Vol.12 (2022) No. 5 ISSN: 2088-5334 The Modified Structural Quasi Score Estimator for Poisson Regression Parameters with Covariate Measurement Error Fevi Novkaniza a,* , Khairil Anwar Notodiputro b , I Wayan Mangku c , Kusman Sadik b a Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Indonesia, Depok, West Java, 16424, Indonesia b Department of Statistics, IPB University, Bogor, West Java, 16680, Indonesia c Department of Mathematics, IPB University, Bogor, West Java, 16680, Indonesia Corresponding author: * fevi.novkaniza@sci.ui.ac.id Abstract—This article proposed the Modified Structural Quasi Score (MSQS) estimators for Poisson regression parameters when a covariate is subject to measurement error. We study the situation when the true covariate in the Poisson regression model is unobserved, and the surrogate for this covariate is related to the true covariate by the additive measurement error model. We assumed that true covariate as a random variable with unknown density function distribution and its observable values as surrogates, which also has Poisson distribution. We applied the Empirical Bayes Deconvolution (EBD) method for estimating the true covariate density with a finite discrete support set. To estimate Poisson regression parameters, we construct an MSQS estimating equation based on proper functions for the mean and variance of the Poisson distributed surrogate. The MSQS estimator for the Poisson regression parameter is the root of the quasi-score function based on the quasi-likelihood method. We did some simulation scenarios for assessing the MSQS estimator by assuming the true covariate comes from Gamma distribution as a conjugate before Poison distribution. We compute the standard error of the mean, standard deviation, and bias of the MSQS estimator for various sample sizes to examine the estimator's appropriateness. The simulation showed that a combination of the finite discrete support set of surrogates based on the range values and smaller-scale parameter of Gamma distribution yields smaller values of bias estimator and the estimated standard deviation. Keywords— Covariate; measurement error; Poisson; quasi score; surrogate. Manuscript received 6 Jun. 2020; revised 28 Mar. 2021; accepted 16 Apr. 2021. Date of publication 31 Oct. 2022. IJASEIT is licensed under a Creative Commons Attribution-Share Alike 4.0 International License. I. INTRODUCTION The Poisson regression model is one of the generalized linear models used for analyzing count data where the response variable is a non-negative integer [1]-[5]. This model is relevant for the analysis of count data in social and natural sciences, such as infometric [6], transportation [7], insurance [8], predictors of length of stay among HIV patients [9] and other application. In the Poisson regression model, the response variable  has a Poisson distribution with a rate parameter  that depends on a covariate : log       and the regression parameters  ,   are needed to estimate. In this article, we restrict to the case of one unobserved covariate  measured by error, and we called it surrogate . If covariate  is measured without error and Poisson regression parameters are produced by maximum likelihood (ML) estimation, it will be consistent and asymptotically efficient. However, if a covariate is measured with error, the ML estimator ignoring the covariate measurement error will lead to inconsistent estimators and bias. In this article, we consider the case event count as the response variable, and we work with unobserved covariate  measured with error. Because  it is an unobserved covariate, we use observable covariate  as a surrogate, and it modeled via additive measurement error model, , where  is measurement error and independent of ,  and identically distributed samples from a known density. Many methods have been proposed and mostly depend on the distribution of  is known [10]–[13]. However, the methods do not utilize the distribution of covariate  measured with error (functional measurement error) [14]-[18]. In contrast to the functional measurement error model, structural measurement error assumed  is a random variable and has distribution. If this distribution can be specified parametrically, it is possible to calculate them directly. We can get the conditional distribution of  given  as the posterior distribution if we get the prior distribution of  , denoted by . We used the Empirical Bayes 1875