The bivariate regression model and its application

The paper studied a bivariate regression model (BRM) and its application. The maximum power and minimum size are used to choose the eligible tests using non-sample prior information (NSPI). In the simulation study on real data, we used Wilk’s lamda to determine the best model of the BRM. The result showed that the power of the pre-test-test (PTT) on the NSPI is a significant choice of the tests among unrestricted test (UT) and restricted test (RT), and the best model of the BRM is Y(1) = −894 + 46X and Y(2) = 78 + 0.2X with significant Wilk’s lamda 0.88 < 0.90 (Wilk’s table).


Introduction
The paper studied a bivariate regression model (BRM) and its application with and without nonsample prior information (NSPI). In the context of the testing hypothesis with NSPI, the power and size of the tests unrestricted test (UT), restricted test (RT) and pre-test-test (PTT) are used. Many authors have already studied about UT, RT, PTT in testing intercept using non-sample prior information (NSPI) such as Pratikno (2012), Khan and Pratikno (2013), Yunus and Khan (2011), Saleh and Sen (1983) and Khan, et al. (2016). Some previous research related to preliminary testing and NSPI are also found in Han and Bancroft (1968), Judge and Bock (1978), and Saleh (2006), Tamura (1965), and Sen (1978, 1982). Other authors such as Khan (2005Khan ( , 2008, Khan and Saleh (1997, 2005, 2008, Khan and Hoque (2003), Saleh (2006) and Yunus (2010) contributed to this research topic in the context of estimation area, and Tamura (1965), Sen (1978, 1982), and Yunus and Khan (2011) discussed testing hypothesis with NSPI on the non parametric model. All the authors used R-code in R-package especially mvtnorm and cubature function, respectively. To choose the best test, we choose a maximum power and minimum size of the tests (UT, RT, PTT) on multivariate simple regression model (MSRM), especially on a bivariate simple regression model (BSRM). Note that the definition of the power is probability reject H0 under Ha in testing In the simulation on real data, the BRM (or BSRM with single predictor) requires the correlation between responses. Here, the Wilk's Lamda and mean deviation error (MDE) are used to check the best model of the BRM and or BSRM. Following Rencher (2002), the general model of the BSRM is given by With Y is a bivariate response, X is predictors and ε is error term. To test intercept with NSPI and create the best model, the power and size of the tests and MDE are used, respectively. A Simulation study is done on climate data, namely temperature (X), rainfall (Y1), and humidity (Y2). The research is done in some steps: (1) we firstly review some previous research related to the power of the tests (UT, RT, PTT) on testing intercept with NSPI on regression model, (2) we do graphical analysis of the power of the tests in a simulation study with generating data from R, and (3) we finally create and analyse the best model of the BSRM for climate data in Cilacap, January 2009 to February 2014).
The research presented the introduction in Section 1. Testing with NSPI on the BSRM is given in Section 2. A simulation study to determine the best model of the BSRM on real data is then obtained in Section 3, and Section 4 described conclusion.

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We then present the power and size of the tests in Figure 1.

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They start from the same value, that is 0.05, and they then increase as 2  increases. We also see that the size of the PTT maximum and UT minimum for 2 0   (Figure 1,(ii)). It is clear that PTT is not a significant choice, but we still recommend PTT as a choice, as well as RT. This condition does not follow the previous research and theory. Furthermore, we recommend changing the distribution of the generated data for both predictor and error.

A Simulation Study on Real Data
Following Pratikno (2012), the MSRM is a model of multiple responses with a single predictor, and for p = 2 the MSRM become the BSRM. Furthermore, the general model of the MSRM of n observation of random variables with single predictor and vector response, (Xi, Yi ), i=1,2,...,n, is given by  To create the eligible model on the BSRM, we must check the correlation between responses. Let, ,  

Result and discussion
To set the model of the BSRM above, we try and analyze two types of data as follows. Firstly, we use data of water discharge (X), sedimentation rate (Y1) and outflow (Y2). In this case, the independence of the response variable () 12 Y , Y is tested using Bartlett test in the equation (3). The result showed that they are not significant correlation. Therefore, we then used another data (climate data) to simulate the best model of the BSRM. Here, the responses have a significant correlation From this data, we obtain