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Given X with uniform distribution in the interval [μ,μ+θ].

  1. Suppose θ is given. Find the posterior distribution with prior distribution on your own.
  2. From that, find the Bayesian estimator with quadratic loss function.
  3. If θ is not given, how to find μ.

Actually, this is a tough question in my last exam and I don't know how to approach it. I have read some papers and I solved it based on what I got.

My solution:

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And I don’t know how to proceed. Is it correct with this approach? Please give me some explanations. Thank you very much.

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Good try:

  1. The posterior density is indeed proportional to $\mathbb{I}_{xn-\theta\le\mu\le x1}$, hence a Uniform $U(xn-\theta,x1)$. Your prior $\pi(\mu)=1$ is not Uniform but flat, as it is improper;
  2. The Bayes estimator for the quadratic loss is indeed the posterior mean, which for a Uniform $U(xn-\theta,x1)$ equals$$\frac{xn+x1-\theta}{2}$$
  3. If $\theta$ is unknown, you first need to set a prior on $\theta$. For instance, I would take the scale improper prior $\pi(\theta)\propto 1/\theta$. You then need to find the marginal posterior of $\mu$ given $x$ by integrating out $\theta$:$$\pi(\mu|\mathbf{x})=\int_0^\infty p(\theta,\mu|\mathbf{x})\,\text{d}\theta$$The joint posterior $p(\theta,\mu|\mathbf{x})$ can be derived from$$p(\theta,\mu|\mathbf{x})\propto\pi(\theta)\pi(\mu)f(\mathbf{x}|\mu,\theta)=\frac{1}{\theta}\frac{1}{\theta^n}\mathbf{I}_{xn-\theta\le\mu\le x1}$$
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    $\begingroup$ Thanks a lot for your answer. Now I can understand thoroughly about Bayesian estimator and Prior Function. So, for each unknown parameter, we will set a prior on it and find the joint posterior. Thank you very much ^^. $\endgroup$ – Sang Huynh Sep 24 '16 at 12:04
  • $\begingroup$ @SangHuynh: thanks. If you think this is acceptable as a definite answer, do not forget to check it! But you may also prefer to wait for a deeper or broader answer. $\endgroup$ – Xi'an Sep 24 '16 at 13:26

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