Let that $\mathbf{s}=(s_1,s_2) \sim Unif(S)$, where $S$ is some spatial area. Suppose $y=h(\mathbf{s})=1-[exp(exp(\beta_0+\beta_1(\mathbf{s}-\mathbf{x})^T(\mathbf{s}-\mathbf{x})))]^{-1}$. We have that $\frac{\partial h}{\partial \mathbf{s}}=2\beta_1e^{\beta_0+\beta_1\cdot Tr[(\mathbf{s}-\mathbf{x})^T(\mathbf{s}-\mathbf{x})]}exp[-(\beta_0+\beta_1\cdot Tr[(\mathbf{s}-\mathbf{x})^T(\mathbf{s}-\mathbf{x})])](\mathbf{s}-\mathbf{x})$. How can I obtain the distribution for $h(\mathbf{s})$ analytically?

My idea:

I was thinking of using Theorem 2.1.5 in Casella and Berger: $$f_y(y)=f_\mathbf{s}(h^{-1}(y))|\frac{d}{d\mathbf{s}}h^{-1}(y)|$$

However, I'm not sure if this theorem extends to the multivariate case that I have.

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    $\begingroup$ Given that $S$ is only some vague "spatial area," could you explain what it might mean to write the distribution of $h(s)$ "analytically"? $\endgroup$ – whuber Jul 27 at 20:19
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    $\begingroup$ The expression will be nasty even in that case, because level sets of $y$ are circles (concentric around $x$) while the boundaries are portions of lines. That makes this an unappetizing question. Sometimes the original statistical problem will suggest useful approximations. Could you provide some background? $\endgroup$ – whuber Jul 27 at 21:38
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    $\begingroup$ How far is $x$ from $S$? The problem of approximating $h$ becomes three separate ones depending on whether $x$ is far from $S$ (compared to its diameter), near $S$, or within $S.$ $\endgroup$ – whuber Jul 27 at 22:11
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    $\begingroup$ You will need to evaluate the integral numerically. $\endgroup$ – whuber Jul 28 at 13:44
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    $\begingroup$ Clearly the integral depends on the details of the boundary of $S:$ in that sense it's obviously messy except, perhaps, when the boundary is a circle centered at $x.$ It remains possible that an analytical solution exists for the unit square case, but I doubt anyone would want to work it out. Your best bet would be to apply Mathematica to the problem to see if it can produce anything for a specific, extremely simple $x$ (such as the center of the square or one of its corners). If it can't even make progress in that case, the problem is hopeless. $\endgroup$ – whuber Jul 28 at 15:15

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