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| ===Rank Regression on X===
| | #REDIRECT [[The_Lognormal_Distribution]] |
| Performing a rank regression on X requires that a straight line be fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized.
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| Again, the first task is to bring our <math>cdf</math> function into a linear form. This step is exactly the same as in regression on Y analysis and all the equations apply in this case too. The deviation from the previous analysis begins on the least squares fit part, where in this case we treat <math>x</math> as the dependent variable and <math>y</math> as the independent variable. The best-fitting straight line to the data, for regression on X (see Chapter [[Parameter Estimation]]), is the straight line:
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| ::<math>x=\widehat{a}+\widehat{b}y</math>
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| The corresponding equations for and <math>\widehat{b}</math> are:
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| ::<math>\hat{a}=\overline{x}-\hat{b}\overline{y}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{x}_{i}}}{N}-\hat{b}\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{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 }}}\,y_{i}^{2}-\tfrac{{{\left( \underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{y}_{i}} \right)}^{2}}}{N}}</math>
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| where:
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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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| and 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, solve the linear equation for the unknown <math>y</math> , which corresponds to:
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| ::<math>y=-\frac{\widehat{a}}{\widehat{b}}+\frac{1}{\widehat{b}}x</math>
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| Solving for the parameters we get:
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| ::<math>a=-\frac{\widehat{a}}{\widehat{b}}=-\frac{{{\mu }'}}{\sigma'}</math>
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| and:
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| ::<math>b=\frac{1}{\widehat{b}}=\frac{1}{\sigma'}</math>
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| The correlation coefficient is evaluated as before using equation in the [[The Correlation Coefficient Calculation|previous section]].
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| '''Example 3:'''
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| {{Example: Lognormal Distribution RRX}}
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