Integrated likelihood inference in semiparametric regression models
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Integrated likelihood inference in semiparametric regression models H. He · T. A. Severini
Received: 19 December 2013 / Accepted: 18 April 2014 © Sapienza Università di Roma 2014
Abstract Consider a linear semiparametric regression model with normal errors in which the mean function depends on two parameters, a p-dimensional regression parameter, which is the parameter of interest, and an unknown function, which is a nuisance parameter. We consider estimation of the parameter of interest using an integrated likelihood function, in which the nuisance parameter is eliminated from the likelihood function by averaging with respect to some distribution. Here we take this distribution to be a Gaussian process with a given covariance function, which may depend on additional parameters. Likelihood inference based on the resulting integrated likelihood is considered and the properties of the score statistic based on the integrated likelihood, the maximum integrated likelihood estimator, and the integrated likelihood ratio statistic are presented. The methodology is illustrated on two examples. Keywords estimation
Gaussian process · Likelihood inference · Likelihood ratio test · Semiparametric
1 Introduction Consider the following model. Let Y1 , Y2 , . . . , Yn denote real-valued random variables of the form Y j = x j β + γ (z j ) + j , j = 1, . . . , n where x1 , . . . , xn are observed constants in p , 1 , . . . , n are unobserved real-valued random variables such that the random vector = (1 , . . . , n )T has a multivariate normal distribution
H. He Heslington, York, UK T. A. Severini (B) Northwestern University, Evanston, IL, USA e-mail: [email protected]
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H. He, T. A. Severini
with mean vector 0 and covariance matrix φ , φ ∈ is an unknown parameter, β is an unknown parameter vector taking values in p , z 1 , . . . , z n are observed constants, taking values in a set Z , and γ is an unknown real-valued function on Z . Our goal is inference about the parameters β and φ in the presence of the nuisance parameter γ . The parameter γ takes values in , an appropriate set of functions on Z . In simple cases, may be a subset of differentiable real-valued functions on Z ; in other cases, might include additional requirements on γ . The likelihood function for this model is given by |φ |
− 21
1 −1 T exp − (Y − Xβ − g) φ (Y − Xβ − g) ; 2
(1.1)
here Y = (y1 , . . . , yn )T , X is the n × p matrix with jth row x T j , and g = (γ (z 1 ), . . . , T γ (z )) . Hence, in order to proceed with likelihood inference for (β, φ), some method of n
dealing with the nuisance parameter γ is needed. Since the maximum likelihood estimator of γ is not consistent in general, the profile likelihood function, which is formed maximizing the likelihood with respect to γ for fixed values of the other parameters, does not have the usual properties here; see, e.g., [24] for further discussion. The purpose of this paper is to consider inference based on an integrated likelihood function, in which γ is remov
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