Template:Weibull parameters Rank Regression on Y

Rank Regression on Y
Performing rank regression on Y requires that a straight line mathematically be fitted to a set of data points such that the sum of the squares of the vertical deviations from the points to the line is minimized. This is in essence the same methodology as the probability plotting method, except that we use the principle of least squares to determine the line through the points, as opposed to just eyeballing it. The first step is to bring our function into a linear form. For the two-parameter Weibull distribution, the (cumulative density function) is:


 * $$ F(t)=1-e^{-\left( \frac{t}{\eta }\right) ^{\beta }} $$

Taking the natural logarithm of both sides of the equation yields:


 * $$\ln[ 1-F(t)] =-( \frac{t}{\eta }) ^{\beta } $$


 * $$ \ln{ -\ln[ 1-F(t)]} =\beta \ln ( \frac{t}{ \eta }) $$

or:


 * $$ \ln \{ -\ln[ 1-F(t)]\} =-\beta \ln (\eta )+\beta \ln (t) $$

Now let:


 * $$ y = \ln \{ -\ln[ 1-F(t)]\} $$


 * $$ a = − βln(\eta) $$

and:


 * $$ b= \beta$$

which results in the linear equation of:


 * $$y=a+bx$$

The least squares parameter estimation method (also known as regression analysis) was discussed in Chapter Parameter Estimation and the following equations for regression on Y were derived:


 * $$ \hat{a}=\frac{\sum\limits_{i=1}^{N}y_{i}}{N}-\hat{b}\frac{ \sum\limits_{i=1}^{N}x_{i}}{N}=\bar{y}-\hat{b}\bar{x} $$

and:


 * $$ \hat{b}={\frac{\sum\limits_{i=1}^{N}x_{i}y_{i}-\frac{\sum \limits_{i=1}^{N}x_{i}\sum\limits_{i=1}^{N}y_{i}}{N}}{\sum \limits_{i=1}^{N}x_{i}^{2}-\frac{\left( \sum\limits_{i=1}^{N}x_{i}\right) ^{2}}{N}}} $$

In this case the equations for yi and xi are:


 * $$ y_{i}=\ln \left\{ -\ln [1-F(t_{i})]\right\}, $$

and:


 * xi = ln(ti).

The $$ F(t_{i})s $$ are estimated from the median ranks.

Once $$ \hat{a} $$ and $$ \hat{b} $$ are obtained, then $$ \hat{\beta } $$ and $$ \hat{\eta } $$ can easily be obtained from previous equations.

The Correlation Coefficient

The correlation coefficient is defined as follows:


 * $$ \rho ={\frac{\sigma _{xy}}{\sigma _{x}\sigma _{y}}} $$

where, σx y = covariance of and, σx = standard deviation of , and σy = standard deviation of. The estimator of ρ is the sample correlation coefficient, $$ \hat{\rho} $$, given by:


 * $$ \hat{\rho}=\frac{\sum\limits_{i=1}^{N}(x_{i}-\overline{x})(y_{i}-\overline{y} )}{\sqrt{\sum\limits_{i=1}^{N}(x_{i}-\overline{x})^{2}\cdot \sum\limits_{i=1}^{N}(y_{i}-\overline{y})^{2}}}$$

Example 3: