## distributions.Loglogistic


Loglogistic (Fisk) distribution with shape [alpha](models.ElasticNet.md#prospicio.models.ElasticNet.alpha) and scale


Usage


``` python
distributions.Loglogistic()
```


[theta](reserving.ClarkFit.md#prospicio.reserving.ClarkFit.theta) (the median): `F(x) = (x/theta)**alpha / (1 + (x/theta)**alpha)`, as SciPy's `fisk`. Its [cdf](risk.Gpd.md#prospicio.risk.Gpd.cdf) is Clark's loglogistic growth curve. The mean is infinite for `alpha <= 1` and the variance for `alpha <= 2`; limited and layer moments always exist.


## Parameters


`shape: float`  

`scale: float`  


## Examples

``` python
>>> from prospicio.distributions import Loglogistic
>>> Loglogistic(1.0, 2.0).cdf(3.0)
```

0.6


## Attributes

| Name | Description |
|----|----|
| [scale](#scale) | Scale [theta](reserving.ClarkFit.md#prospicio.reserving.ClarkFit.theta), the median. |
| [shape](#shape) | Shape [alpha](models.ElasticNet.md#prospicio.models.ElasticNet.alpha). |

------------------------------------------------------------------------


#### scale


Scale [theta](reserving.ClarkFit.md#prospicio.reserving.ClarkFit.theta), the median.


`scale: float`


------------------------------------------------------------------------


#### shape


Shape [alpha](models.ElasticNet.md#prospicio.models.ElasticNet.alpha).


`shape: float`


## Methods

| Name | Description |
|----|----|
| [cdf()](#cdf) | Distribution function `P(X <= x)`. |
| [layer()](#layer) | Expected loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs [attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment). |
| [layer_second_moment()](#layer_second_moment) | Second moment of the loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs |
| [layer_variance()](#layer_variance) | Variance of the loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs [attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment). |
| [lev()](#lev) | Limited expected value `E[min(X, limit)]`. |
| [mean()](#mean) | Mean of the distribution (`inf` if it does not exist). |
| [quantile()](#quantile) | Quantile: the smallest [x](pricing.TabulatedCurve.md#prospicio.pricing.TabulatedCurve.x) with `P(X <= x) >= p`. |
| [sample()](#sample) | [n](distributions.Binomial.md#prospicio.distributions.Binomial.n) draws from stream `stream` of the generator keyed by |
| [std()](#std) | Standard deviation of the distribution. |
| [stop_loss()](#stop_loss) | Expected excess over a retention, `E[max(X - retention, 0)]`. |
| [survival()](#survival) | Survival function `P(X > x)`, accurate far into the tail. |
| [variance()](#variance) | Variance of the distribution (`inf` if it does not exist). |

------------------------------------------------------------------------


#### cdf()


Distribution function `P(X <= x)`.


Usage


``` python
cdf(x)
```


##### Parameters


`x: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### layer()


Expected loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs [attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment).


Usage


``` python
layer(limit, attachment)
```


##### Parameters


`limit: float`  
`inf` for an unlimited layer.

`attachment: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### layer_second_moment()


Second moment of the loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs


Usage


``` python
layer_second_moment(limit, attachment)
```


[attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment).


##### Parameters


`limit: float`  

`attachment: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### layer_variance()


Variance of the loss to the layer [limit](reinsurance.Layer.md#prospicio.reinsurance.Layer.limit) xs [attachment](reinsurance.Layer.md#prospicio.reinsurance.Layer.attachment).


Usage


``` python
layer_variance(limit, attachment)
```


##### Parameters


`limit: float`  

`attachment: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### lev()


Limited expected value `E[min(X, limit)]`.


Usage


``` python
lev(limit)
```


##### Parameters


`limit: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### mean()


Mean of the distribution (`inf` if it does not exist).


Usage


``` python
mean()
```


##### Returns


`float`  


------------------------------------------------------------------------


#### quantile()


Quantile: the smallest [x](pricing.TabulatedCurve.md#prospicio.pricing.TabulatedCurve.x) with `P(X <= x) >= p`.


Usage


``` python
quantile(p)
```


##### Parameters


`p: float`  
Probability in `[0, 1]`.


##### Returns


`float`  


##### Raises


`ValueError`  
If [p](models.Elpd.md#prospicio.models.Elpd.p) is outside `[0, 1]`.


------------------------------------------------------------------------


#### sample()


[n](distributions.Binomial.md#prospicio.distributions.Binomial.n) draws from stream `stream` of the generator keyed by


Usage


``` python
sample(n, seed, stream=0)
```


[seed](aggregate.EventSet.md#prospicio.aggregate.EventSet.seed).


##### Parameters


`n: int`  

`seed: int`  

`stream: int = ``0`  


##### Returns


`list of float`  


------------------------------------------------------------------------


#### std()


Standard deviation of the distribution.


Usage


``` python
std()
```


##### Returns


`float`  


------------------------------------------------------------------------


#### stop_loss()


Expected excess over a retention, `E[max(X - retention, 0)]`.


Usage


``` python
stop_loss(retention)
```


##### Parameters


`retention: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### survival()


Survival function `P(X > x)`, accurate far into the tail.


Usage


``` python
survival(x)
```


##### Parameters


`x: float`  


##### Returns


`float`  


------------------------------------------------------------------------


#### variance()


Variance of the distribution (`inf` if it does not exist).


Usage


``` python
variance()
```


##### Returns


`float`