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| ===Rank Regression on Y===
| | #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.
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| The least squares parameter estimation method (regression analysis) was discussed in Chapter 3, and the following equations for rank regression on Y (RRY) were derived:
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| ::<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>
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| :and:
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| ::<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>
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| In our case, the equations for <math>{{y}_{i}}</math> and <math>{{x}_{i}}</math> are:
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| ::<math>{{y}_{i}}=\ln [1-F({{T}_{i}})]</math>
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| :and:
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| ::<math>{{x}_{i}}={{T}_{i}}</math>
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| 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 Eqns. (ae) and (be).
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| For the one-parameter exponential, Eqns. (aae) and (bbe) become:
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| ::<math>\begin{align}
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| \hat{a}= & 0, \\
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| \hat{b}= & \frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}{{y}_{i}}}{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,x_{i}^{2}}
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| \end{align}</math>
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| ====The Correlation Coefficient====
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| The estimator of <math>\rho </math> is the sample correlation coefficient, <math>\hat{\rho }</math>, given by:
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| ::<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>
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| {{2 parameter exponential distribution example}}
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