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| ===Rank Regression on Y===
| | #REDIRECT [[The Lognormal Distribution]] |
| Performing a 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, or regression analysis, was discussed in Chapter 3 and the following equations for regression on Y were derived, and are again applicable:
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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}}={{\Phi }^{-1}}\left[ F(T_{i}^{\prime }) \right]</math>
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| :and:
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| ::<math>{{x}_{i}}=T_{i}^{\prime }</math>
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| where the <math>F(T_{i}^{\prime })</math> is estimated from the median ranks. Once <math>\widehat{a}</math> and <math>\widehat{b}</math> are obtained, then <math>\widehat{\sigma }</math> and <math>\widehat{\mu }</math> can easily be obtained from Eqns. (aln) and (bln).
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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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| '''Example 2:'''
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| {{Example: Lognormal Distribution RRY}}
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