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Here is the following density :

$ (1 - \vert x-\theta \vert \mathbb{1}_{[\theta-1 ; \theta+1]} (x) $

Using the moments method, I found $ \widehat{\theta}_{n}^{(2)} = \overline{X}_{n} $.

1- I want an estimator $\widehat{\theta}_{n}^{(1)}$ using the median. How to find it ?

2- What is an intuitive explanation of this estimator : $\widehat{\theta}_{n}^{(3)} = max(X_i) -1$ ?

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