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Confidence balls in Gaussian regression. (English) Zbl 1093.62051
From the paper: In the present paper, we consider the statistical model \[ Y_i= f_i+\sigma\varepsilon_i,\quad i=1,\dots,n, \] where \(f= (f_1,\dots,f_n)'\) is an unknown vector, \(\sigma\) a positive number and \(\varepsilon_1,\dots, \varepsilon_n\) a sequence of i.i.d. standard Gaussian random variables. For some \(\beta\in(0,1)\), the aim of this paper is to build a non-asymptotic Euclidean confidence ball for \(f\) with probability of coverage \(1-\beta\) from observations of \(Y=(Y_1,\dots, Y_n)'\). This statistical model includes, as a particular case, the functional regression model \[ Y_i=F(x_i)+\sigma\varepsilon_i,\quad i=1, \dots,n, \] where \(F\) is an unknown function on some interval, say \([0,1]\), and the \(x_i\)’s are some distinct deterministic points in this interval.
Starting from the observation of an \(\mathbb{R}^n\)-Gaussian vector of mean \(f\) and covariance matrix \(\sigma^2I_n\) \((I_n\) is the identity matrix), we propose a method for building a Euclidean confidence ball around \(f\), with prescribed probability of coverage. For each \(n\), we describe its non-asymptotic property and show its optimality with respect to some criteria.

MSC:
62G15 Nonparametric tolerance and confidence regions
62G08 Nonparametric regression and quantile regression
62G05 Nonparametric estimation
62G10 Nonparametric hypothesis testing
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