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\title{The Generalized Estimating Equations for the Unknown Correlation Structure of the Data}
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             \author[1]{Md. Soyebur  Rahman}

             \affil[1]{  }

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\date{\small \em Received: 6 December 2015 Accepted: 3 January 2016 Published: 15 January 2016}

\maketitle


\begin{abstract}
        


In many study the data are taken different period of time and the information about them is gathered relating to an event of concern at different time periods. The data are taken different time period are correlated. Regression analysis based on the Generalized Estimating Equation (GEE) is an increasing important method of such data. The Generalized Estimating Equation is an important and widely used approach in such analysis. Since the true correlation is unknown GEE offers to take a working correlation for analysis. In our study we consider four common correlation structure namely, independent, exchangeable, pair wise, autoregressive.In the study the data are taken from the Dhaka stock exchange (DSE) this data are highly correlated. At first we apply different methods of estimating parameter the we apply GEE for estimating the parameters. Finally we get the GEE gives better estimate than any other method.

\end{abstract}


\keywords{generalized estimating equation, GEE, OLS, GLM.}

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\let\tabcellsep& 	 	 		 
\section[{I. Introduction}]{I. Introduction}\par
he term Generalized Estimating Equations indicates that an estimating equation is not the result of a likelihood-based derivation, but that it is obtained by generalizing other estimating equation. \hyperref[b3]{Liang and Zeger (1986)}, Zeger and \hyperref[b3]{Liang (1986)} and prentice (1988) developed a most recent method of estimating the parameters of the marginal model. They present a class of estimating equations that take the correlation into account to increase the efficiency. This class of estimating equations is called Generalized Estimating Equations (GEE) and the correlation matrix is called working correlation matrix. The name working is used in the sense that it is an approximate correlation matrix of i Y 's (the response variable).The estimates of ? obtained by GEE are consistent and in addition consistent variance estimates can be obtained under the weak assumption the weighted average of the estimated correlation matrices converge to a fixed matrix. 
\section[{II. Data Analysis}]{II. Data Analysis}\par
The stock exchange data are highly correlated from one company to other company.\par
In our study, we have used stock exchange data because, we want to compare the different method of estimation. We have taken six company and their closing price and the general index in 2009 of DSE. In our calculation we have taken general index as an explanatory variable and the different company closing price as a independent variable. We want to estimate the following model 0 1 1 3 3 6 6 ... y X X X ? ? ? ? = + + + +\textbf{(1) where,} 
\section[{III. Summary and Conclusion}]{III. Summary and Conclusion}\par
Selection of correlation structure is an important issue in Generalized Estimating Equation (GEE). We have earlier stated that there are four correlation structure namely, independent, exchangeable, autoregressive and pair wise. One needs to select under which correlation structure is unknown GEE works well or provides efficient estimate in several situation.\par
We have concentrated on standard error as a measure of accuracy for an estimator of the parameter. So in this case, we look only on the standard error under different correlation structure of different methods of estimation procedure.\par
In this study we consider stock exchange data, this data are highly correlated.\par
In the previous situation, we see that the standard error of the parameters of the GEE method is lowest than any other method. From this study we may conclude that from different method of estimation the GEE parameters are gives the efficient estimate and best approach.\begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.08564231738035265\textwidth}P{0.47263853904282116\textwidth}P{0.038539042821158685\textwidth}P{0.0663727959697733\textwidth}P{0.049779596977329976\textwidth}P{0.016593198992443325\textwidth}P{0.03051007556675063\textwidth}P{0.041215365239294706\textwidth}P{0.04870906801007556\textwidth}}
\tabcellsep \tabcellsep \tabcellsep \multicolumn{3}{l}{Using the GLM procedure}\tabcellsep \\
\tabcellsep \tabcellsep \tabcellsep \multicolumn{2}{l}{The GLM Procedure}\tabcellsep \tabcellsep \\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep Standard\tabcellsep \tabcellsep \\
Parameter\tabcellsep \multicolumn{2}{l}{Estimate}\tabcellsep \tabcellsep Error\tabcellsep \multicolumn{2}{l}{t Value}\tabcellsep Pr > |t|\\
Intercept\tabcellsep \multicolumn{2}{l}{-97.97753227}\tabcellsep \tabcellsep 244.4865048\tabcellsep \tabcellsep -0.40\tabcellsep 0.6890\\
x1\tabcellsep \multicolumn{2}{l}{-27.79733308}\tabcellsep \tabcellsep 7.7704859\tabcellsep \tabcellsep -3.58\tabcellsep 0.0004\\
X2\tabcellsep \multicolumn{2}{l}{-7.88169809}\tabcellsep \tabcellsep 1.0654488\tabcellsep \tabcellsep -7.40\tabcellsep <.0001\\
X3\tabcellsep \multicolumn{2}{l}{2.29819026}\tabcellsep \tabcellsep 0.3979271\tabcellsep \tabcellsep 5.78\tabcellsep <.0001\\
X4\tabcellsep \multicolumn{2}{l}{13.56079439}\tabcellsep \tabcellsep 1.5965023\tabcellsep \tabcellsep 8.49\tabcellsep <.0001\\
X5\tabcellsep \multicolumn{2}{l}{3.20755189}\tabcellsep \tabcellsep 0.2794746\tabcellsep \tabcellsep 11.48\tabcellsep <.0001\\
X6\tabcellsep \multicolumn{2}{l}{1.63484104}\tabcellsep \multicolumn{3}{l}{0.8332389 Using GEE procedure The GENMOD Procedure}\tabcellsep 1.96\tabcellsep 0.0509\tabcellsep Year 2016\\
\tabcellsep \tabcellsep \multicolumn{4}{l}{Analysis of Initial Parameter Estimates}\tabcellsep \\
\tabcellsep \tabcellsep \tabcellsep \multicolumn{4}{l}{Standard Wald 95\% Confidence Chi-}\\
Intercept X Variable 1 X Variable 2 X Variable 3 Parameter T Intercept x1 X2 X3 X4 X5 X6 Scale\tabcellsep \multicolumn{7}{l}{Calculation of the model (1) in the different methods as follows: Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% -101.01708 244.3919221 -0.413341 0.679737 -582.5174 380.48328 -27.610049 7.774982161 -3.55114 0.000464 -42.9283 -12.291798 -7.917902 1.068852888 -7.40785 2.34E-12 -10.02375 -5.8120506 2.2945252 0.397909548 5.766449 2.56E-08 1.510565 3.0784856 DF Estimate Error Limits Square Pr > ChiSq 1 -97.9775 240.8947 -570.122 374.1674 0.17 0.6842 1 -27.7973 7.6563 -42.8035 -12.7912 13.18 0.0003 1 -7.8817 1.0498 -9.9393 -5.8241 56.37 <.0001 1 2.2982 0.3921 1.5297 3.0667 34.36 <.0001 1 13.5608 1.5730 10.4777 16.6439 74.32 <.0001 1 3.2076 0.2754 2.6678 3.7473 135.68 <.0001 1 1.6348 0.8210 0.0257 3.2440 3.97 0.0464 1 162.6532 7.4241 148.7342 177.8748}\tabcellsep Global Journal of Computer Science and Technology ( ) Volume XVI Issue I Version I\\
X Variable 4\tabcellsep 13.603066\tabcellsep \multicolumn{2}{l}{1.598677955}\tabcellsep 8.508947\tabcellsep 2.16E-15\tabcellsep 10.45336\tabcellsep 16.752778\\
X Variable 5\tabcellsep 3.2163438\tabcellsep \multicolumn{2}{l}{0.279808985}\tabcellsep 11.49478\tabcellsep 1.61E-24\tabcellsep 2.665065\tabcellsep 3.7676227\\
X Variable 6\tabcellsep 1.6285946\tabcellsep \multicolumn{2}{l}{0.833022998}\tabcellsep 1.955042\tabcellsep 0.051774\tabcellsep -0.012625\tabcellsep 3.2698144\end{longtable} \par
  {\small\itshape [Note: y = DSE general index 2009 1 X = Aims first guaranteed mutual fund 2 X =ACI pharmaceuticals Limited 3 X =Glasco Smith-kline Bangladesh Limited 4 X =Beximco Pharmaceuticals Limited 5 X =Al-Arafah Islami Bank Limited 6 X =Bata Shoe Company (Bangladesh) Limited. G Author ?: Jahangirnagar University, Savar, Dhaka. e-mail: soyebur.rahman@gmail.com Author ? : Lecturer, Department of Statistics, Mawlana Bhashani Science and Technology University. Author ?: Assistant Professor, Department of Statistics, Mawlana Bhashani Science and Technology University.]} 
\caption{\label{tab_0}}\end{figure}
 			\footnote{© 2016 Global Journals Inc. (US)} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\end{document}
