Mathematical and statistical functions for the Negative Binomial distribution, which is commonly used to model the number of successes, trials or failures before a given number of failures or successes.

## Value

Returns an R6 object inheriting from class SDistribution.

## Details

The Negative Binomial distribution parameterised with number of failures before successes, $$n$$, and probability of success, $$p$$, is defined by the pmf, $$f(x) = C(x + n - 1, n - 1) p^n (1 - p)^x$$ for $$n = {0,1,2,\ldots}$$ and probability $$p$$, where $$C(a,b)$$ is the combination (or binomial coefficient) function.

The Negative Binomial distribution can refer to one of four distributions (forms):

1. The number of failures before K successes (fbs)

2. The number of successes before K failures (sbf)

3. The number of trials before K failures (tbf)

4. The number of trials before K successes (tbs)

For each we refer to the number of K successes/failures as the size parameter.

## Distribution support

The distribution is supported on $${0,1,2,\ldots}$$ (for fbs and sbf) or $${n,n+1,n+2,\ldots}$$ (for tbf and tbs) (see below).

## Default Parameterisation

NBinom(size = 10, prob = 0.5, form = "fbs")

N/A

N/A

## References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.

Other discrete distributions: Arrdist, Bernoulli, Binomial, Categorical, Degenerate, DiscreteUniform, EmpiricalMV, Empirical, Geometric, Hypergeometric, Logarithmic, Matdist, Multinomial, WeightedDiscrete

Other univariate distributions: Arcsine, Arrdist, Bernoulli, BetaNoncentral, Beta, Binomial, Categorical, Cauchy, ChiSquaredNoncentral, ChiSquared, Degenerate, DiscreteUniform, Empirical, Erlang, Exponential, FDistributionNoncentral, FDistribution, Frechet, Gamma, Geometric, Gompertz, Gumbel, Hypergeometric, InverseGamma, Laplace, Logarithmic, Logistic, Loglogistic, Lognormal, Matdist, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull, WeightedDiscrete

## Super classes

distr6::Distribution -> distr6::SDistribution -> NegativeBinomial

## Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

alias

Alias of the distribution.

packages

Packages required to be installed in order to construct the distribution.

## Active bindings

properties

Returns distribution properties, including skewness type and symmetry.

## Methods

Inherited methods

### Method new()

Creates a new instance of this R6 class.

NegativeBinomial$new( size = NULL, prob = NULL, qprob = NULL, mean = NULL, form = NULL, decorators = NULL ) #### Arguments size (integer(1)) Number of trials/successes. prob (numeric(1)) Probability of success. qprob (numeric(1)) Probability of failure. If provided then prob is ignored. qprob = 1 - prob. mean (numeric(1)) Mean of distribution, alternative to prob and qprob. form character(1)) Form of the distribution, cannot be changed after construction. Options are to model the number of, • "fbs" - Failures before successes. • "sbf" - Successes before failures. • "tbf" - Trials before failures. • "tbs" - Trials before successes. Use $description to see the Negative Binomial form.

decorators

(character())
Decorators to add to the distribution during construction.

### Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation $$E_X(X) = \sum p_X(x)*x$$ with an integration analogue for continuous distributions.

#### Arguments

which

(character(1) | numeric(1)
Ignored if distribution is unimodal. Otherwise "all" returns all modes, otherwise specifies which mode to return.

### Method variance()

The variance of a distribution is defined by the formula $$var_X = E[X^2] - E[X]^2$$ where $$E_X$$ is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

#### Arguments

...

Unused.

### Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment, $$k_X = E_X[\frac{x - \mu}{\sigma}^4]$$ where $$E_X$$ is the expectation of distribution X, $$\mu$$ is the mean of the distribution and $$\sigma$$ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

#### Arguments

t

(integer(1))
t integer to evaluate function at.

...

Unused.

### Method cf()

The characteristic function is defined by $$cf_X(t) = E_X[exp(xti)]$$ where X is the distribution and $$E_X$$ is the expectation of the distribution X.

#### Arguments

z

(integer(1))
z integer to evaluate probability generating function at.

...

Unused.

### Method clone()

The objects of this class are cloneable with this method.

#### Usage

NegativeBinomial\$clone(deep = FALSE)

#### Arguments

deep

Whether to make a deep clone.