CONFIDENCE INTERVAL OF THE PARAMETER ON MULTIPREDICTOR BIRESPONSE LONGITUDINAL DATA ANALYSIS USING LOCAL LINEAR ESTIMATOR FOR MODELING OF CASE INCREASE AND CASE FATALITY RATES COVID-19 IN INDONESIA: A THEORETICAL DISCUSSION

In this paper, we describe a theoretical discussion about confidence intervals for longitudinal data based on local linear estimator. The confidence interval represents the range of possible values in the estimating process. The confidence intervals for the parameter in nonparametric regression can be used to determine the predictor variables that have a significant effect on the response variable. In this research, we theoretically discuss estimation of the confidence interval of the parameter on multipredictor biresponse nonparametric regression model for longitudinal data based on local linear estimator which is applied to data of the case increase and case fatality rates COVID-19 in Indonesia. The estimation result can be used to determine the predictor variable, e.g. temperature which has a significant effect on the case increase and case fatality rates COVID-19 in Indonesia so that it can be advised to the ministry of health to control the case increase and case fatality rates COVID-19 in Indonesia.

In real life, there are usually many cases involving regression models with two response variables that are correlated with each other and influenced by more than one predictor variable, so that the problem can be solved by using multipredictor biresponse nonparametric regression.
However, these researchers are only limited to study point estimation.
One of the most important parts of statistical inference is the confidence interval. The confidence interval represents the range of possible values in which, with some certainty, we can find the statistical size of the population [32]. Confidence intervals for parameters in nonparametric regression can be used to determine the predictor variables that have a significant effect on the response variable. The conclusion is decided by looking at whether the parameter confidence interval contains a zero value. If the confidence interval contains a value of zero, then the predictor variable has no significant effect on the response variable. There are several researchers who have discussed confidence intervals. They are, for examples, [33] used spline estimator; [34] used local polynomial estimator for estimating confidence intervals.
Although much research has been studied on confidence intervals however the research was 3 MULTIPREDICTOR BIRESPONSE LONGITUDINAL DATA ANALYSIS applied to cross-section data. Therefore, the aim of this research is to create a confidence interval of parameter for nonparametric regression model with two response variables and more than one predictor variable which is applied to longitudinal data, namely data of case increase and case fatality rates COVID-19 in Indonesia. Hence, in this research, we discuss theoretically how to estimate the confidence interval of the parameter on nonparametric regression model by using a local linear estimator. The linear local estimator has the advantage of estimating the function at each point so that the model we get is closer to the actual data pattern then can be used to construct a confidence interval of case increase and case fatality rates COVID-19 in Indonesia.

PRELIMINARIES
Multi-predictor biresponse nonparametric regression model on longitudinal data is a nonparametric regression model that describes the relationship pattern of two correlated response variables and more than one predictor variable with a dataset collected from several subjects in a certain period. Suppose we have a paired longitudinal data (1) ( ,. For two response variables, equation (2) can be written as follows: Hence, equation (3) can be expressed as follows: Furthermore, the estimated parameter of multi-predictor biresponse nonparametric regression model in equation (4) is obtained using the weighted least square method, and therefore we have the estimated parameter as follows [30]:

MAIN RESULTS
The first step to construct the confidence interval of the parameter on multi-predictor biresponse and since y is a linear combination of ,  then y is also to follow a normal distribution. Hence, we have:  (9) var ( ) Next, from equation (8) and (9), we get: Then, we will construct the confidence interval of 0 ( ) , 1,2, ,2( 1) g x g p  =+ using pivotal quantity. The pivotal quantity of the parameter on multi-predictor biresponse nonparametric regression model is U(x) as follows: Next, by substituting equation (8) and (9) into equation (11), we get: Here, U(x) represents the pivotal quantity for the parameter of regression model 0 () x  , however in the case application it is often that 2  is unknown, so we estimate it using mean square error (MSE). Therefore, we get pivotal quantity as follows: Based on equation (14), the pivotal quantity in the equation (13) can be written as follows: The next step is determining the distribution of pivotal quantity. If the numerator and denominator are divided by the root of the population variance 2  then we will obtain the pivotal quantity which follows a t-student distribution with degree of freedom 2R-(2( 1)) p + and therefore it can be written as follows: where B and A are as follows: ( ) Based on equation (17) and (18) (19) . Based on equation (19), (20) and (21)   Therefore, the pivotal quantity in (23) can be written as follows:  ,2R-(2( 1)) 2 2 ( ) ( ) ( ) 1 The confidence interval for parameter of nonparametric regression model in equation (28) can be used to determine the predictor variables that have a significant effect on the response variable which is applied to data of the case increase and case fatality rates COVID-19 in Indonesia.

CONCLUSIONS
Theoretically, based on equation (28) we can obtain the confidence interval of case increase and case fatality rates COVID-19 in Indonesia and then we can determine the predictor variables that have a significant effect on the case increase and case fatality rates COVID-19 in Indonesia. The estimation result can be advised to the ministry of health to control the case increase and case fatality rates COVID-19 in Indonesia.