Following this question:Why the Pitman estimator is given by the sample mean of X and Y?.
Let $X_1, \dots ,X_n$ be i.i.d. random variables having distribution $P(X_j = \theta) = P(X_j = θ + 1) =1/2,$ where $\theta\in Z$ is an integer valued unknown location parameter. Find the minimum risk location equivariant estimator of $\theta$ for the squared error loss function.
It seems that the answer is the median of $\{X_1,\dots, X_n\}$. I try to get the posterior distribution of $\theta$ but I cannot do that. The likelihood function $$ L(\theta, X)=\prod_{i=1}^n p(X_i|\theta)=? $$
It seems that the posterior distribution is also maximized when $\theta$ is equal to the most common value among the $X_i$. Therefore, the minimum risk location equivariant estimator is $\hat{\theta}=mode\{X_1,\dots,X_n\}$.
