Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ are the densities of $P_X$ and $P_W$ respectively, then $f_W = f_X * f_U$ where $*$ is the convolution operator and $f_U$ is another density. One way to see this is to suppose that $W = X + U$ with $W\sim f_W$, $X\sim f_X$, $U\sim f_U$ where $X$ and $U$ are independent. We define the mapping $E_W$ in the following way,

$$ \begin{align} E_W\phi(w) & = E(\phi(X)|W)\\ & = \int \phi(x) f_{X|W}(x|w)dx\\ & = \int k(x,w)\phi(x)f_X(x)dx\\ & = \langle k(\cdot,w), \phi\rangle_X \end{align} $$

where $f_{X|W}(x|w) = f_{X,W}(x,w)/f_W(w)$ is the conditional density and

$$ \begin{align} k(x,w) & = \frac{f_{X,W}(x,w)}{f_{X}(x)f_{W}(w)}\\ & = \frac{f_{U}(w-x)}{f_{W}(w)}, \end{align} $$

the Hilbert - Schmidt Kernel. Now I understand that for $E_W$ to be a compact operator from $L_2(\mathbb{R},P_X)$ to $L_2(\mathbb{R}, P_W)$, we require that

$$ \int |k(x,w)|^2f_X(x)f_W(w)dxdw <\infty, $$

which is something I would like to use in my work. My $\textbf{question}$ is then when is this assumption reasonable? Is this a standard assumption used in many areas of mathematical statistics? In particular, what about the measurement error case (i.e. the scenario I have depicted above)?



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