Consistency of logistic classifier in abstract Hilbert spaces (Q1711588)

Let $E$ be an infinite-dimensional separable Hilbert space. Let $X$ denote an $E$-valued random variable and $Y$ a further random variable, taking values in $\{-1, +1\}$, where $X$ and $Y$ are defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume a logistic model of the form \[ \mathbb{P}(Y = +1 | X = x) = p_{\theta_0}(x) = \frac{1}{1 + e^{-\langle \theta_0, x \rangle}}, \] where $\langle \cdot, \cdot \rangle$ denotes the inner product in $E$ and $\theta_0$ is an unknown element of $E$. Furthermore, let $(X_1, Y_1), \ldots, (X_n, Y_n)$ be a sample of independent $E \times \{-1, +1\}$-valued observables, such that $(X_1, Y_1)$ is distributed as $(X, Y)$. \par The authors are concerned with estimating $\theta_0$ by means of the quasi-maximum likelihood method, along some fixed sequence $(E_k)_{k}$ of linear subspaces of $E$ with $\dim E_k = k$, where $k = k_n$ depends on the sample size $n$. Two sets of conditions are derived for the consistency of the resulting estimator $\hat{\theta}_{k_n, n}$ as $n$ tends to infinity. The first set of conditions refers to the distribution of $X$, and the second set of conditions refers to the growth rate of $k_n$ as a function of $n$. Finally, a simulation study is presented to illustrate the necessity of the conditions.