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===Rank Regression on Y for Exponential Distribution===
#REDIRECT [[The_Exponential_Distribution]]
Performing a rank regression on Y requires that a straight line be fitted to the set of available data points such that the sum of the squares of the vertical deviations from the points to the line is minimized.
The least squares parameter estimation method (regression analysis) was discussed in [[Parameter Estimation|Chapter 4]], and the following equations for rank regression on Y (RRY) were derived:
 
 
::<math>\hat{a}=\bar{y}-\hat{b}\bar{x}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}}{N}</math>
 
and:
 
::<math>\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 }}}\,x_{i}^{2}-\tfrac{{{\left( \underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}} \right)}^{2}}}{N}}</math>
 
In our case, the equations for <math>{{y}_{i}}</math> and <math>{{x}_{i}}</math> are:
 
::<math>{{y}_{i}}=\ln [1-F({{t}_{i}})]</math>
 
and:
 
::<math>{{x}_{i}}={{t}_{i}}</math>
 
 
and the <math>F({{t}_{i}})</math> is estimated from the median ranks. Once <math>\hat{a}</math> and <math>\hat{b}</math> are obtained, then <math>\hat{\lambda }</math> and <math>\hat{\gamma }</math> can easily be obtained from above equations.
For the one-parameter exponential, equations for estimating ''a'' and ''b'' become:
 
::<math>\begin{align}
  \hat{a}= & 0, \\
  \hat{b}= & \frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}{{y}_{i}}}{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,x_{i}^{2}} 
\end{align}</math>
 
 
'''The Correlation Coefficient'''
 
The estimator of <math>\rho </math> is the sample correlation coefficient, <math>\hat{\rho }</math>, given by:
 
::<math>\hat{\rho }=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,({{x}_{i}}-\overline{x})({{y}_{i}}-\overline{y})}{\sqrt{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{({{x}_{i}}-\overline{x})}^{2}}\cdot \underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{({{y}_{i}}-\overline{y})}^{2}}}}</math>
 
{{2 parameter exponential distribution example}}

Latest revision as of 09:07, 9 August 2012

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