# ShiftedLoglogistic: Shifted Log-Logistic Distribution Class In distr6: The Complete R6 Probability Distributions Interface

 ShiftedLoglogistic R Documentation

## Shifted Log-Logistic Distribution Class

### Description

Mathematical and statistical functions for the Shifted Log-Logistic distribution, which is commonly used in survival analysis for its non-monotonic hazard as well as in economics, a generalised variant of Loglogistic.

### Details

The Shifted Log-Logistic distribution parameterised with shape, β, scale, α, and location, γ, is defined by the pdf,

f(x) = (β/α)((x-γ)/α)^{β-1}(1 + ((x-γ)/α)^β)^{-2}

for α, β > 0 and γ >= 0.

### Value

Returns an R6 object inheriting from class SDistribution.

### Distribution support

The distribution is supported on the non-negative Reals.

### Default Parameterisation

ShiftLLogis(scale = 1, shape = 1, location = 0)

N/A

N/A

### Super classes

`distr6::Distribution` -> `distr6::SDistribution` -> `ShiftedLoglogistic`

### Public fields

`name`

Full name of distribution.

`short_name`

Short name of distribution for printing.

`description`

Brief description 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

#### Public methods

Inherited methods

#### Method `new()`

Creates a new instance of this R6 class.

##### Usage
```ShiftedLoglogistic\$new(
scale = NULL,
shape = NULL,
location = NULL,
rate = NULL,
decorators = NULL
)```
##### Arguments
`scale`

`numeric(1))`
Scale parameter of the distribution, defined on the positive Reals. `scale = 1/rate`. If provided `rate` is ignored.

`shape`

`(numeric(1))`
Shape parameter, defined on the positive Reals.

`location`

`(numeric(1))`
Location parameter, defined on the Reals.

`rate`

`(numeric(1))`
Rate parameter of the distribution, defined on the positive Reals.

`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) = ∑ p_X(x)*x

with an integration analogue for continuous distributions.

##### Usage
`ShiftedLoglogistic\$mean(...)`
`...`

Unused.

#### Method `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).

##### Usage
`ShiftedLoglogistic\$mode(which = "all")`
##### Arguments
`which`

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

#### Method `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)`.

##### Usage
`ShiftedLoglogistic\$median()`

#### 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.

##### Usage
`ShiftedLoglogistic\$variance(...)`
`...`

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.

##### Usage
`ShiftedLoglogistic\$pgf(z, ...)`
##### 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
`ShiftedLoglogistic\$clone(deep = FALSE)`
##### Arguments
`deep`

Whether to make a deep clone.

### References

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

Other continuous distributions: `Arcsine`, `BetaNoncentral`, `Beta`, `Cauchy`, `ChiSquaredNoncentral`, `ChiSquared`, `Dirichlet`, `Erlang`, `Exponential`, `FDistributionNoncentral`, `FDistribution`, `Frechet`, `Gamma`, `Gompertz`, `Gumbel`, `InverseGamma`, `Laplace`, `Logistic`, `Loglogistic`, `Lognormal`, `MultivariateNormal`, `Normal`, `Pareto`, `Poisson`, `Rayleigh`, `StudentTNoncentral`, `StudentT`, `Triangular`, `Uniform`, `Wald`, `Weibull`
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`, `Matdist`, `NegativeBinomial`, `Normal`, `Pareto`, `Poisson`, `Rayleigh`, `StudentTNoncentral`, `StudentT`, `Triangular`, `Uniform`, `Wald`, `Weibull`, `WeightedDiscrete`