Pareto Distribution

stamox.distribution.ppareto(q: Union[Float, ArrayLike], scale: Union[Float, ArrayLike], alpha: Union[Float, ArrayLike], lower_tail: bool = True, log_prob: bool = False, dtype = <class 'jax.numpy.float64'>) -> ArrayLike

Computes the cumulative distribution function of the Pareto distribution.

Parameters:

Name	Type	Description	Default
q	Union[Float, ArrayLike]	The value at which to evaluate the CDF.	required
scale	Union[Float, ArrayLike]	The scale parameter of the Pareto distribution.	required
alpha	Union[Float, ArrayLike]	The shape parameter of the Pareto distribution.	required
lower_tail	bool	Whether to compute the lower tail of the CDF. Defaults to True.	True
log_prob	bool	Whether to return the log probability. Defaults to False.	False
dtype	jnp.dtype	The dtype of the output. Defaults to jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	The cumulative distribution function of the Pareto distribution evaluated at q.

Examples:

>>> ppareto(0.2, 0.1, 2.0)
Array(0.75, dtype=float32, weak_type=True)

---

stamox.distribution.qpareto(p: Union[Float, ArrayLike], scale: Union[Float, ArrayLike], alpha: Union[Float, ArrayLike], lower_tail: Bool = True, log_prob: Bool = False, dtype = <class 'jax.numpy.float64'>) -> ArrayLike

Computes the quantile function of the Pareto distribution.

Parameters:

Name	Type	Description	Default
p	Union[Float, ArrayLike]	Quantiles to compute.	required
scale	Union[Float, ArrayLike]	Scale parameter of the Pareto distribution.	required
alpha	Union[Float, ArrayLike]	Shape parameter of the Pareto distribution.	required
lower_tail	Bool	Whether to compute the lower tail probability. Defaults to True.	True
log_prob	Bool	Whether to compute the log probability. Defaults to False.	False
dtype	jnp.dtype	The dtype of the output. Defaults to jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	The quantiles of the Pareto distribution.

Examples:

>>> qpareto(0.2, 0.1, 2.0)
Array([0.1118034], dtype=float32, weak_type=True)

---

stamox.distribution.dpareto(x: Union[Float, ArrayLike], scale: Union[Float, ArrayLike], alpha: Union[Float, ArrayLike], lower_tail = True, log_prob = False, dtype = <class 'jax.numpy.float64'>) -> ArrayLike

Computes the density of the Pareto distribution.

Parameters:

Name	Type	Description	Default
x	Union[Float, ArrayLike]	The value at which to evaluate the density.	required
scale	Union[Float, ArrayLike]	The scale parameter of the Pareto distribution.	required
alpha	Union[Float, ArrayLike]	The shape parameter of the Pareto distribution.	required
lower_tail	bool	Whether to compute the lower tail probability. Defaults to True.	True
log_prob	bool	Whether to return the log probability. Defaults to False.	False
dtype	jnp.dtype	The dtype of the output. Defaults to jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	The density of the Pareto distribution evaluated at x.

Examples:

>>> dpareto(0.2, 0.1, 2.0)
Array([2.4999998], dtype=float32, weak_type=True)

---

stamox.distribution.rpareto(key: PRNGKeyArray, sample_shape: Optional[Sequence[int]] = None, scale: Union[Float, ArrayLike] = None, alpha: Union[Float, ArrayLike] = None, lower_tail: Bool = True, log_prob: Bool = False, dtype = <class 'jax.numpy.float64'>) -> ArrayLike

Generate random variable following a Pareto distribution.

Parameters:

Name	Type	Description	Default
key	KeyArray	A random number generator key.	required
sample_shape	Optional[Shape]	The shape of the samples to be drawn. Defaults to None.	None
scale	Union[Float, ArrayLike]	The scale parameter of the Pareto distribution.	None
alpha	Union[Float, ArrayLike]	The shape parameter of the Pareto distribution.	None
lower_tail	Bool	Whether to calculate the lower tail probability. Defaults to True.	True
log_prob	Bool	Whether to return the log probability. Defaults to False.	False
dtype	jnp.dtype	The dtype of the output. Defaults to jnp.float_.	<class 'jax.numpy.float64'>

Returns:

Type	Description
ArrayLike	random variable following a Pareto distribution.

Examples:

>>> rpareto(jax.random.PRNGKey(0), sample_shape=(2, 3), scale=0.1, alpha=2.0)
Array([[0.15330292, 0.10539087, 0.19686179],
        [0.30740616, 0.15743963, 0.13524036]], dtype=float32)