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:

    “I herewith commission you to carry out all preparations with regard to ... a total solution of the Jewish question in those territories of Europe which are under German influence.... I furthermore charge you to submit to me as soon as possible a draft showing the ... measures already taken for the execution of the intended final solution of the Jewish question.”
    —Hermann Goering (1893–1946)

    “I can’t quite define my aversion to asking questions of strangers. From snatches of family battles which I have heard drifting up from railway stations and street corners, I gather that there are a great many men who share my dislike for it, as well as an equal number of women who ... believe it to be the solution to most of this world’s problems.”
    —Robert Benchley (1889–1945)

    “Give a scientist a problem and he will probably provide a solution; historians and sociologists, by contrast, can offer only opinions. Ask a dozen chemists the composition of an organic compound such as methane, and within a short time all twelve will have come up with the same solution of CH4. Ask, however, a dozen economists or sociologists to provide policies to reduce unemployment or the level of crime and twelve widely differing opinions are likely to be offered.”
    —Derek Gjertsen, British scientist, author. Science and Philosophy: Past and Present, ch. 3, Penguin (1989)