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| ===Rank Regression on X===
| | #REDIRECT [[The Normal Distribution]] |
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| As was mentioned previously, 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 fitted line is minimized.
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| Again, the first task is to bring our function, the probability of failure function for normal distribution, into a linear form. This step is exactly the same as in regression on Y analysis. All other equations apply in this case as they did for the regression on Y. The deviation from the previous analysis begins on the least squares fit step 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 for the data, for regression on X, is the straight line:
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| ::<math>x=\widehat{a}+\widehat{b}y</math>
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| The corresponding equations for <math>\widehat{a}</math> 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}}) \right]</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> values are estimated from the median ranks. Once <math>\widehat{a}</math> and <math>\widehat{b}</math> are obtained, solve the above linear equation for the unknown value of <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 }\Rightarrow \mu =\widehat{a}</math>
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| and:
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| ::<math>b=\frac{1}{\widehat{b}}=\frac{1}{\sigma }\Rightarrow \sigma =\widehat{b}</math>
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| The correlation coefficient is evaluated as before. | |
| <br>
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| '''Example 3:'''
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| {{Example: Normal Distribution RRX}}
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