Negative Binomial Distribution

The Negative Binomial distribution is a discrete probability distribution that models the number of failures before a specified number of successes occurs in a sequence of independent Bernoulli trials. It is defined by parameters $r > 0$ (number of successes) and $0 < p \leq 1$ (probability of success).

Usage

Nbinom(size = 1, prob = 0.5)

# S4 method for class 'Nbinom,numeric'
d(distr, x, log = FALSE)

# S4 method for class 'Nbinom,numeric'
p(distr, q, lower.tail = TRUE, log.p = FALSE)

# S4 method for class 'Nbinom,numeric'
qn(distr, p, lower.tail = TRUE, log.p = FALSE)

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

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

# S4 method for class 'Nbinom'
median(x)

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

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

# S4 method for class 'Nbinom'
sd(x)

# S4 method for class 'Nbinom'
skew(x)

# S4 method for class 'Nbinom'
kurt(x)

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

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

llnbinom(x, size, prob)

# S4 method for class 'Nbinom,numeric'
ll(distr, x)

enbinom(x, size, type = "mle", ...)

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

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

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

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

# S4 method for class 'Nbinom'
avar_me(distr)

Arguments

size: number of trials (zero or more).
prob: numeric. Probability of success on each trial.
distr: an object of class Nbinom.
x: For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Nbinom. For the log-likelihood and the estimation functions, x is the sample of observations.
log, log.p: logical. Should the logarithm of the probability be returned?
q: numeric. Vector of quantiles.
lower.tail: logical. If TRUE (default), probabilities are $P(X \leq x)$, otherwise $P(X > x)$.
p: numeric. Vector of probabilities.
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 negative binomial distribution is: $$ P(X = k) = \binom{k + r - 1}{k} (1 - p)^k p^r, \quad k \in \mathbb{N}_0.$$

Examples

# -----------------------------------------------------
# Negative Binomial Distribution Example
# -----------------------------------------------------

# Create the distribution
N <- 10 ; p <- 0.4
D <- Nbinom(N, p)

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

d(D, 0:4) # density function
#> [1] 0.0001048576 0.0006291456 0.0020761805 0.0049828332 0.0097165246
p(D, 0:4) # distribution function
#> [1] 0.0001048576 0.0007340032 0.0028101837 0.0077930168 0.0175095415
qn(D, c(0.4, 0.8)) # inverse distribution function
#> [1] 13 20
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#>   [1] 0.0638953912 0.0644631755 0.0499115616 0.0329101076 0.0244856421
#>   [6] 0.0513778730 0.0681223171 0.0513778730 0.0604342270 0.0156364106
#>  [11] 0.0671491411 0.0554572907 0.0585707753 0.0585707753 0.0038031980
#>  [16] 0.0681223171 0.0681223171 0.0244856421 0.0681223171 0.0638953912
#>  [21] 0.0513778730 0.0499115616 0.0428148942 0.0191622678 0.0232269913
#>  [26] 0.0638953912 0.0670901608 0.0499115616 0.0604342270 0.0681223171
#>  [31] 0.0001048576 0.0638953912 0.0604342270 0.0428148942 0.0163237614
#>  [36] 0.0499115616 0.0604342270 0.0441323282 0.0513778730 0.0156364106
#>  [41] 0.0163237614 0.0062980310 0.0644631755 0.0513778730 0.0428148942
#>  [46] 0.0101035268 0.0191622678 0.0499115616 0.0681223171 0.0329101076
#>  [51] 0.0585707753 0.0638953912 0.0038031980 0.0029237085 0.0383951255
#>  [56] 0.0191622678 0.0499115616 0.0329101076 0.0329101076 0.0670901608
#>  [61] 0.0554572907 0.0670901608 0.0244856421 0.0383951255 0.0670901608
#>  [66] 0.0681223171 0.0644631755 0.0644631755 0.0671491411 0.0670901608
#>  [71] 0.0670901608 0.0585707753 0.0670901608 0.0681223171 0.0638953912
#>  [76] 0.0585707753 0.0644631755 0.0329101076 0.0671491411 0.0671491411
#>  [81] 0.0428148942 0.0604342270 0.0244856421 0.0585707753 0.0163237614
#>  [86] 0.0604342270 0.0671491411 0.0604342270 0.0585707753 0.0513778730
#>  [91] 0.0499115616 0.0062980310 0.0022326501 0.0671491411 0.0644631755
#>  [96] 0.0441323282 0.0671491411 0.0554572907 0.0513778730 0.0513778730

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

mean(D) # Expectation
#> [1] 15
median(D) # Median
#> [1] 14
mode(D) # Mode
#> [1] 13
var(D) # Variance
#> [1] 37.5
sd(D) # Standard Deviation
#> [1] 6.123724
skew(D) # Skewness
#> [1] 0.6531973
kurt(D) # Excess Kurtosis
#> [1] 0.6266667
entro(D) # Entropy
#> [1] 1.048722
finf(D) # Fisher Information Matrix
#>     prob 
#> 104.1667 

# List of all available moments
mom <- moments(D)
mom$mean # expectation
#> [1] 15

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

ll(D, x)
#> [1] -325.7134
llnbinom(x, N, p)
#> [1] -325.7134

enbinom(x, N, type = "mle")
#> $prob
#> [1] 0.4025765
#> 
enbinom(x, N, type = "me")
#> $prob
#> [1] 0.4025765
#> 

mle(D, x)
#> $prob
#> [1] 0.4025765
#> 
me(D, x)
#> $prob
#> [1] 0.4025765
#> 
e(D, x, type = "mle")
#> $prob
#> [1] 0.4025765
#> 

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

vnbinom(N, p, type = "mle")
#>   prob 
#> 0.0096 
vnbinom(N, p, type = "me")
#>   prob 
#> 0.0096 

avar_mle(D)
#>   prob 
#> 0.0096 
avar_me(D)
#>   prob 
#> 0.0096 

v(D, type = "mle")
#>   prob 
#> 0.0096