Let \widehat{\mathbf{p}} denote the MLE

Let \widehat{\mathbf{p}} denote the MLE for a categorical statistical model ( \{ a_1, \ldots , a_ K \} , \{
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Let \mathbf{p}^0 be the discrete pmf that we wish to test the goodness of fit for an observed sequence of iid samples. Let
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Suppose that K = 3, and let E =\{ 1, 2, 3\}. Let X_1, \ldots , X_ n \stackrel{iid}{\sim } \mathbf{P}_{\mathbf{p}} for some
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Let X_1, \ldots , X_ n \stackrel{iid}{\sim } \mathbf{P} for some distribution \, \mathbf{P}\,, and let F denote its cdf. Let F_
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Let X_1, \ldots , X_ n \stackrel{iid}{\sim } \mathbf{P} for some distribution \, \mathbf{P}\,, and let F denote its cdf. Let F_
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As above, under the statistical model ( \{ 1,2,3 \} , \{ \mathbf{P}_{\mathbf{p}} \} _{\mathbf{p} \in \Delta _3}), we haveL_{12}(
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As in the previous problem, we consider the matrixH = I_ n - \frac{1}{n} \mathbf{1} \mathbf{1}^ T and for simplicity let n = 3.
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Let X_1, \ldots , X_ n be iid samples with cdf F, and let F^0 denote the cdf of \textsf{Unif}(0,1). Recall thatF^0(t) = t \cdot
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Let X and Y be real-valued random variables, both distributed according to a distribution \, \mathbf{P}.\, (We make no
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Given the maximum likelihood estimators \, \widehat{\beta }_0 \,, \, \widehat{\beta }_1 \,, what are the associated predicted
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