Coverage of credible intervals in nonparametric monotone regression
This paper concerns a nonparametric regression model for a response variable \(Y\) with respect to a predictor variable \(X\in [0, 1]\) given by \(Y = f (X)+\varepsilon\), where \(f\) is a monotone increasing function on \([0, 1]\) and \(\varepsilon\) is a mean-zero random error with finite variance \(\sigma^2\). Therefore, it develop an easy-to-use Bayesian method for constructing credible intervals for the function value \(f(x_0)\) at an interior point \(x_0\), and obtain the asymptotic frequentist coverage of the resulting interval. Works towards the circular law conjecture have been done by many authors including Mehta, Edelman, Girko, Bai, Götze and Tikhomirov, Pan and Zhou and the present authors (see the references) under some assumptions on the distribution. For example, Mehta (complex Gaussian distribution) and Edelman (real Gaussian distribution). The authors present results on coverage of credible after introducing some notation, assumptions and the prior on \(f\) and \(\sigma\). Belongs those assumptions, we have that the isotonic regression estimator of \(f\) is obtained by minimizing the sum of squares \(\sum_i^n = (Y_i-f(X_i))^2\) subject to the constraint that \(f\) is nondecreasing on \([0, 1]\). The resulting estimator of \(f\) is a nondecreasing step function, constant on the pieces \((X_i-1,X_i]\), with \(X_0\) defined to be zero. The estimated value of \(f(x_0)\) is given by the left derivative of the greatest convex minorant of the graph of the line segments connecting \(\{(0,0), (1/n,Y_1/n), (2/n, (Y_1 +Y_2)/n),\dots , (1, (\sum_{i=1}^n Y_i)/n)\}\), at the point \(i(x_0)/n\), where \(i(x_0)\) is the integer such that \(X_{i-1} < x_0 \leq X_i\). By using the assumptions of Theorem 3.1 and Proposition 3.2 and Theorem 3.3, the authors study coverage of credible intervals for the value of the regression function at an interior point through simulations.
scientific article; zbMATH DE number 7367871
scientific article