plan for add power tuning to quantile point estimate

[Shinning-Zhou]

Hi @aangelopoulos Thank you for your clearly code.

I was wondering if you have any plans to add power tuning to quantile estimation?

Thanks

[Michael-Howes]

I am interested in developing this method if this feature request is still open. @Shinning-Zhou, @aangelopoulos, @tijana-zrnic

[aangelopoulos]

I think that's a great idea! :)

Best, Anastasios Angelopoulos https://people.eecs.berkeley.edu/~angelopoulos/

On Mon, Oct 6, 2025 at 1:38 PM Michael-Howes @.***> wrote:

Michael-Howes left a comment (aangelopoulos/ppi_py#23) https://github.com/aangelopoulos/ppi_py/issues/23#issuecomment-3373991061

I am interested in developing this method if this feature request is still open. @Shinning-Zhou https://github.com/Shinning-Zhou, @aangelopoulos https://github.com/aangelopoulos, @tijana-zrnic https://github.com/tijana-zrnic

— Reply to this email directly, view it on GitHub https://github.com/aangelopoulos/ppi_py/issues/23#issuecomment-3373991061, or unsubscribe https://github.com/notifications/unsubscribe-auth/AGBYOUVJSP4LK4HS3I2GN3D3WLHLLAVCNFSM6AAAAABVGNES6KVHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMZTGNZTHE4TCMBWGE . You are receiving this because you were mentioned.Message ID: @.***>

[Michael-Howes]

Here is my proposal. Please let me know what you think.

In the existing code, quantile estimation is done by inverting a test for the hypothesis $H_y : P(Y \le y) = q$. The hypothesis is tested by applying a CLT to the statistic:

$$T_y = \frac{1}{N}\sum_{i=1}^N I[f(\widetilde{X}i)\le y] - \frac{1}{n}\sum{i=1}^n (I[f(X_i) \le y] - I[Y_i \le y]).$$

One approach for power tuning would be to replace $T_y$ with $T_{y,\lambda}$

$$T_{y,\lambda} = \frac{1}{N}\sum_{i=1}^N \lambda I[f(\widetilde{X}i)\le y] - \frac{1}{n}\sum{i=1}^n (\lambda I[f(X_i) \le y] - I[Y_i \le y]).$$

Then, the results for PPI++ for estimating the mean give the optimal choice of $\lambda$, a consistent estimator $\hat{\lambda}$ and a CLT for $T_{y,\hat{\lambda}}$.

To invert this test and construct confidence intervals, there will be a family of tuning parameters $\hat{\lambda}_y$. I don't think this family would cause issues with validity. My understanding is that test inversion only involves a point-wise CLT.

How does the above proposal sound? It is different to power tuning as it appears in the PPI++ paper. That method of power tuning would require working with the pinball loss. I have thought a little bit about this but it appears tricky since the second derivative of the population pinball loss involves the density of $Y$. Let me know if you have previously thought about power tuning for quantile estimation.

[Michael-Howes]

Actually it seems like the best way to do power tuning for quantile estimation would be to use predict-then-debias as illustrated in https://github.com/aangelopoulos/ppi_py/blob/main/examples/tree_cover_ptd.ipynb.

Using the bootstrap means that the optimal tuning parameters can be estimated without any density estimation. Happy to hear thoughts.