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Can I get some help on interpreting this equation? Is it saying which ever section separated by the commma is bigger would be the answer? How do you intepret Sigma pi= 1? Thanks!

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    $\begingroup$ Well one answer is $\Sigma_{i=1}^n p_i$ What does $U_i(\cdot,\cdot)$ refer to? If you reduce that ambiguity we may be able to help more. $\endgroup$ – Carl Apr 25 '20 at 4:37
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$U$ is the score function. The solution to the sum of scores equal to 0 is a generalized estimating equation. $p$ is a set of possible probability weights.

You are trying to find the set of $p$ that maximizes their product, since $n$ is a constant and can be factored outside of the product. Without the constraint that $\beta$ is actually a solution to the GEE, we'd expect $p=1/n$ i.e. the unweighted GEE to be the solution. But if $\beta$ is set to something different, there may be a set of weights for which that is the solution.

$R$ is commonly referred to as a risk function, which is the expected loss. As an example, for quadratic loss, $R$ is the MSE of an estimator.

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The supremum is maximizing the quantity $\prod_{i=1}^n np_i$ subject to all the constraints after the colon, and separated by commas. So, it's maximizing $\prod_{i=1}^n np_i$ subject to \begin{align*} \sum_{i=1}^np_iU_i(\beta,t)&=0,\;\text{and}\\ p_i&\ge 0\;\forall\,1\le i\le n,\;\text{and}\\ \sum_{i=1}^n p_i&=1. \end{align*} The last two conditions merely imply that the $p_i$ form a probability distribution.

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