In general it's possible to estimate a given multivariate density function $f(x)$ using some non-parametric methods such as kernels or histogram. What are the conditions on a function $g$ such that the estimation of $g(f(x))$ will be consistent as well?
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$\begingroup$ It all depends on the convergence properties of $\hat f_n$ the estimator of $f$. Under strong uniform consistency, the property is preserved by continuous transforms. $\endgroup$– Xi'anCommented Jan 18, 2016 at 7:19
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$\begingroup$ It's hard for me to see why this is tagged "regression" or "multiple-regression". I removed those tags and added "kernel-smoothing" and "histogram". $\endgroup$– Nick CoxCommented Jan 18, 2016 at 9:04
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1 Answer
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Let $x$ be a fixed vector, the estimate of its density, $\hat{f}_{n}\left(x\right)$.
Let $g\left(\cdot\right)$ be a differentiable function where the derivative of $g$ is bounded in an open neighborhood of $f\left(x\right)$. From the mean value theorem:
$g\left(\hat{f}_{n}\left(x\right)\right) - g\left(f\left(x\right)\right) = g'\left(y_{n}\right) \times \left[ \hat{f}_{n}\left(x\right) - f\left(x\right)\right]$, where $y_{n}$ is a value between $\hat{f}_{n}\left(x\right)$ and $f\left(x\right)$.
Because $\hat{f}_{n}\left(x\right) \overset{p}\rightarrow f\left(x\right)$. With probability $1$ in a large enough sample, $y_{n}$ will be close enough to $f\left(x\right)$ that $g'$ is bounded. Say $|g'| \le M$ in the neighborhood of $f\left(x\right)$.
$\text{pr}\left\{|g'\left(y_{n}\right) \times \left[ \hat{f}_{n}\left(x\right) - f\left(x\right)\right]| > \epsilon\right\} \le \text{pr}\left\{ M \times |\hat{f}_{n}\left(x\right) - f\left(x\right)| > \epsilon\right\} \rightarrow 0$.
Where the limit holds because $\hat{f}_{n}\left(x\right) \overset{p}\rightarrow f\left(x\right)$.
