Penalized empirical likelihood for partially linear errors-in-variables models
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Penalized empirical likelihood for partially linear errors‑in‑variables models Xia Chen1 · Liyue Mao1 Received: 5 December 2018 / Accepted: 22 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we study penalized empirical likelihood for parameter estimation and variable selection in partially linear models with measurement errors in possibly all the variables. By using adaptive Lasso penalty function, we show that penalized empirical likelihood has the oracle property. That is, with probability tending to one, penalized empirical likelihood identifies the true model and estimates the nonzero coefficients as efficiently as if the sparsity of the true model was known in advance. Also, we introduce the penalized empirical likelihood ratio statistic to test a linear hypothesis of the parameter and prove that it follows an asymptotic Chi-square distribution under the null hypothesis. Some simulations and an application are given to illustrate the performance of the proposed method. Keywords Penalized empirical likelihood · Measurement error · Variable selection · Partially linear models Mathematics Subject Classification 62F12 · 62G05
1 Introduction Let (X, T, Y) denote a triple of random variables and vectors. Consider the following partially linear model (1)
Y = XT 𝜷 + g(T) + e
where X is a p-dimensional random vector, T is a random variable defined on [0, 1] and 𝜷 is an unknown p-dimensional parameter vector. The function g(⋅) is
* Xia Chen [email protected] 1
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China
13
Vol.:(0123456789)
X. Chen, L. Mao
an unknown smoothing function defined in [0, 1], and e is the random error with E(e|X, T) = 0. Measurement error data are often encountered in many fields, including engineering, economics, physics, biology, biomedical sciences and epidemiology (see, e.g., Gustafson 2005; Carroll et al. 2006). For example, in acquired immunodeficiency syndrome (AIDS) studies, virologic and immunologic markers, such as plasma concentrations of human immunodeficiency virus (HIV) 1 RNA and CD4+ cell counts, are measured with errors. Statistical inference methods for various measurement error situations have been well established over the past several decades. For the case that only X is measured with errors and T can be measured directly, Cui and Li (1998), Liang et al. (1999) and He and Liang (2000) studied the asymptotic normality of the estimators of the parameter and the convergence rate of the estimate of the nonparametric function in model (1). Jin and Tong (2015) studied the corrected-loss estimation for errors-in-variables partially linear model. Xu et al. (2017) investigated hypothesis tests in partial linear errors-in-variables models with missing response. The empirical likelihood inference for parameter 𝛽 in semilinear model can be found in Cui and Kong (2006). For the case that only T is measured with errors and X can be measured directly, Liang (2000) studied the ge
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