Estimating the Unrestricted and Restricted Liu Estimators for the Poisson Regression Model: Method and Application
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Estimating the Unrestricted and Restricted Liu Estimators for the Poisson Regression Model: Method and Application Kristofer Månsson1 · B. M. Golam Kibria2 Accepted: 16 July 2020 © The Author(s) 2020
Abstract This paper considers both unrestricted and restricted Liu estimators in the presence of multicollinearity for the Poisson regression model. It also considers some new estimators of the shrinkage parameter for both unrestricted and restricted Liu estimators. Based on a simulation study and its empirical application, we found that the restricted estimator outperforms the unrestricted one. Further, the restricted Liu estimator also outperforms both the unrestricted Liu and restricted Liu estimators. Hence, this new method is a preferred option when the coefficient vector β may belong to a linear sub-space defined by Rβ = r. Keywords Liu estimator · Maximum likelihood · Monte Carlo simulations · MSE · Multicollinearity · Poisson regression · Restricted estimator Mathematics Subject Classification Primary 62J07 · Secondary 62J02
1 Introduction In empirical applications in economics when modeling the gene expression data, the dependent variables are often in the form of non-negative integers of counts, for example, in analyzing the determinants of the number of patents by a firm or in biostatistics. In such a situation, a common distributional assumption for the dependent variable distribution. More specifically, it is distributed ) ( y)i is that it follows (a Poisson as Po 𝜇i , where 𝜇i = exp 𝐱𝐢� 𝛃 , 𝐱𝐢 is the ith row of 𝐗 which is a n × (p + 1) data matrix with p explanatory variables and 𝛃 is a (p + 1) × 1 vector of coefficients. The parameter for the unrestricted Poisson regression model is usually estimated using the iterative weighted least squares (IWLS) algorithm as: * Kristofer Månsson [email protected] 1
Department of Economics, Finance and Statistics, Jönköping University, Jönköping, Sweden
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Department of Mathematics and Statistics, Florida International University, Miami, FL, USA
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Vol.:(0123456789)
K. Månsson, B. M. G. Kibria
)−1 ( ( ) ̂𝐳 , ̂ 𝐗� 𝐖̂ 𝛃ML = 𝐗� 𝐖𝐗
(1)
[ ] ̂ = diag 𝜇̂ i and 𝐳̂ is a vector where the ith element equals where (𝐖 ) yi −𝜇̂ i ẑ i = log 𝜇̂ i + 𝜇̂ . Hence, it is an iterative maximum likelihood (ML) procedure. i
As has been shown by Kaçiranlar and Dawoud (2018), Månsson et al. (2012) and Türkan and Özel (2016) among others this estimator is sensitive to multicollinearity. A common solution to this problem is applying the ridge regression method (Hoerl and Kennard 1970). Another widely used method is applying the Liu type estimator introduced by Liu (1993) and introduced in the context of the Poisson regression model by Månsson et al. (2012). This method has certain advantages over the ridge regression method since it is a linear function of the biasing parameter d. Therefore, this method has become more popular during recent years (see, for example, Akdeniz and Kaciranlar 1995; Kaciranlar 2003). The unrestricted Liu type estimator for the Poisson regre
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