Timeline for Is quantile regression a maximum likelihood method?
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| Mar 1, 2023 at 22:40 | comment | added | Zhanxiong | @Ggjj11 No, like quantile loss function, the squared loss function also does not rely on the normality assumption, it is derived from $\bar{Y} = \operatorname{argmin}_u E[(Y - u)^2]$ for any distribution of $Y$. The relationship between normal and squared loss function is precisely analogous to the relationship between asymmetric Laplace and quantile loss. | |
| Mar 1, 2023 at 22:37 | comment | added | Ggjj11 | If you want, I could rephrase "What is the rationale of the quantile loss function (3) in quantile regression (of arbitrary complex models)? Does it rely on the specification of the distributional form of response variable? And, specifically, is the quantile loss (somehow) a (log) likelihood function?". In the case you find my wording strange, please think about how the mean squared error is derived from a likelihood function assuming a normal distribution in MLE. Surely you understand. | |
| Mar 1, 2023 at 22:24 | comment | added | Zhanxiong | If I understand your real intention correctly, you may have asked your question like this: "How is the quantile loss function $(3)$ discovered?" or "What is the rationale of the quantile loss function $(3)$ in quantile regression? Does it rely on the specification of the distributional form of response variable?" Either alternative is much clearer than what your original post is (which unnecessarily brought the MLE topic in). | |
| Mar 1, 2023 at 22:16 | comment | added | Zhanxiong | The point of mentioning asymmetric Laplace distribution in the answer is that when the conditional distribution of $y$ is asymmetric Laplace, then the negative log-likelihood coincided with the quantile loss. Your statement "my question is if the quantile loss function is a negative (log) likelihood function which is to be minimized" is again ill-posed -- if you do not specify the conditional distribution, how are you able to even write down the "negative (log) likelihood function"??? I hope you have carefully read through my answer (and you are not the one who downvoted :)). | |
| Mar 1, 2023 at 22:12 | comment | added | Zhanxiong | @Ggjj11 As the link in your comment demonstrated, the quantile loss is derived without requiring any specific parametric distributional form of $y$. The key observation in discovering the quantile loss is that in one-sample problem, the population quantile minimizes the expected check function, i.e., $Q_\tau(Y) = \operatorname{argmin}_{q} E[\rho_\tau(Y - q)]$. | |
| Mar 1, 2023 at 22:01 | comment | added | Ggjj11 | @Zhanxiong my question is if the quantile loss function is a negative (log) likelihood function which is to be minimized. In the quantile loss derivation stats.stackexchange.com/a/252043/298651 no explicit assumption about a probability distribution is made (and therefore no likelihood is computed). Could you please add more information here. If the quantile loss is (implicitly??) assuming the asymmetric Laplace distribution, what is so special about this distribution that it appears in the context of quantiles? | |
| Mar 1, 2023 at 7:14 | comment | added | Stephan Kolassa | Common terms for the loss function used in quantile regression are "pinball loss", or "quantile, "linlin", "hinge", "tick" or "newsvendor loss". Any of these search terms will likely get more hits than "least $L^1$ estimation" or "asymmetric Laplace loss" or similar. | |
| Mar 1, 2023 at 5:41 | history | edited | Zhanxiong | CC BY-SA 4.0 |
Add more clarifications by elaborating the underlying model of MLE.
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| Mar 1, 2023 at 4:11 | comment | added | Thomas Lumley | The asymmetric Laplace distribution has a quantile that is the MLE of a parameter: en.wikipedia.org/wiki/… | |
| Mar 1, 2023 at 2:51 | comment | added | Zhanxiong | @BigBendRegion I couldn't follow your question very well. Can you elaborate? | |
| Mar 1, 2023 at 2:44 | comment | added | BigBendRegion | Do you or does anyone else know what distributions correspond to quantiles other than the median? | |
| Feb 28, 2023 at 23:31 | history | edited | Zhanxiong | CC BY-SA 4.0 |
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| Feb 28, 2023 at 23:09 | history | edited | Zhanxiong | CC BY-SA 4.0 |
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| Feb 28, 2023 at 23:04 | history | answered | Zhanxiong | CC BY-SA 4.0 |