Estimation in Simple Linear Regression

Abstract

Let us first review the basic theory on crisp simple linear regression. Our development, throughout this chapter and the next two chapters, follows Sections 7.8 and 7.9 in [1]. We have some data (x _i , y _i), 1 ≤ i ≤ n, on two variables x and Y. Notice that we start with crisp data and not fuzzy data. Most papers on fuzzy regression assume fuzzy data. The values of x are known in advance and Y is a random variable. We assume that there is no uncertainty in the x data. We can not predict a future value of Y with certainty so we decide to focus on the mean of Y, E(Y). We assume that E(Y) is a linear function of x, say $E\left( Y \right) = a + b\left( {x - \overline x } \right)$ Here $\overline x$ is the mean of the x-values and not a fuzzy set. Our model is

$$\mathop Y\nolimits_i = a + b\left( {\mathop x\nolimits_i - \overline x } \right) + \mathop \in \nolimits_i $$

(22.1)

where ∈ _i are independent and N(0, σ ²) with σ ² unknown. The basic regression equation for the mean of Y is $y = a + b\left( {x - \overline x } \right)$ and now we wish to estimate the values of a and b. Notice our basic regression line is not y = a + bx,and the expression for the estimator of a will differ between the two models.