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}

Read more about this topic:  Deming Regression

Famous quotes containing the word solution:

    “Any solution to a problem changes the problem.”
    —R.W. (Richard William)

    “Coming out, all the way out, is offered more and more as the political solution to our oppression. The argument goes that, if people could see just how many of us there are, some in very important places, the negative stereotype would vanish overnight. ...It is far more realistic to suppose that, if the tenth of the population that is gay became visible tomorrow, the panic of the majority of people would inspire repressive legislation of a sort that would shock even the pessimists among us.”
    —Jane Rule (b. 1931)

    “What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
    —Boris Pasternak (1890–1960)