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A full understanding of this issue requires a theory of integration over probability distributions, not just functions. However, even in such an abstract theory it's possible to visualize the integrals as areas under curves. The universal principle is that in any "reasonable" theory of integration, it should be possible to integrate by parts. Consider the ...

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A couple of notes: Most often the p-values given next to regression coefficients are based on the Wald test statistic, which is the estimated value of the coefficient divided by its standard error. This tests the null hypothesis that the specific coefficient is zero, and all other coefficients are non-zero. You could test the same hypothesis also with a ...

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For independent observables $\{x_{it}\}$ and $\{a_{it}\}$, I would write the joint conditional likelihood as \begin{aligned} P(\{x_{it}\},\{a_{it}\}|\{\lambda_i\},\{\xi_i\},\beta) = \prod_i TN(a_{it}|\lambda_i,\beta) Pr(x_{it}|\lambda_i,\xi_i) \end{aligned} Some comments: Notice that I've only included mention of the parameters whose values are ...

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