## Consistency of logistic classifier in abstract Hilbert spaces.(English)Zbl 1418.62075

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 $$(X, Y)$$.
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.

### MSC:

 62F12 Asymptotic properties of parametric estimators 62H30 Classification and discrimination; cluster analysis (statistical aspects) 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 62J12 Generalized linear models (logistic models)

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### References:

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