Bayesian robust estimation of partially functional linear regression models using heavy-tailed distributions
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Bayesian robust estimation of partially functional linear regression models using heavy-tailed distributions Guodong Shan1 · Yiheng Hou2 · Baisen Liu2 Received: 26 June 2018 / Accepted: 6 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Functional linear regression (FLR) is a popular method that studies the relationship between a scalar response and a functional predictor. A common estimation procedure for the FLR model is using maximum likelihood by assuming normal distributions for measurement errors; however this method may make inferences vulnerable to the presence of outliers. In this article, we introduce a robust estimation method of partially functional linear model by considering a class of scale mixtures of normal (SMN) distributions for measurement errors. Due to intractable closed form of likelihood function with the SMN distributions, a Bayesian framework is adopted and an MCMC algorithm is developed to carry out posterior inference on model parameters. The finite sample performance of our proposed method is evaluated by using some simulation studies and a real dataset. Keywords Functional data · Outliers · Scale mixtures of normal distributions · Metropolis-Hastings
1 Introduction With the rapid development of modern data collection and storage technology, the data collected in many fields often appear in the form of functions. For such data with functional characteristics, traditional multivariate data analysis methods are found having many limits in the specific applications. In contrast, functional data analysis (FDA) is becoming an active tool to deal with this type of data and has received
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00180020-00975-3) contains supplementary material, which is available to authorized users.
B
Baisen Liu [email protected]
1
School of Science, Changchun University, Changchun 130002, China
2
School of Statistics, Dongbei University of Finance and Economics, Dalian 116025, China
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G. Shan et al.
more and more attentions in recent years. Ramsay and Silverman (2005) provided fundamental theories and methodologies of FDA. Other valuable references of FDA refer to Ferraty and Vieu (2006), Horváth and Kokoszka (2007) and Hsing and Eubank (2015). Recently, Kokoszka and Reimherr (2017) provided a systematic and accessible exposition of the methodology and the required mathematical framework for FDA. In the area of FDA, one of the most popular methods is the scalar-on-function FLR where the scalar response Y depends on a linear projection of the functional predictor X (t). Here, X (t) is a square integrable stochastic process over a compact T interval I = [0, T ], that is, EX 2 = E{ 0 X (t)2 dt} (Ramsay and Silverman 2005). Without loss of generality, we suppose throughout that I = [0, 1]. For the response Y , the traditional FLR takes the following representation
1
Y =α+
β(t)X (t)dt + ε,
(1)
0
where α is an unknown intercept, β(t) is a squared integral slope functi
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