Mathematical and statistical functions for the WeightedDiscrete distribution, which is commonly used in empirical estimators such as Kaplan-Meier.

## Value

Returns an R6 object inheriting from class SDistribution.

## Details

The WeightedDiscrete distribution is defined by the pmf, $$f(x_i) = p_i$$ for $$p_i, i = 1,\ldots,k; \sum p_i = 1$$.

Sampling from this distribution is performed with the sample function with the elements given as the x values and the pdf as the probabilities. The cdf and quantile assume that the elements are supplied in an indexed order (otherwise the results are meaningless).

The number of points in the distribution cannot be changed after construction.

## Distribution support

The distribution is supported on $$x_1,...,x_k$$.

## Default Parameterisation

WeightDisc(x = 1, pdf = 1)

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, NegativeBinomial

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, NegativeBinomial, Normal, Pareto, Poisson, Rayleigh, ShiftedLoglogistic, StudentTNoncentral, StudentT, Triangular, Uniform, Wald, Weibull

## Super classes

distr6::Distribution -> distr6::SDistribution -> WeightedDiscrete

## 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.

## Active bindings

properties

Returns distribution properties, including skewness type and symmetry.

## Methods

Inherited methods

### Method new()

Creates a new instance of this R6 class.

#### Arguments

n

(integer(1))
Ignored.

### 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. If distribution is improper (F(Inf) != 1, then E_X(x) = Inf).

#### 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. If distribution is improper (F(Inf) != 1, then var_X(x) = Inf).

#### 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. If distribution is improper (F(Inf) != 1, then k_X(x) = Inf).

#### Arguments

base

(integer(1))
Base of the entropy logarithm, default = 2 (Shannon entropy)

...

Unused.

### Method mgf()

The moment generating function is defined by $$mgf_X(t) = E_X[exp(xt)]$$ where X is the distribution and $$E_X$$ is the expectation of the distribution X. If distribution is improper (F(Inf) != 1, then mgf_X(x) = Inf).

#### Arguments

t

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

...

Unused.

### Method pgf()

The probability generating function is defined by $$pgf_X(z) = E_X[exp(z^x)]$$ where X is the distribution and $$E_X$$ is the expectation of the distribution X. If distribution is improper (F(Inf) != 1, then pgf_X(x) = Inf).

#### Arguments

deep

Whether to make a deep clone.

## Examples

x <- WeightedDiscrete$new(x = 1:3, pdf = c(1 / 5, 3 / 5, 1 / 5)) WeightedDiscrete$new(x = 1:3, cdf = c(1 / 5, 4 / 5, 1)) # equivalently
#> WeightDisc(3)

# d/p/q/r
x$pdf(1:5) #> [1] 0.2 0.6 0.2 0.0 0.0 x$cdf(1:5) # Assumes ordered in construction
#> [1] 0.2 0.8 1.0 1.0 1.0
x$quantile(0.42) # Assumes ordered in construction #> [1] 2 x$rand(10)
#>  [1] 1 2 2 2 1 2 3 2 2 3

# Statistics
x$mean() #> [1] 2 x$variance()
#> [1] 0.4

summary(x)
#> Weighted Discrete Probability Distribution.
#> Parameterised with:
#>
#>        Id Support       Value            Tags
#>    <char>  <char>      <list>          <list>
#> 2:    pdf [0,1]^n 0.2,0.6,0.2 required,linked
#> 3:      x     ℝ^n       1,2,3 required,unique
#>
#>
#> Quick Statistics
#> 	Mean:		2
#> 	Variance:	0.4
#> 	Skewness:	0
#> 	Ex. Kurtosis:	-0.5
#>
#> Support: {1, 2, 3} 	Scientific Type: ℝ
#>
#> Traits:		discrete; univariate
#> Properties:	asymmetric; platykurtic; no skew