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I have a problem with the scale uniform interval estimator (Example 9.1.6, page 419, Casella-Berger). Let $X_1,\dots,X_n \sim \text{IID U}(0,\theta)$ be our observed data. We are interested in an interval estimator of $\theta$.

Consider the interval estimator $[Y + c; Y + d]$ where $Y = \max (X_1,\dots,X_n)$ and $0<c<d$ (note that $\theta$ is necessarily larger than $y$). I have no problem with the solution but I don't understand why we consider the interval $Y + c$ instead of $Y -c$.

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I think it should be 0 $\le$ c $<$ d for $[Y + c; Y + d]$, and $1 \le a < b$ for $[aY;bY]$, $0 < a < b$ doesn't make sense for any of these. Since $0 < x < \theta$ consequently $0 < max(x_1,\dots,x_n) < \theta$ then $0 < y < \theta$. Also $a0 < ay < a\theta$ shows that the minor value that 'a' can assume is 1 to preserve $0 < y < \theta$, and 'b' needs to be greater then 'a' so the interval makes sense, hence we have $1 \le a < b$ for $[aY;bY]$. Using the same logic for $[Y + c; Y + d]$ you will get $0\le c < d$. So, if $d > c$ and $\theta$ is larger then $Y$, you already have the basic relations you need for a possible interval where $Y + c < Y + d$, if intead you take $Y - c < Y + d$, you will work with desnecessary informations since you are trying to increase the size of the interval.

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Welcome to SO!

I am not sure I understand your question correctly, but if $Y=\max_i X_i$, then the upper limit of the uniform distribution $\theta$ necessarily must be $\theta \ge Y$. Hence looking for anything less than $Y$ does not make sense...

I am not familiar with this particular problem, but it seems to be a continuous version of the well known "German Tank Problem".

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At the moment this is not really a solution at all, since there is no specification of how you get $c$ and $d$ from the observed data. At the moment the "solution" is merely saying that the interval estimator will a lower bound that is above $y$. You have already noted in your question that $\theta$ is always larger than $y$, so obviously it makes sense to use an interval of this form.

The next step in solving the problem would be to specify $c$ and $d$ as functions of the data so that you get a specific interval estimate for a given set of data. The standard way to do this would be to use the sufficient statistic $Y$ to form a "pivotal quantity" that can be used to form an appropriate confidence interval. In any case, however you form the interval estimator, you will want to check that it has good statistical properties (e.g., correct coverage probability, consistency, etc.).

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