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

Returns an R6 object inheriting from class SDistribution.

The Matdist distribution is defined by the pmf, $$f(x_{ij}) = p_{ij}$$ for \(p_{ij}, i = 1,\ldots,k, j = 1,\ldots,n; \sum_i p_{ij} = 1\).

This is a special case distribution in distr6 which is technically a vectorised distribution
but is treated as if it is not. Therefore we only allow evaluation of all functions at
the same value, e.g. `$pdf(1:2)`

evaluates all samples at '1' and '2'.

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 distribution is supported on \(x_{11},...,x_{kn}\).

Matdist(matrix(0.5, 2, 2, dimnames = list(NULL, 1:2)))

N/A

N/A

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

Other discrete distributions:
`Bernoulli`

,
`Binomial`

,
`Categorical`

,
`Degenerate`

,
`DiscreteUniform`

,
`EmpiricalMV`

,
`Empirical`

,
`Geometric`

,
`Hypergeometric`

,
`Logarithmic`

,
`Multinomial`

,
`NegativeBinomial`

,
`WeightedDiscrete`

Other univariate distributions:
`Arcsine`

,
`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`

,
`NegativeBinomial`

,
`Normal`

,
`Pareto`

,
`Poisson`

,
`Rayleigh`

,
`ShiftedLoglogistic`

,
`StudentTNoncentral`

,
`StudentT`

,
`Triangular`

,
`Uniform`

,
`Wald`

,
`Weibull`

,
`WeightedDiscrete`

`distr6::Distribution`

-> `distr6::SDistribution`

-> `Matdist`

`name`

Full name of distribution.

`short_name`

Short name of distribution for printing.

`description`

Brief description of the distribution.

`properties`

Returns distribution properties, including skewness type and symmetry.

`new()`

Creates a new instance of this R6 class.

`Matdist$new(pdf = NULL, cdf = NULL, decorators = NULL)`

`pdf`

`numeric()`

Probability mass function for corresponding samples, should be same length`x`

. If`cdf`

is not given then calculated as`cumsum(pdf)`

.`cdf`

`numeric()`

Cumulative distribution function for corresponding samples, should be same length`x`

. If given then`pdf`

calculated as difference of`cdf`

s.`decorators`

`(character())`

Decorators to add to the distribution during construction.`x`

`numeric()`

Data samples,*must be ordered in ascending order*.

`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).

`median()`

Returns the median of the distribution. If an analytical expression is available
returns distribution median, otherwise if symmetric returns `self$mean`

, otherwise
returns `self$quantile(0.5)`

.

`mode()`

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

`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).

`skewness()`

The skewness of a distribution is defined by the third standardised moment, $$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$$ 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. If distribution is improper (F(Inf) != 1, then sk_X(x) = Inf).

`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).

`entropy()`

The entropy of a (discrete) distribution is defined by $$- \sum (f_X)log(f_X)$$ where \(f_X\) is the pdf of distribution X, with an integration analogue for continuous distributions. If distribution is improper then entropy is Inf.

`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).

`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. If distribution is improper (F(Inf) != 1, then cf_X(x) = Inf).

`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).

```
x <- Matdist$new(pdf = matrix(0.5, 2, 2, dimnames = list(NULL, 1:2)))
Matdist$new(cdf = matrix(c(0.5, 1), 2, 2, TRUE, dimnames = list(NULL, c(1, 2)))) # equivalently
#> Matdist()
# d/p/q/r
x$pdf(1:5)
#> [,1] [,2]
#> 1 0.5 0.5
#> 2 0.5 0.5
#> 3 0.0 0.0
#> 4 0.0 0.0
#> 5 0.0 0.0
x$cdf(1:5) # Assumes ordered in construction
#> [,1] [,2]
#> 1 0.5 0.5
#> 2 1.0 1.0
#> 3 1.0 1.0
#> 4 1.0 1.0
#> 5 1.0 1.0
x$quantile(0.42) # Assumes ordered in construction
#> [,1] [,2]
#> [1,] 1 1
x$rand(10)
#> [,1] [,2]
#> [1,] 2 2
#> [2,] 2 2
#> [3,] 1 1
#> [4,] 2 2
#> [5,] 2 2
#> [6,] 1 2
#> [7,] 2 2
#> [8,] 2 2
#> [9,] 1 2
#> [10,] 2 2
# Statistics
x$mean()
#> [1] 1.5 1.5
x$variance()
#> [1] 0.25 0.25
summary(x)
#> Matrix Probability Distribution.
#> Parameterised with:
#>
#> Id Support Value Tags
#> 1: cdf [0,1]^n required,linked
#> 2: pdf [0,1]^n 0.5,0.5,0.5,0.5 required,linked
#> 3: x ℤ immutable
#>
#>
#> Quick Statistics
#> Mean: 1.5, 1.5
#> Variance: 0.25, 0.25
#> Skewness: 00
#> Ex. Kurtosis: -2-2
#>
#> Support: {1, 2} Scientific Type: ℝ^n
#>
#> Traits: discrete; univariate
#> Properties: asymmetric; platykurtic platykurtic; no skew no skew
```