aub_htp.pdf.zolotarev#
Functions
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Helper used in some parameterizations. |
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Build V(θ) kernel for Zolotarev integral forms. - α ≠ 1: use Type-B (Zolotarev) representation components. V(θ) = [cos(αθ0)]^{1/(α-1)} * [cos(αθ0+(α-1)θ)/cos θ] * [cos θ / sin(α(θ+θ0))]^{α/(α-1)} - α = 1: use the α=1 kernel with exp term. Returns a function V(θ). |
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Small-x series for g(x; α, θ) used in near-zero expansions. |
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CF-integral pdf in Nolan S0 parameterization around zero. |
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Zolotarev one-point pdf evaluator via 1D quadrature. - α ≠ 1: uses Type-B integral on θ ∈ [-θ0, π/2] j is the scale Jacobian; integrand uses V(θ) * exp(-x^{α/(α-1)} V(θ)). - α = 1, β ≠ 0: uses α=1 kernel on θ ∈ [-π/2, π/2] with exponential tilt exp(-π x V(θ) / (2β)). Returns (value, estimated error) from quad multiplied by Jacobian. |
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Direct CF-based integral for pdf near zero using quad_vec. |
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Batch wrapper over generate_pdf_one_point using Zolotarev integral. |
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Enforce x ≥ 0 by symmetry. |
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Simple upper bound estimate for the neglected tail of the g-series. R_N(x) ≤ |x|^N / N! * Γ((N+1)/α) / (απ). |
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Zolotarev θ0 pivot for stable laws. |
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Convert (α, β) to Nolan's S0 'theta' style (often called θ_C here). |
- aub_htp.pdf.zolotarev.calculate_K(alpha)#
Helper used in some parameterizations. K(α) = α - 1 + sign(1 - α)
- aub_htp.pdf.zolotarev.calculate_V(alpha, beta, theta0)#
Build V(θ) kernel for Zolotarev integral forms. - α ≠ 1: use Type-B (Zolotarev) representation components.
- V(θ) = [cos(αθ0)]^{1/(α-1)} * [cos(αθ0+(α-1)θ)/cos θ] *
[cos θ / sin(α(θ+θ0))]^{α/(α-1)}
α = 1: use the α=1 kernel with exp term.
Returns a function V(θ).
- aub_htp.pdf.zolotarev.g_approx_small_x(x, alpha, theta, N=5)#
Small-x series for g(x; α, θ) used in near-zero expansions. g(x) ≈ (1/(απ)) Σ_{n=0}^{N-1} x^n/n! * Γ((n+1)/α) * sin[ (π/2)(n+1)(1-θ) ] - Vectorized over x. - Prints (n+1)/α for quick debugging/inspection.
- aub_htp.pdf.zolotarev.generate_pdf_NolanS0(X, alpha, beta)#
CF-integral pdf in Nolan S0 parameterization around zero. - Uses generate_pdf_one_point_around_zero for vector X.
- aub_htp.pdf.zolotarev.generate_pdf_one_point(x, alpha, beta)#
Zolotarev one-point pdf evaluator via 1D quadrature. - α ≠ 1: uses Type-B integral on θ ∈ [-θ0, π/2]
j is the scale Jacobian; integrand uses V(θ) * exp(-x^{α/(α-1)} V(θ)).
α = 1, β ≠ 0: uses α=1 kernel on θ ∈ [-π/2, π/2] with exponential tilt exp(-π x V(θ) / (2β)).
Returns (value, estimated error) from quad multiplied by Jacobian.
- aub_htp.pdf.zolotarev.generate_pdf_one_point_around_zero(X, alpha, beta)#
Direct CF-based integral for pdf near zero using quad_vec. f(x) = (1/π) ∫_0^∞ e^{-t^α} cos(x t − β tan(πα/2) t^α) dt (S0-style) - Vectorized over X using quad_vec. - High accuracy tolerances for stability around x≈0.
- aub_htp.pdf.zolotarev.generate_pdf_zolotarev_1(X, alpha, beta)#
Batch wrapper over generate_pdf_one_point using Zolotarev integral. - Evaluates at each x in X and collects only the value (drop error).
- aub_htp.pdf.zolotarev.reflect_if_negative(x, beta, theta0)#
Enforce x ≥ 0 by symmetry. - If x < 0, reflect: x -> -x, β -> -β, θ0 -> -θ0, and mark flipped=True. - Else return inputs unchanged, flipped=False.
- aub_htp.pdf.zolotarev.remainder_bound(x, alpha, theta, N=10)#
Simple upper bound estimate for the neglected tail of the g-series. R_N(x) ≤ |x|^N / N! * Γ((N+1)/α) / (απ)
- aub_htp.pdf.zolotarev.theta0_stable(alpha, beta)#
Zolotarev θ0 pivot for stable laws. - For α ≠ 1: θ0 = arctan(β tan(πα/2)) / α - For α = 1: special-case pivot (here set to π/2)
- aub_htp.pdf.zolotarev.transform_to_C_parametrization(alpha, beta)#
Convert (α, β) to Nolan’s S0 ‘theta’ style (often called θ_C here). - α ≠ 1: θ_C = (2/π) arctan(β tan(πα/2)) - α = 1: use a fixed θ_C = 2/π * arctan(2/π) (placeholder convention) Returns (α_C, θ_C) with α_C = α.