Student-t and Cauchy copulas #
For every positive real number ν of degrees of freedom, mix a centered Gaussian
vector by G^(-1/2), where the independent precision G has gamma shape and rate
ν/2. No covariance moments or integrality of ν are required. The matrix R
is a dispersion/correlation parameter; it is not asserted to be a covariance
matrix of the resulting Student law when its moments do not exist.
A gamma law bundled as a probability measure.
Equations
- ProbabilityTheory.gammaProbability a r ha hr = ⟨ProbabilityTheory.gammaMeasure a r, ⋯⟩
Instances For
noncomputable def
ProbabilityTheory.Copula.studentTLaw
{d : ℕ}
(R : Matrix (Fin d) (Fin d) ℝ)
(ν : ℝ)
(hν : 0 < ν)
:
The standard multivariate Student-t law used for the copula construction.
Equations
- ProbabilityTheory.Copula.studentTLaw R ν hν = ProbabilityTheory.Copula.gaussianScaleMixtureLaw R (ProbabilityTheory.gammaProbability (ν / 2) (ν / 2) ⋯ ⋯) fun (t : ℝ) => (√t)⁻¹
Instances For
noncomputable def
ProbabilityTheory.Copula.studentT
{d : ℕ}
(R : Matrix (Fin d) (Fin d) ℝ)
(hR : R.PosSemidef)
(hdiag : ∀ (i : Fin d), R i i = 1)
(ν : ℝ)
(hν : 0 < ν)
:
Copula d
Student-t copulas, for every positive real number of degrees of freedom.
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
ProbabilityTheory.Copula.isSklarCopula_studentT
{d : ℕ}
(R : Matrix (Fin d) (Fin d) ℝ)
(hR : R.PosSemidef)
(hdiag : ∀ (i : Fin d), R i i = 1)
(ν : ℝ)
(hν : 0 < ν)
:
IsSklarCopula (studentTLaw R ν hν) (studentT R hR hdiag ν hν)
noncomputable def
ProbabilityTheory.Copula.cauchy
{d : ℕ}
(R : Matrix (Fin d) (Fin d) ℝ)
(hR : R.PosSemidef)
(hdiag : ∀ (i : Fin d), R i i = 1)
:
Copula d
The Cauchy copula is the Student-t copula with one degree of freedom.
Equations
- ProbabilityTheory.Copula.cauchy R hR hdiag = ProbabilityTheory.Copula.studentT R hR hdiag 1 ProbabilityTheory.Copula.cauchy._proof_1