Abstract
The ridge regression estimator is a commonly used procedure to deal with multicollinear data. This paper proposes an estimation procedure for high-dimensional multicollinear data that can be alternatively used. This usage gives a continuous estimate, including the ridge estimator as a particular case. We study its asymptotic performance for the growing dimension, i.e., p→∞ when n is fixed. Under some mild regularity conditions, we prove the proposed estimator’s consistency and derive its asymptotic properties. Some Monte Carlo simulation experiments are executed in their performance, and the implementation is considered to analyze a high-dimensional genetic dataset.
Highlights
Consider the multiple regression model given by Y = Xβ + e, (1)Academic Editor: Jin-Ting ZhangReceived: 10 November 2021Accepted: 24 November 2021Published: 28 November 2021Publisher’s Note: MDPI stays neutral where Y = (y1, . . . , yn )> is a vector of n responses, X = (x1, . . . , xn )> is an n × p design matrix, with the ith predictor xi ∈ R p, β = ( β 1, . . . , β p )> is the coefficients vector, and e is an n-vector of unobserved errors
As c increases, the quadratic bias (QB) of the proposed estimator explodes for the heavier tail distribution. This may be seen as a disadvantage of the proposed estimators, but even for large values of c, the relative mean square error (RMSE) stays the same, evidence of relatively small variance for the heavier tail distribution
This section assesses the performance of the proposed estimators using the mean prediction error (MPE) and mean squared error (MSE) criteria of a data set adopted from Metzeler et al [16], in which the information for 79 patients was collected
Summary
Liu [8] developed a similar estimator; it is linear for the tuning parameter via the following optimization problem, for the case p < n: minp S( β) + (d β − β)> (d β − β). Proposed a two-parameter estimator by solving the following optimization problem:. P > n, the LS estimator (2) cannot be obtained, so it is not possible to use the two-parameter ridge estimator in Equation (6). Developing a high-dimensional two-parameter version of this estimator and studying its asymptotic performance is interesting and worthwhile. In this paper, we propose a high-dimensional version of Ozkale and Kaciranlar’s estimator and give the asymptotic properties. The paper’s organization is as follows: In Section 2, a high-dimensional twoparameter estimator is proposed, and its asymptotic characteristics are discussed.
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