International Journal of Academic and Applied Research (IJAAR) ISSN: 2000-005X Vol. 3 Issue 5 , May, 2019 Pages: 14-18 http://www.ijeais.org/ijaar 14 Estimation of the Bi-response Poisson Regression Model Based on Local Linear Approach Darnah 1 , Moh. Imam Utoyo 2 and Nur Chamidah 3 1Department of Mathematics, Mulawarman University, Makassar 75123, Indonesia correspondence author: darnah.98@gmail.com 2,3Department of Mathematics, Airlangga University, Surabaya 60115, Indonesia 2m.i.utoyo@fst.unair.ac.id, 3nur-c@fst.unair.ac.id Abstract: Bi-response Poisson regression model based on the bivariate Poisson distribution and the response variable is a count data that correlate. The function of the bi-response Poisson regression model can be estimate using the parametric approach and the nonparametric approach. In this paper, we discuss the nonparametric approach using local linear estimator. The parameters of the bi-response Poisson regression model are estimated by the method of maximum likelihood. Keywords: bi-response, Poisson regression, local linear. 1. INTRODUCTION The regression model can be used to describe the relation between the response variable and the predictor variable. The response variable of the regression model allows for continue data and count data. Some the regression model of the count data are Poisson regression model, Generalized Poisson regression model, and Negative Binomial regression model. According to Zamani, et al (2015), Poisson regression model has been widely used for modeling count data with covariates. The Poisson regression model is the basic framework for count data analysis and it arises from the evidence that the number of occurrences of a given event over a specific time period depends on some covariates (Santos & Neves, 2008). The Poisson regression model based on the Poisson distribution. The Poisson distribution satisfies the equal-dispersion property because the conditional mean of the response variable equals its conditional variances (Famoye, 1993). Poisson regression modeling often involves not only one response variable but also can two or more response variables because of a phenomenon involving multiple response variables. Poisson regression analysis involving one response variable with one or more predictor variables is called Poisson regression. Poisson regression involving two response variables correlated with one or more predictor variables is called bi-response Poisson regression. The extension of the bi-response Poisson regression is the multiresponse Poisson regression, the Poisson regression involving more than two response variables with one or more predictor variables. The function of the Poisson regression model can be estimated using two approaches, there are the parametric approach and the nonparametric approach. The parametric approach used if the function of the regression model is known except for finitely many unknown parameters and the relation between the response variable and the predictor variable is assumed to have a specific type curve. If we used the parametric regression approach to this condition then consequently, giving misleading inference about the regression model (Chamidah & Saifuddin, 2013). One collection of procedures that can be used for this purpose are nonparametric regression approach. These approaches give estimate of regression function that allow great flexibility in the possible form of the regression curve because the estimate get from data. A nonparametric regression model generally only assumes that the regression curve satisfies smooth properties that are continuity and differentiability (Eubank, 1999). In this research, we will be studied about bi-response Poisson regression estimation based on local linear approach. 2. BI-RESPON POISSON REGRESSION MODEL Let the three random variables 1 V , 2 V , and U to follow three independent Poisson distributions with the positive parameters 1 2 , ,  and  respectively. According to Jung & Winkelmann (1993), new random variables 1 Y and 2 Y can be constructed by 1 1 Y V U  and 2 2 Y V U  where 1 Y and 2 Y are Poisson random variables. The mean 1 Y and 2 Y are 1 1 ( )E Y    and 2 2 ( )E Y    . The density probability function of 1 Y and 2 Y given by: 1 1 2 2 1 2P(Y = y , Y = y ) ( , )f y y   1 2 1 2 1 2 1 2 0 exp ( ) ; ! ( )! ( )! s k y k y kk k y k y k                 (1) where 1 2y , y 0,1,2, and 1 2s = min(y , y ) . The covariance between 1 Y and 2 Y is: 1 2 1 2 (Y , ) (V , V )Cov Y Cov U U   ( )Var U International Journal of Academic and Applied Research (IJAAR) ISSN: 2000-005X Vol. 3 Issue 5 , May, 2019 Pages: 14-18 http://www.ijeais.org/ijaar 15  (2) The correlation between 1 Y and 2 Y is: 1 2 1 2 (Y , ) ( )( ) Corr Y         (3) The parameter  is non-negative (Berchout & Flug, 2004). Following the standard approach in univariate Poisson regression we model the marginal expectation of 1 Y and 2 Y , respectively, as a loglinear function of exogenous variables. (Jung and Winkelmann, 1993). exp( ); 1, 2 dan 1, 2, ..., T ri ri r x r i n      (4) The likelihood function for the observed random sample is given by: 1 2 2 11 ( , , ) = exp( exp( )) T ri r i n ri x B        and the log likelihood function is: 1 2 L = log ( , , )   1 1 2 2 1 1 1 = exp( ) exp( ) log ; T T i i i n n n i i i n x x B           (5) where 1 2 1 1 2 2 1 2 0 [exp( ) ] [exp( ) ] ; ! ( )! ( )! T Ti i i i i i i i k y k y kks x x B k y k y k              min( ), 1, 2 i ri s y r  The maximum likelihood estimates of the parameters can be obtained by solving equations: log log 0 and = 0; r = 1, 2. L L r       3. LOCAL LINEAR ESTIMATOR IN SINGLE RESPON Local linear estimator is special case in local polynomial estimator. Local polynomial estimator is one of the smoothing methods that can be used as a regression function approach (.)m (Fan & Gijbels, 1996). Local polynomial estimators can estimate regression functions 0 ( )m x and their derivatives: ( ) 0 0 0 '( ), ''( ), , ( ) p m x m x m x Suppose the regression function (.)m has a derivative (p + 1) at the point 0 x , the regression function (.)m can be approached locally with the degree of polynomial p. By Taylor expansion, for x in a neighborhood of, we have: 0 0 0 ( ) ( ) '( )( )m x m x m x x x    2 ( ) 0 0 0 0 ''( )( ) ( )( ) 2! ! p p m x x x m x x x p     (6) or ( ) 0 0 0 0 0 0 ( ) ( ) ( ) ( )( ) ; ! j j j j p p j j m x m x x x x x x j         0 0 ( , )x x h x h   where ( ) 0 0 ( ) ( ) ; 0,1,2, , . ! j j m x x j p j    (7) Suppose we taking n-pairs data sample ( , ) i i x y , the estimates of  based on Weighted Least Square (WLS) procedure by minimizing function: 2 0 0 0 1 0 { ( )( ) } ( ); 1, 2, , j i j i h i pn i j y x x x K x x i n        (8) where (.)K is Kernel function and h is bandwidth. Local polynomial estimators can estimate regression functions 0 ( )m x and their derivatives. According to Equation (7), local polynomial estimators of derivatives jth in regression function ( ) 0 ( ) j m x is: ( ) 0 0  ( ) ! ( ); 0,1,2, , ; j j m x j x j p  (9) For j=0 , the estimate of regression function at the point 0x is: 0 0 0  ( ) ( )m x x . (10) Local linear estimator we obtain if the degree of polynomial (p) equal one (p=1). Local linier estimator can be written is: 2 0 0 1 0 0 0 1 { ( ) ( )( )} ( ) i i h i n i y x x x x K x x       (11) 4. STUDY OF BI-RESPONSE POISSON REGRESSION USING LOCAL LINEAR ESTIMATOR Suppose we have pair observational data ( , ) i ri x y , 1, 2 dan 1, 2, ..., r i n  which is distribute independently with x is vector of covariates and ri y is the count bi-response that follows the Poisson distribution. The probability density function of ri y given by equation (1). Generally, equation (4) can be wrote: exp[ ( )]; 1, 2 dan 1, 2, ..., ri r i m x r i n     (12) The function (.) r m in equation (5) is a smooth function. Assume that the function (.) r m has a ( 1) th p  continuous derivative at the point 0 x . We approximate the function (.) r m by Taylor expansion with order one or 1p  , for data point i x in a neighborhood of 0 x with 0 0 ( , ) i x x h x h   and h is a bandwidth: ( ) ( ) 0 0 1 0 0 ( ) ( ) ( )( ); 1, 2. r r r i i i i m x x x x x r     or International Journal of Academic and Applied Research (IJAAR) ISSN: 2000-005X Vol. 3 Issue 5 , May, 2019 Pages: 14-18 http://www.ijeais.org/ijaar 16 ( ) 0 0 ( ) ( ) ( ); 1, 2. T r r i i m x x x x r  where  01 ( ) T i i x x x  and  ( ) ( ) ( )0 0 1 0( ) ( ) T r r r i i x x   Let 0 ( ) h i K x x is a Kernel weight and h is bandwidth, the local likelihood function can be obtained from Eq (1): 0 1 2 0 1 2 1 ( ) ( , , , ) = ( , ) h i i i i i n i K x x x f y y      (13) The log local likelihood function is: 1 2 0 1 2 0 L( , , , )= ln ( , , , ) i i i i x x        1 2 1 2 0 1 2 1 2 1 0 = ( ) ( ) ln ! ( )! ( )! i i i i h i i i i i y k y kk n s i k K x x k y k y k                            (14) Next, by subtituting Equation (12) into Equation (14), we have:    1 2 0 0 1 2 1 L( , , , ) ( ) exp ( ) exp ( ) i i h i i i n i m m x K x x m x m x            1 21 2 1 2 0 exp ( ) exp ( ) ln ! ( )! ( )! i i i i i i y k y k k s k m x m x k y k y k                   (15) Let (1) (2) 0 ; i ik ik s k D D D    Where (1) 1 1 0 0(1) 1 [exp( ( ) ( )) ] ! ( )! T i i ik i y kk x x x D k y k       and ( 2) 2 2 0 0( 2) 2 [exp( ( ) ( )) ] ( )! T i i ik i y k x x x D y k       The Equation (15) can be written : (1) (2) 0 0 0 0 1 ( ( ) ( ), , ) ( ) h i n i L x x x K x x        ( )0 0 2 1 exp ( ) ( ) ln T r i i r x x x D        (16) Parameter estimation  , (1) 0 ( )x , dan (1) 0 ( )x in biresponse Poisson regression model can be obtained by local likelihood maximum method given by following Theorem 1. Theorem 1 By assuming bi-variate Poisson distribution for response variable ri y , these considerations yield the conditional local weighted log-likelihood in Equation (16), the maximum likelihood estimator for parameter can be found from a solution of maximum likelihood equation:     (1) (1) 0 0 1 0 1 0 (1) (1) 2 0 0 1 0 1 0 exp ( ) ( ) ( ) 1 exp ( ) ( ) i i i i h i i i i n s i k x x x yL K x x x x x                               2 ( 2) ( 2) 0 0 1 0 exp ( ) ( ) i i i i k k y x x x                 (17) and  (1)0(1) 1 ( ) exp T h i i i n i L K x x x x             (1)1 (1) 0 exp exp T i i i T i s k y k x x x                  (18)  ( 2)0( 2) 1 exp( ) Ti ih i n i x x L K x x               ( 2)1 2 ( 2) 0 exp exp T i i i i T i s k y k y k x x x                (19) Proof: The likelihood function in Equation (16) will have maximum value when the first derivative of parameter  , (1) 0 ( )x , and (2) 0 ( )x equal to zero. The partial derivatives of equation (11) is: 0 1 1 ( ) 1 i h i i n i L D K x x D                  (20) i D    in equation (20) can be obtained using equation (15) and equation (16) as follows: (1) ( 2) ( 2) (1) 0 i ik ik ik ik s k D D D D D                  (21) where   11 (1) (1) 0 0 1 1 exp ( ) ( ) !( )! ik T i ik i ky x x xD k y k                 (1)0 0 1exp ( ) ( )Ti ik x x x y    (22)    2( 2) ( 2) 2 0 0 2 1 exp ( ) ( ) ( )! iT i i ik i y k k y x x xD y k              (23) By substituting equation (15), equation (16), equation (22), and equation (23) to equation (21), we have:   1(1)0 0 1 20 exp ( ) ( ) !( )!( )! ik T ii i i y k s k x x x k y k y k D                       International Journal of Academic and Applied Research (IJAAR) ISSN: 2000-005X Vol. 3 Issue 5 , May, 2019 Pages: 14-18 http://www.ijeais.org/ijaar 17   2(2)0 0exp ( ) ( ) iTi y k x x x          (1) 0 0 1 2 (1) 2 (2) 0 0 0 0 exp ( ) ( ) exp ( ) ( ) exp ( ) ( ) T i i i T T i i k x x x y k y x x x x x x                              (24) According to equation (14) and equation (23), equation (20) can be written: (1) 0 0 1 0 0 (1) 2 0 0 1 0 exp ( ) ( ) ( ) 1 exp ( ) ( ) T i i h T i n s i k k x x x yL K x x x x x                           2 ( 2) 0 0 exp ( ) ( ) i T i k y x x x           (25)  ( )0 00( ) 0 1 exp ( ) ( )( ) ( ) T r i ih ir n i x x x x L K x x x            ( ) 0 1 ( ) ;i r i D D x      (26) (1) 0( ) iD x   and (1) 0( ) iD x   in equation (26) can be obtained based on equation (14) :    1(1) 1 0 0 (1) 0 1 1 0 exp ( ) ( ) ( ) ! ( )! iT k i i i i y k s k y k x x xD x k y k                      2( 2) 0 0(1) 0 0 0 2 exp ( ) ( ) exp ( ) ( ) ( ) ( )! iT iT i i i y k x x x x x x x x y k                  (27)    1(1) 1 0 0 (2) 0 1 0 exp ( ) ( ) ( ) ! ( )! iT k i i i i y k s k y k x x xD x k y k                      2( 2) 2 0 0 2 1 exp ( ) ( ) ( )! iT i i i y k y k x x x y k            (2)0 0 0exp ( ) ( ) ( )Ti ix x x x x    (28) According to equation (27), equation (26) can be expressed as:  (1)0 0 0 0(1) 0 1 ( ) exp ( ) ( ) ( ) ( ) T h i i i n i L K x x x x x x x x              (1)1 0 0 0 (1) 0 0 0 exp ( ) ( ) ( ) exp ( ) ( ) T i i i T i s k y k x x x x x x x x                  (29) According to equation (28), equation (26) can be expressed as:  ( 2 )0 0 0 0( 2) 0 1 ( ) exp ( ) ( ) ( ) ( ) T h i i i n i L K x x x x x x x x               ( 2 )1 2 0 0 0 ( 2) 0 0 0 exp ( ) ( ) ( ) exp ( ) ( ) T i i i i T i s k y k y k x x x x x x x x                   (30) The likelihood function in equation (16) has maximum value when equation (25), equation (29), and equation (30) equals zero, we can be written as : (1) 0 0 1 0 (1) 2 0 0 1 0 exp ( ) ( ) ( ) 1 exp ( ) ( ) T i i h i T i n s i k k x x x y K x x x x x                        2 ( 2) 0exp ( ) ( )0 0 T i k y i x x x           (31)  (1)0 0 0 0 1 ( ) exp ( ) ( ) ( ) T h i i i n i K x x x x x x x         (1)1 0 0 0 (1) 0 0 0 exp ( ) ( ) ( ) 0 exp ( ) ( ) T i i i T i s k y k x x x x x x x x                  (32) (34)  (2)1 0 0 0 0 1 ( ) exp ( ) ( ) ( ) T h i i n i K x x x x x x x          ( 2)1 2 0 0 0 ( 2) 0 0 0 exp ( ) ( ) ( ) 0 exp ( ) ( ) T i i i i T i s k y k y k x x x x x x x x                   (33) The first partial derivative obtained in Equation (31) to Equation(33) are nonlinear in parameters so that an iteration process is required to obtain the solution. The commonly used numerical method is the Newton-Raphson method. In the Newton-Raphson method, there is a Hessian matrix whose element is a partial derivative of the two functions of ln likelihood to the  , (1) ( ) T  , and (2) ( ) T  parameters. 5. CONCLUSION Estimator bi-response Poisson regression model using local linear approach can be obtained from the first partial derivative of the likelihood function. The first partial derivative are nonlinear in parameters so that we required an iteration process to obtain the solution, it is the NewtonRaphson method. International Journal of Academic and Applied Research (IJAAR) ISSN: 2000-005X Vol. 3 Issue 5 , May, 2019 Pages: 14-18 http://www.ijeais.org/ijaar 18 REFERENCES [1] Berkhout, P. & Plug, E. 2004. A Bivariate Poisson Count Data Model using Conditional Probabilities. Statistica Neerlandica, 58(3):349-364. [2] Fan, J. & Gijbels, I. 1996. Local Polynomial Modelling and Its Application. Chapman and Hall. London. [3] Chamidah, N. & Saifuddin, T. 2013. Estimation of Children Growth Curve Based on Kernel Smoothing in Multi-Response Nonparametric Regression. Applied Mathematical Science, 7(37): 1839 – 1847. [4] Famoye, F. 1993. Restricted Generalized Poisson Regression Model. Journal of Communication in Statistics-Theory and Methods, 22(5): 1335-1354. [5] Eubank, R. 1999. Spline Smoothing and Nonparametric Regression Second Edition. Marcel Dekker. New York. [6] Jung, R. C. & Winkelmann, R. 1993. Two Aspects of Labor Mobility: A Bivariate Poisson Regression Approach. Empirical Economics, 18:543-556. [7] ZASantos, J. A. & Neves, M. M. 2008. A Local Maximum Likelihood Estimator for Poisson Regression. Metrika, 68: 257-270. [8] Zamani, H., Faoughi, P., & Ismail, N. 2016. Bivariate Generalized Poisson Regression Model: Application On Health Care Data. Journal of Empirical Economics, 115.