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| ===Rank Regression on Y===
| | #REDIRECT [[The Normal Distribution]] |
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| Performing rank regression on Y requires that a straight line 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.
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| The least squares parameter estimation method (regression analysis) was discussed in [[Parameter Estimation]], and the following equations for regression on Y were derived:
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| ::<math>\begin{align}\hat{a}= & \bar{b}-\hat{b}\bar{x} \\
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| =& \frac{\sum_{i=1}^N y_{i}}{N}-\hat{b}\frac{\sum_{i=1}^{N}x_{i}}{N}\\
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| \end{align}
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| </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 the case of the normal distribution, the equations for <math>{{y}_{i}}</math> and <math>{{x}_{i}}</math> are:
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| ::<math>{{y}_{i}}={{\Phi }^{-1}}\left[ F({{t}_{i}}) \right]</math>
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| and:
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| ::<math>{{x}_{i}}={{t}_{i}}</math>
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| where the values for <math>F({{T}_{i}})</math> are estimated from the median ranks. Once <math>\widehat{a}</math> and <math>\widehat{b}</math> are obtained, <math>\widehat{\sigma }</math> and <math>\widehat{\mu }</math> can easily be obtained from above equations.
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| '''The Correlation Coefficient'''
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| The estimator of the sample correlation coefficient, <math>\hat{\rho }</math> , is 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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| '''Example 2:'''
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| {{Example: Normal Distribution RRY}}
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