Deming regression: Difference between revisions

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== 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''):

: <math>\begin{align}

& \overline{x} &= \fractfrac{1}{n}\sum x_i, \quad& \overline{y} &= \fractfrac{1}{n}\sum y_i, \\

& s_{xx} &= \tfrac{1}{n}\sum (x_i-\overline{x})^2 &&= \overline{x^2} - \overline{x}^2, \\

& s_{xy} &= \tfrac{1}{n}\sum (x_i-\overline{x})(y_i-\overline{y}) &&= \overline{x y} - \overline{x} \, \overline{y}, \\

& s_{yy} &= \tfrac{1}{n}\sum (y_i-\overline{y})^2 &&= \overline{y^2} - \overline{y}^2.

\end{align}\,</math>

Finally, the least-squares estimates of model's parameters will be{{sfn|Glaister|2001}}

: <math>\begin{align}

& \hat\beta_1 = \frac{s_{yy}-\delta s_{xx} + \sqrt{(s_{yy}-\delta s_{xx})^2 + 4\delta s_{xy}^2}}{2s_{xyy}}, \\

& \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).