[ENH] Rayleigh Distribution with analytical energy calculation

[KaranSinghDev]

Reference Issues/PRs

#22

What does this implement/fix? Explain your changes.

This PR implements a native, high-performance version of the Rayleigh Distribution (skpro.distributions.Rayleigh).

I have implemented the following analytical methods to bypass Monte Carlo and _approx_derivative fallbacks:

• _pdf and _log_pdf- $$f(x) = \frac{x}{\sigma^2} e^{-x^2 / (2\sigma^2)}, \quad x \ge 0$$
• _cdf and _ppf (Inverse CDF) - $$F(x) = 1 - e^{-x^2 / (2\sigma^2)}$$
• _mean - $$\sigma \sqrt{\frac{\pi}{2}}$$
• _var - $$\frac{4-\pi}{2} \sigma^2$$
• _energy_self: $\mathbb{E}[|X - Y|] = \sigma \sqrt{\pi} (\sqrt{2} - 1)$

Does your contribution introduce a new dependency? If yes, which one?

No

What should a reviewer concentrate their feedback on?

• Formula Consistency: I used the σ parameterization to align exactly with scipy.stats.
• Performance: I implemented the exact analytical formula for _energy_self() to avoid the slow Monte Carlo default.
• Broadcasting: Verified that the implementation handles scalar, 1D, and 2D parameter inputs via the _bc_params broadcasting logic.

Did you add any tests for the change?

I relied on existing distributions suite. I implemented get_test_params with multiple parameter sets (scalar, array, and DataFrame inputs). The Rayleigh class was picked up by the suite, passing 194 automated tests locally.

Any other comments?

I chose this distribution as I am aware of it and I have used it before in signal processing and believe it will be useful in the current list of distributions (even though it is low priority). I have also performed numerical verification against scipy.stats.rayleigh to ensure accuracy in the PDF, CDF, and Mean/Variance calculations.

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For new estimators

• I've added the estimator to the API reference - in docs/source/api_reference/taskname.rst, follow the pattern.
• I've added one or more illustrative usage examples to the docstring, in a pydocstyle compliant Examples section.
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[KaranSinghDev]

I have updated the PR for the workflow failing earlier .

[fkiraly]

Nice!

For exact energy tag, the cross-term is still missing, _energy_x - do you also have that, or do you leave that for now?

[KaranSinghDev]

@fkiraly Yes I was working on the cross-term to make the energy tag fully exact.

I have added the closed-form analytical solution for _energy_x and the piecewise solution I implemented for $X \sim \text{Rayleigh}(\sigma)$ is:

• For $x \le 0$: $\mathbb{E}[|X - x|] = \sigma \sqrt{\frac{\pi}{2}} - x$
• For $x > 0$: $\mathbb{E}[|X - x|] = x + \sigma \sqrt{\frac{\pi}{2}} - \sigma \sqrt{2\pi} \cdot \text{erf}\left(\frac{x}{\sigma \sqrt{2}}\right)$

I verified these formulas against scipy.integrate.quad across both negative and positive x
values. The analytical implementation overlays with the numerical integration

[fkiraly]

Nice!

I think there does not exist a reference for these formulae, so it would be great if we could display them in docstrings.

For that, this PR (abandoned) needs to be fixed. Maybe if you have time, you could have a look, and then add your explicit formulae?
#698

Alternatively, there is already a markdown where we are collecting these, in the distributions folder, where you can add them.

[KaranSinghDev]

@fkiraly Sure I will do it .