Could you please help me with the proof for proposition 1?

[xlnwel]

Hi @ku2482

Thanks for the code. May I ask you a question?

The author gives proposition 1 and its proof as follows:

I'm quite confused about how they compute the third step, which involves the integral over a quantile function. Could you please help me with that?

[toshikwa]

Hi @xlnwel

I wrote the rough derivation of the first half(①). Note that

• we can remove absolute value symbols as we assume non-decreasing property.
• derivative of f(x/2) equals to 1/2 f'(x/2).

If you have any questions, feel free to ask me.

[xlnwel]

Hi @ku2482,

Thank you for the derivation, which helps me understand most of the proof. However, I still got several questions. The following derivative is what I wrote

The second and third parts are consistent with yours, but the first and fourth are different. I computed the first and fourth parts from

but you directly moved the partial derivative inside. May I ask you why?

This affects how to compute

which I calculate as follows

This is quite different from the results of proposition 1. Where did I make mistakes?

[toshikwa]

Hi.

Oh, you're right. Partial derivative should be placed as the second image you sent.

However, results would be the same. First and fourth parts are constants, which don't depend on omega. Because tau_hat is the midpoint and the signs of first and forth parts are opposite, these terms should disappear before we apply the partial derivative.

[xlnwel]

Hi, @ku2482

I know that. But could you please help me with the following calculation? I somehow compute the following part wrongly, and I cannot find the error

Thanks in advance.

[toshikwa]

Hi, @xlnwel

I'm not sure where exactly confuses you. Let me assure you that my derivation only calculates only first half of the whole derivative (①) as below, is it clear?

[toshikwa]

I wrote down the derivation of latter half of the whole derivative. Please tell me where exactly is bothering you.

Maybe, you wrongly removed the absolute value symbol, didn't you? Note that

• F_inv(\hat \tau_i) > F_inv(\omega)when \tau_i < \omega < \hat \tau_i
• F_inv(\hat \tau_i) < F_inv(\omega)when \hat \tau_i < \omega < \tau_{i+1}

Is everything clear?

[xlnwel]

Hi,

Everything is crystal clear now. Thank you so much for the proof:-)