Template:Lognormal distributionrank regression on x

Rank Regression on X
Performing a rank regression on X requires that a straight line be fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized.

Again, the first task is to bring our $$cdf$$  function into a linear form. This step is exactly the same as in regression on Y analysis and all the equations apply in this case too. The deviation from the previous analysis begins on the least squares fit part, where in this case we treat $$x$$  as the dependent variable and  $$y$$  as the independent variable. The best-fitting straight line to the data, for regression on X (see Chapter Parameter Estimation), is the straight line:


 * $$x=\widehat{a}+\widehat{b}y$$

The corresponding equations for   and  $$\widehat{b}$$  are:


 * $$\hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}$$

and:


 * $$\hat{b}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}{{y}_{i}}-\tfrac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}}{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,y_{i}^{2}-\tfrac{N}}$$

where:


 * $${{y}_{i}}={{\Phi }^{-1}}\left[ F(t_{i}^{\prime }) \right]$$

and:


 * $${{x}_{i}}=t_{i}^{\prime }$$

and the $$F(t_{i}^{\prime })$$  is estimated from the median ranks. Once $$\widehat{a}$$  and  $$\widehat{b}$$  are obtained, solve the linear equation for the unknown  $$y$$, which corresponds to:


 * $$y=-\frac{\widehat{a}}{\widehat{b}}+\frac{1}{\widehat{b}}x$$

Solving for the parameters we get:


 * $$a=-\frac{\widehat{a}}{\widehat{b}}=-\frac{\sigma'}$$

and:


 * $$b=\frac{1}{\widehat{b}}=\frac{1}{\sigma'}$$

The correlation coefficient is evaluated as before using equation in the previous section.

Example 3: