## distributions.Custom A loss severity defined by your own distribution function: the slow Usage ``` python distributions.Custom() ``` path for a distribution the library does not have. Give the cdf, and the quantile function if you have one (sampling inverts the cdf by bisection otherwise, about a hundred cdf calls per draw). The mean, variance, limited expected values and layer moments are computed by Gauss-Legendre quadrature of the survival function between the distribution's own quantiles, ignoring the probability above the `1 - 1e-12` quantile. A [Custom](distributions.Custom.md#prospicio.distributions.Custom) goes anywhere a severity does (layers, compound distributions, simulated events, copula marginals, mixtures); calculations that meet one run single-threaded, since every value calls back into Python. ## Parameters `cdf: callable` `cdf(x) -> float`: `P(X <= x)` for `x >= 0`, in `[0, 1]` and non-decreasing. Losses are non-negative. `quantile: callable` `quantile(p) -> float`: the smallest [x](pricing.TabulatedCurve.md#prospicio.pricing.TabulatedCurve.x) with `cdf(x) >= p`. `name: str = ``"custom"` Shown in errors and `repr`. ## Raises `ValueError` If a callable raises or returns a value out of range, or the cdf never reaches `1 - 1e-12` at a finite loss. An error in a later call makes that value `nan`; [last_error](distributions.Custom.md#prospicio.distributions.Custom.last_error) says why. ## Examples ``` python >>> import math >>> from prospicio.distributions import Custom >>> d = Custom(lambda x: 1 - math.exp(-x / 100), name="exponential") >>> round(d.mean(), 6) ``` 100.0 ``` python >>> round(d.lev(50), 6) == round(100 * (1 - math.exp(-0.5)), 6) ``` True ## Attributes | Name | Description | |----|----| | [has_quantile](#has_quantile) | Whether a quantile function was given. | | [last_error](#last_error) | The first error raised by a callable since construction, or `None`. | | [name](#name) | The name given at construction. | | [upper](#upper) | The `1 - 1e-12` quantile, where the moment integrals stop. | ------------------------------------------------------------------------ #### has_quantile Whether a quantile function was given. `has_quantile: bool` ------------------------------------------------------------------------ #### last_error The first error raised by a callable since construction, or `None`. `last_error: str | None` ------------------------------------------------------------------------ #### name The name given at construction. `name: str` ------------------------------------------------------------------------ #### upper The `1 - 1e-12` quantile, where the moment integrals stop. `upper: 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`