# Distribution of estimator of multiple (spatially related) proportions

Consider a spatial random process $$Z(s)$$, where $$s$$ denotes the spatial location.

Our objective is to delineate the zone $$\mathcal{Z}$$ where the probability that $$Z$$ exceed a given threshold $$\zeta$$ is above a specified probability $$r$$:

$$\mathcal{Z} = \{s | \mathrm{Pr} (Z(s) > \zeta) > r\} .$$

We know a sample $$\{z_1, \dots, z_N\}$$ of $$N$$ realisations of $$Z(s)$$. /I dropped the dependence in $$s$$ for notational brevity, but the $$z_i$$ are functions of the location $$s$$./

A straightforward estimate of the exceedance probability is the average of the $$N$$ images $$\{1_\zeta(z_1), \dots, 1_\zeta(z_N)\}$$ by the indicator function $$1_\zeta : z \mapsto$$ (1 if $$z> \zeta$$ and 0 otherwise).

Now, we would like to account for the uncertainty of this estimate when drawing the estimated zone of probable exceedance $$\tilde{\mathcal{Z}}$$.

The probability $$r$$ is small, say $$5\%$$ or possibly $$1\%.$$ The sample size is expected to range from a few decades to a few hundreds.

This problem similar to the question of “Confidence interval for a proportion” (Confidence interval for a proportion) except that we consider several dependent variables, namely the $$Z(s_1), \dots, Z(s_n)$$ for a set of locations $$s_1, \dots, s_n$$.

How to display (visualise) this added uncertainty is the subject of another question (Display uncertainty on spatialy distributed proportions (visualisation)).

A possible way of doing this, if I understand your question well, is to deal with your process $$Z$$ as a gaussian process.
To do so, you need to give yourself a covariance kernel, i.e. a function $$k$$ so that you can expect the covariance of $$Z$$ to follow: $$Cov(Z(s_1), Z(s_2)) = k(s_1, s_2)$$. A very usual covariance kernel is the radial basis function (gaussian kernel): $$k(s_1, s_2) = exp(-\frac{(s_1-s_2)^2}{2\sigma^2})$$. Depending on the shape of your spatial process, you may want to use another covariance kernel, but what follows remains the same.
Now with this kernel and your data, you are able to build a predictor $$\hat{Z}$$ by gaussian process regression (some standard packages do the job very easily). What is interesting in this technique is that $$\hat{Z}(s)$$ is a gaussian variable at any site $$s$$. The mean is the unbiaised predictor, and you also get a prediction variance allowing you to build confidence intervals. This means that you can compute $$Pr(Z(s)>\xi)$$ at any $$s$$, and thus determine the zone you are looking for.
• $$Z$$ must be continuous. You can try it with non continuous processes but expect strange results
• Most of the usual covariance structures (but not all) will imply that covariance is stationary, i.e. $$k(s_1, s_2) = K(s_1 - s_2)$$. When the process is defined over a too large space, the regressor simply can't simultaneously obey to all local covariance structures.