Deming Regression - Solution

Solution

The solution can be expressed in terms of the second-degree sample moments. That is, we first calculate the following quantities (all sums go from i = 1 to n):

\begin{align} & \overline{x} = \frac{1}{n}\sum x_i, \quad \overline{y} = \frac{1}{n}\sum y_i, \\ & s_{xx} = \tfrac{1}{n-1}\sum (x_i-\overline{x})^2, \\ & s_{xy} = \tfrac{1}{n-1}\sum (x_i-\overline{x})(y_i-\overline{y}), \\ & s_{yy} = \tfrac{1}{n-1}\sum (y_i-\overline{y})^2. \end{align}

Finally, the least-squares estimates of model's parameters will be

\begin{align} & \hat\beta_1 = \frac{s_{yy}-\delta s_{xx} + \sqrt{(s_{yy}-\delta s_{xx})^2 + 4\delta s_{xy}^2}}{2s_{xy}} \\ & \hat\beta_0 = \overline{y} - \hat\beta_1\overline{x}, \\ & \hat{x}_i^* = x_i + \frac{\hat\beta_1}{\hat\beta_1^2+\delta}(y_i-\hat\beta_0-\hat\beta_1x_i). \end{align}

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