The indicative function and the smoothness

General
1

For quantile regression, we know that indicator functions are often involved in parameter estimation.Now When I’m dealing with a model,the question is following:
we have equation:I\left(Y-X_{i}^{T} \beta<0\right)
The indication function is not smooth in the process，so I’m going to smooth out the indicative function,Which is the following:
\Phi_{h}\left(Y-X_{i}^{T} \beta\right)

what I should do in stan ?

2

If you need a smooth alternative to the step function

f\left(x\right)=\begin{cases} 0 & x<0\\ 1 & x>0 \end{cases}

then one choice is the inverse logit function.

inv_logit(Y - X*beta)
3

Thank you for you reply .I want to use gaussian kernel function to smooth the Indicator function,what I should do ?Thank you very much .

4

The convolution of Gaussian and indicator function is Phi(x). It is also available in Stan by default.

5

For the gaussian kernel function ,how to choose a bandwith for the smooth.Thank you so much .

6

Depends on the data I guess. You could try and make it a parameter. Larger the bandwidth, the smoother the function, but too large bandwidth changes the model.
You could try and make it a parameter. The sampler can probably figure it out.

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