Biased Adjusted Poisson Ridge Estimators-Method and Application
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RESEARCH PAPER
Biased Adjusted Poisson Ridge Estimators-Method and Application Muhammad Qasim1 • Kristofer Månsson1 • Muhammad Amin2 • B. M. Golam Kibria3 • Pa¨r Sjo¨lander1 Received: 14 April 2020 / Accepted: 31 August 2020 Ó The Author(s) 2020
Abstract Ma˚nsson and Shukur (Econ Model 28:1475–1481, 2011) proposed a Poisson ridge regression estimator (PRRE) to reduce the negative effects of multicollinearity. However, a weakness of the PRRE is its relatively large bias. Therefore, as a ¨ zel (J Appl Stat 43:1892–1905, 2016) examined the performance of almost unbiased ridge estimators remedy, Tu¨rkan and O for the Poisson regression model. These estimators will not only reduce the consequences of multicollinearity but also decrease the bias of PRRE and thus perform more efficiently. The aim of this paper is twofold. Firstly, to derive the mean square error properties of the Modified Almost Unbiased PRRE (MAUPRRE) and Almost Unbiased PRRE (AUPRRE) and then propose new ridge estimators for MAUPRRE and AUPRRE. Secondly, to compare the performance of the MAUPRRE with the AUPRRE, PRRE and maximum likelihood estimator. Using both simulation study and real-world dataset from the Swedish football league, it is evidenced that one of the proposed, MAUPRRE (k^q4 ) performed better than the rest in the presence of high to strong (0.80–0.99) multicollinearity situation. Keywords Maximum likelihood estimator Multicollinearity Poisson ridge regression Modified almost unbiased ridge estimators Mean square error
1 Introduction
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s40995-020-00974-5) contains supplementary material, which is available to authorized users. & Kristofer Ma˚nsson [email protected] Muhammad Qasim [email protected] Muhammad Amin [email protected] B. M. Golam Kibria [email protected] Pa¨r Sjo¨lander [email protected] 1
Department of Economics, Finance and Statistics, Jo¨nko¨ping University, Jo¨nko¨ping, Sweden
2
Department of Statistics, University of Sargodha, Sargodha, Pakistan
3
Department of Mathematics and Statistics, Florida International University, Miami, FL, USA
The Poisson regression model (PRM) is a special form of the generalized linear models and is used when the dependent variable is collected in terms of counts of nonnegative integers. A PRM adopts a Poisson distribution for the dependent variable and assumes the log of its expected value can be modeled by a linear combination of relevant parameters. The model is commonly applied for counts such as the occurrence rate of an event (counts) per unit of time. These counts must be independent to facilitate that one count will not make another event to be more or less likely. Instead, the probability of a count per unit of time is related to independent variables such as, e.g., the time of day. Examples of likely Poisson processes could be the number of infected patients per day at a clinic, a country’s number of bankruptcies per
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