Example
Let be a discrete random variable that takes values 1,2,..,9 with equal probabilities. The task is to find the median of Y, and hence the value is chosen. The expected loss, L(u), is
Since is a constant, it can be taken out of the expected loss function (this is only true if ). Then, at u=3,
Suppose that u is increased by 1 unit. Then the expected loss will be changed by on changing u to 4. If, u=5, the expected loss is
and any change in u will increase the expected loss. Thus u=5 is the median. The Table below shows the expected loss (divided by ) for different values of u.
| u | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Expected loss | 36 | 29 | 24 | 21 | 20 | 21 | 24 | 29 | 36 |
Read more about this topic: Quantile Regression
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
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