# Are vanishing bias and variance enough for pointwise consistency for KDE-based estimation?

## 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?