For the simple linear regression model under the assumptions

For the simple linear regression model, under the assumptions of Theorem 10.3.3, establish that Cov(Y_i- B_1 - B_2 x_i, Y_j - B_1 - B_2x_j |X_1 = x_1, ..., X_n = x_n) = sigma^2 delta_ij - sigma^2(1/n + (x_i - x) (x_j - x)/sigma_k = 1^n (x_k - x)^2 where delta_i, j = 1 when i = j and is 0 otherwise.

Solution

Ans-

In statistics, linear regression is an approach for modeling the relationship between a scalar dependent variable y and one or moreexplanatory variables (or independent variables) denoted X. The case of one explanatory variable is called simple linear regression. For more than one explanatory variable, the process is called multiple linear regression.^[1] (This term should be distinguished frommultivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.)^[2]

In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimatedfrom the data. Such models are called linear models.^[3] Most commonly, the conditional mean of y given the value of X is assumed to be an affine function of X; less commonly, the median or some other quantile of the conditional distribution of y given X is expressed as a linear function of X. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of ygiven X, rather than on the joint probability distribution of y and X, which is the domain of multivariate analysis.

Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.^[4] This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.

Linear regression has many practical uses. Most applications fall into one of the following two broad categories:

Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the \"lack of fit\" in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares loss function as in ridge regression (L²-norm penalty) and lasso (L¹-norm penalty). Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms \"least squares\" and \"linear model\" are closely linked, they are not synonymous.