distr6 (version 1.3.6)

NegativeBinomial: Negative Binomial Distribution Class

Description

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.

Constructor

NegativeBinomial$new(size = 10, prob = 0.5, qprob = NULL, mean = NULL, form = "fbs", decorators = NULL, verbose = FALSE)

Constructor Arguments

Argument	Type	Details
size	numeric	number of failures/successes.
prob	numeric	probability of success.
qprob	numeric	probability of failure.
mean	numeric	location parameter.
form	character	form of negative binomial, see details.

decorators Decorator decorators to add functionality. See details.

Constructor Details

The Negative Binomial distribution is parameterised with size as a positive whole number, and either prob or qprob as a number between 0 and 1, or mean as a numeric greater than the number of failures/successes (if form is 'tbf' or 'tbs'). These are related via, $$qprob = 1 - prob$$ and the mean formula is dependent on the form. If mean is given then qprob and prob are ignored. If qprob is given then prob is ignored. The additional form argument determines which of the four Negative Binomial distributions should be constructed, this cannot be updated after construction. form should be one of "sbf" (successes before failures), "tbf" (trials before failures), "fbs" (failures before successes) or "tbs" (trials before successes). "fbs" is taken as default if none are supplied or an unrecognised form is given.

Public Variables

Variable	Return
name	Name of distribution.
short_name	Id of distribution.
description	Brief description of distribution.

Public Methods

Accessor Methods	Link
decorators	decorators
traits	traits
valueSupport	valueSupport
variateForm	variateForm
type	type
properties	properties
support	support
symmetry	symmetry
sup	sup
inf	inf
dmax	dmax
dmin	dmin
skewnessType	skewnessType
kurtosisType	kurtosisType

Statistical Methods Link pdf(x1, ..., log = FALSE, simplify = TRUE) pdf cdf(x1, ..., lower.tail = TRUE, log.p = FALSE, simplify = TRUE) cdf quantile(p, ..., lower.tail = TRUE, log.p = FALSE, simplify = TRUE) quantile.Distribution rand(n, simplify = TRUE) rand mean() mean.Distribution variance() variance stdev() stdev prec() prec cor() cor skewness() skewness kurtosis(excess = TRUE) kurtosis entropy(base = 2) entropy mgf(t) mgf cf(t) cf pgf(z) pgf median() median.Distribution iqr() iqr mode(which = "all") mode

Parameter Methods Link parameters(id) parameters getParameterValue(id, error = "warn") getParameterValue setParameterValue(..., lst = NULL, error = "warn") setParameterValue

Validation Methods Link liesInSupport(x, all = TRUE, bound = FALSE) liesInSupport liesInType(x, all = TRUE, bound = FALSE) liesInType

Representation Methods Link strprint(n = 2) strprint print(n = 2) print summary(full = T) summary.Distribution

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 \(p \epsilon [0,1]\), where \(C(a,b)\) is the combination (or binomial coefficient) function.

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).

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, prob is always the probability of success and qprob is the probability of failure. Use $description to see the Negative Binomial form.

References

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

See Also

listDistributions for all available distributions. Binomial for the Binomial distribution and Geometric for the Geometric distribution.

Examples

# NOT RUN {
# Different parameterisations
NegativeBinomial$new(size = 5, prob = 0.2)
NegativeBinomial$new(size = 5, qprob = 0.2)
NegativeBinomial$new(size = 5, mean = 4)

# Different forms of the distribution
NegativeBinomial$new(form = "fbs")
NegativeBinomial$new(form = "sbf")

# Use description to see which form is used
NegativeBinomial$new(form = "tbf")
NegativeBinomial$new(form = "tbs")

x <- NegativeBinomial$new() # Default is size = 10, prob = 0.5 and failures before successes

# Update parameters (form cannot be updated)
x$setParameterValue(qprob = 0.2)  # When any parameter is updated, all others are too!
x$parameters()


# d/p/q/r
x$pdf(5)
x$cdf(5)
x$quantile(0.42)
x$rand(4)

# Statistics
x$mean()
x$variance()

summary(x)
# }