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Question:

Is the condition that asymptotic bias and asymptotic variance goes to zero for infinite samples sufficient to guarantee the pointwise consistency of an estimator based on plug-in kernel density estimator (KDE)?

Details:

I am interested in the estimation of a log of ratio between two continuous dentisy functions $f=\log \frac{p_1(x)}{p_2(x)}$ with kernel density estimation (KDE).

Assume that we have two sets of samples $\{x_i^{(1)} \}_{i=1}^{N_1}$ and $\{x_j^{(2)} \}_{j=1}^{N_2}$ from two continuous $D$-dimensional density functions $p_1(x)$ and $p_2(x)$, respectively.

Then, a logarithm of a ratio of densities $\log\frac{p_1(x)}{p_2(x)}=\log p_1(x) - \log p_2 (x)$ at a test point x is estimated with KDEs as plug-in density estimates. The KDEs are defined as $$ \widehat{p}_1(x)=\frac{1}{N_1}\sum_{i=1}^{N_1}K_h(x, x_i^{(1)}),\quad \widehat{p}_2(x)=\frac{1}{N_2}\sum_{j=1}^{N_2}K_h(x, x_j^{(2)}), $$ where $K_h(\cdot, \cdot)$ is a kernel function with a bandwidth $h$. Let's just assume a Gaussian kernel. The resulting estimator is $\widehat{f}(x) =\log \frac{\widehat{p}_1(x)}{\widehat{p}_2(x)}$. I can assume some conditions for $p_1(x)$, $p_2(x)$, and $K_h(\cdot, \cdot)$ so that the logarithm is well defined.

I have obtained the bias, $\mathbb{E}[\widehat{f}]-f$, and the variance, $\mathbb{E}[(\widehat{f}(x)-\mathbb{E}[\widehat{f}(x)])^2]$, of the estimator $\widehat{f}(x)$, both of which goes to zero with the conditions $N\to\infty$, $h\to0$, $Nh^D\to\infty$.

I have found articles which say vanishing bias and variance are enough for pointwise consistency in KDE case. 1) Section 2.6 in Hansen's lecture note 2) Theorem 1 of Wied and Weissbach, 2012.

Therefore, currently, I suspect that the condition of bias and variance going to zero is sufficient for pointwise consistency of my estimator $\widehat{f}$. Is this correct? or if not, what kind of argument is necessary to prove that the estimator is pointwise consistent?

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