -0.7
#> [2,] -0.2  1.6 -1.4
#> [3,] -0.7 -1.4  2.1
entro(D) # Entropy
#> [1] 2.885135
finf(D) # Fisher Information Matrix
#>           prob1     prob2
#> prob1  85.71429 -14.28571
#> prob2 -14.28571  35.71429

# List of all available moments
mom <- moments(D)
mom$mean # expectation
#> [1] 1 2 7

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

ll(D, x)
#> [1] -292.1644
llmultinom(x, N, p)
#> [1] -292.1644

emultinom(x, type = "mle")
#> $prob
#> [1] 0.099 0.189 0.712
#> 
emultinom(x, type = "me")
#> $prob
#> [1] 0.099 0.189 0.712
#> 

mle(D, x)
#> $prob
#> [1] 0.099 0.189 0.712
#> 
me(D, x)
#> $prob
#> [1] 0.099 0.189 0.712
#> 
e(D, x, type = "mle")
#> $prob
#> [1] 0.099 0.189 0.712
#> 

mle("multinom", x) # the distr argument can be a character
#> $prob
#> [1] 0.099 0.189 0.712
#> 

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

vmultinom(N, p, type = "mle")
#>        prob1 prob2
#> prob1 0.0125 0.005
#> prob2 0.0050 0.030
vmultinom(N, p, type = "me")
#>        prob1 prob2
#> prob1 0.0125 0.005
#> prob2 0.0050 0.030

avar_mle(D)
#>        prob1 prob2
#> prob1 0.0125 0.005
#> prob2 0.0050 0.030
avar_me(D)
#>        prob1 prob2
#> prob1 0.0125 0.005
#> prob2 0.0050 0.030

v(D, type = "mle")
#>        prob1 prob2
#> prob1 0.0125 0.005
#> prob2 0.0050 0.030

Usage

Arguments

Value

Details

See also

Examples