Suppose $f(x)$ is a probability density function supported on the real line. Assume that $\hat{f}(x)$ is a consistent estimator of $f(x)$. Can we claim that $\hat{F}(x)$ is a consistent estimator of $F(x)$, which is the corresponding cdf? Based on Scheffe's theorem and Nadaraya's results "Some new estimates for distribution functions" I see that we can make such a claim. Is that always true? Any feedback/discussion/textbook is appreciated. Thanks.
does consistency in terms of density carry on cdf?
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probability-distributions
density-function