BUG: stats.levy_stable: incorrect logpdf values for beta == +-1

[tjhunter]

Describe your issue.

Hello,
I am implementing Levy's alpha-stable distribution for the JAX framework, using scipy as reference and I found some discrepancies.

The tail values of scipy's levy_stable.logpdf are incorrect for alpha in (1,2], beta in {-1, +1} and |x| >= 10. In particular, they do not obey the expected exponential law.

See Nolan (2020), Proposition 3.1 page 100 for the correct tail approximation. I have implemented it (in jax) here:
https://github.com/tjhunter/levy-stable-jax/blob/b3e3e05cc572b68742a8d1f0223fd88230836f30/src/levy_stable_jax/distribution.py#L169

Here is an example. In blue is the theoretical tail behaviour (implemented by levy-stable-jax), in orange scipy

Link to the notebook that created the plot:
https://github.com/tjhunter/levy-stable-jax/blob/main/notebooks/scipy_issues.ipynb

My suggestion to resolve this issue would be to bypass the quadrature-based implementation in C for |beta| == 1 and to directly implement the tail approximation as in `levy-stable-jax1. Happy to discuss a possible PR.

Reproducing Code Example

import seaborn as sns
from scipy.stats import levy_stable as sp_levy_stable
import numpy as np

sp_levy_stable.parameterization = "S0"
(alpha, beta) = (1.6, 1.0)
xs = np.linspace(-15, 10, 100)
sns.lineplot(x=xs,            y=sp_levy_stable.logpdf(xs,alpha=alpha, beta=beta))

Error message

N/A

SciPy/NumPy/Python version and system information

1.13.0 1.26.4 sys.version_info(major=3, minor=11, micro=9, releaselevel='final', serial=0)
Build Dependencies:
  blas:
    detection method: pkgconfig
    found: true
    include directory: /usr/local/include
    lib directory: /usr/local/lib
    name: openblas
    openblas configuration: USE_64BITINT=0 DYNAMIC_ARCH=1 DYNAMIC_OLDER= NO_CBLAS=
      NO_LAPACK= NO_LAPACKE= NO_AFFINITY=1 USE_OPENMP= ZEN MAX_THREADS=64
    pc file directory: /usr/local/lib/pkgconfig
    version: 0.3.26.dev
  lapack:
    detection method: pkgconfig
    found: true
    include directory: /usr/local/include
    lib directory: /usr/local/lib
    name: openblas
    openblas configuration: USE_64BITINT=0 DYNAMIC_ARCH=1 DYNAMIC_OLDER= NO_CBLAS=
      NO_LAPACK= NO_LAPACKE= NO_AFFINITY=1 USE_OPENMP= ZEN MAX_THREADS=64
    pc file directory: /usr/local/lib/pkgconfig
    version: 0.3.26.dev
  pybind11:
    detection method: config-tool
    include directory: unknown
    name: pybind11
    version: 2.12.0
Compilers:
  c:
    commands: cc
    linker: ld.bfd
    name: gcc
    version: 10.2.1
  c++:
    commands: c++
    linker: ld.bfd
    name: gcc
    version: 10.2.1
  cython:
    commands: cython
    linker: cython
    name: cython
    version: 3.0.10
  fortran:
    commands: gfortran
    linker: ld.bfd
    name: gcc
    version: 10.2.1
  pythran:
    include directory: ../../tmp/pip-build-env-ilw0s1wu/overlay/lib/python3.11/site-packages/pythran
    version: 0.15.0
Machine Information:
  build:
    cpu: x86_64
    endian: little
    family: x86_64
    system: linux
  cross-compiled: false
  host:
    cpu: x86_64
    endian: little
    family: x86_64
    system: linux
Python Information:
  path: /opt/python/cp311-cp311/bin/python
  version: '3.11'