M-cross-validation criterion is proposed for selecting a smoothing parameter in a nonparametric median regression model in which a uniform weak convergency rate for the M-cross-validated local median estimate, and the...M-cross-validation criterion is proposed for selecting a smoothing parameter in a nonparametric median regression model in which a uniform weak convergency rate for the M-cross-validated local median estimate, and the upper and lower bounds of the smoothing parameter selected by the proposed criterion are established. The main contribution of this study shows a drastic difference from those encountered in the classical L2-, L1- cross-validation technique, which leads only to the consistency in the sense of the average. Obviously, our results are novel and nontrivial from the point of view of mathematics and statistics, which provides insight and possibility for practitioners substituting maximum deviation for average deviation to evaluate the performance of the data-driven technique.展开更多
Consider the nonparametric regression model Yni = g(xni) + eni, 1≤i≤n, where g is an unknown function to be estimated on [0,1], xni (1≤i≤n) are the fixed design points in the interval [0,1] and {eni,1≤i≤n} is a ...Consider the nonparametric regression model Yni = g(xni) + eni, 1≤i≤n, where g is an unknown function to be estimated on [0,1], xni (1≤i≤n) are the fixed design points in the interval [0,1] and {eni,1≤i≤n} is a triangular array of row iid random variables having median zero. The nearest neighbor median estimator gn. h (xni )=m (Yi(1)(n)…Yi(h)(n) is taken as the estimator of the unknown function g(x). Median cross validation (mcv) criterion is employed to select the smoothing parameter h . Let hn * be the smoothing parameter chosen by mcv criterion. Under mild regularity conditions, the upper and lower bounds of hn* , the rate of convergence and the weak consistency of the median cross-validated estimate gn,hn* (xni) are obtained.展开更多
文摘M-cross-validation criterion is proposed for selecting a smoothing parameter in a nonparametric median regression model in which a uniform weak convergency rate for the M-cross-validated local median estimate, and the upper and lower bounds of the smoothing parameter selected by the proposed criterion are established. The main contribution of this study shows a drastic difference from those encountered in the classical L2-, L1- cross-validation technique, which leads only to the consistency in the sense of the average. Obviously, our results are novel and nontrivial from the point of view of mathematics and statistics, which provides insight and possibility for practitioners substituting maximum deviation for average deviation to evaluate the performance of the data-driven technique.
基金Project supported by the National Natural Science Foundation of China and the Doctoral Foundation of Education of China.
文摘Consider the nonparametric regression model Yni = g(xni) + eni, 1≤i≤n, where g is an unknown function to be estimated on [0,1], xni (1≤i≤n) are the fixed design points in the interval [0,1] and {eni,1≤i≤n} is a triangular array of row iid random variables having median zero. The nearest neighbor median estimator gn. h (xni )=m (Yi(1)(n)…Yi(h)(n) is taken as the estimator of the unknown function g(x). Median cross validation (mcv) criterion is employed to select the smoothing parameter h . Let hn * be the smoothing parameter chosen by mcv criterion. Under mild regularity conditions, the upper and lower bounds of hn* , the rate of convergence and the weak consistency of the median cross-validated estimate gn,hn* (xni) are obtained.
