Robust Quantile Regression Using L2E
Jonathan Lane and David W. Scott 1
Rice University
Interface 2010
June 17, 2010
1
NSF...
Outline
Quantile Regression
Koenker’s Method
MLE
L2E Extention
L2E Theoretical Results
Regression Results
Dealing with Unk...
Quantile Regression
Determining conditional quantiles.
Multiple methods, classical method is the method presented
by Koenk...
Koenker’s Quantile Regression
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Koenker’s Quantile Regression on Contaminated Data
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Koenker and Bassett
Define the loss function for τ ∈ (0, 1):
ρτ (x) =
−(1 − τ)x if x < 0
τx if x ≥ 0
Then we can find that f...
Koenker and Bassett
We can use this loss function to perform quantile regression. For
simple linear regression in two dime...
MLE
We note that if we take a, b > 0, we can reparameterize the loss
function by:
ρa,b(x) =
−ax if x < 0
bx if x ≥ 0
This ...
Plot of Koenker’s Constant Values for τ = .75
−3 −2 −1 0 1 2 3
−0.50.00.51.01.5
x
y
−a = −.5
b = 1.5
Plot of Koenker’s ρ Function for τ = .75
−3 −2 −1 0 1 2 3
01234
x
y
−a = −.5 b = 1.5
Plot of Corresponding Double Exponential Distribution
−3 −2 −1 0 1 2 3
0.00.10.20.30.4
x
y
−a = −.5 b = 1.5
MLE
Note that minimizing Koenker’s loss function is equivalent to
fitting a double exponential distribution to data using M...
MLE
Classical quantile regression, like L1 error, does not have an
analytic solution for MLE, as the derivative of ρa,b is...
S-Curve with Scaling Factor c = 1
−3 −2 −1 0 1 2 3
−0.50.00.51.01.5
x
y
−a = −.5
b = 1.5
For this example, we use a modifie...
S-Curves with Various Scaling Factors
−3 −2 −1 0 1 2 3
−0.50.00.51.01.5
x
y
A tuning parameter, denoted by c, adjusts the ...
Smooth ρ Function with Scaling Factor c = 1
−3 −2 −1 0 1 2 3
01234
x
y
−a = −.5 b = 1.5
Smooth ρ Functions with Various Scaling Factors
−3 −2 −1 0 1 2 3
01234
x
y
Smooth Double Exponential Function with c = 1
−3 −2 −1 0 1 2 3
0.00.10.20.30.4
x
y
−a = −.5 b = 1.5
Smooth Double Exponential Functions with Various
Scaling Factors
−3 −2 −1 0 1 2 3
0.00.10.20.30.4
x
y
MLE Fit on Generated Normal(0,1) Data
−3 −2 −1 0 1 2 3 4
0.00.10.20.3
x
y
qqqq qqqqq qq qqq qq qq qq q q qqq qq q qqq qqq ...
Theoretic MLE Quantiles For c = 1 and N(0, 1) data
log(a,10)
log(b,10)
0.001
0.01
0.025
0.05
0.1
0.25
0.333
0.4
0.444
0.5
...
MLE
Using smooth function allows us to perform quantile
estimation while allowing for analytic solutions
Parametric - an e...
L2E
L2 estimation, or L2E, was developed by Scott (2001) as a robust,
parametric density estimator. To estimate a density ...
L2E Extension
We can apply this method to quantile regression by trying to fit a
double exponential distribution to a sampl...
L2E Extension
Theoretic quantile achieved for most distributions is not b
a+b
(Although this is true for Unif (0, 1))
Assu...
Theoretic L2E Quantiles For Normal(0,1) Data
log(a,10)
log(b,10)
0.001
0.01
0.025
0.05
0.1
0.25
0.333
0.4
0.444
0.5
0.556
...
Theoretic L2E Quantiles For Normal(0,1) Data With
a + b = 2 Line
log(a,10)
log(b,10)
0.001
0.01
0.025
0.05
0.1
0.25
0.333
...
L2E Extension
log(a,10)
log(b,10)
0.001
0.01
0.025
0.05
0.1
0.25
0.333
0.4
0.444
0.5
0.556
0.6
0.6
67
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0.9
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0.9
...
.75 L2E Fit on Generated Normal(0,1) Data
−3 −2 −1 0 1 2 3 4
0.00.10.20.30.4
x
y
qqq q qqqq q q qqq qqq q qq qqq q q qqqq ...
.75 L2E Fit on Generated Normal(0,1) Data with
Contamination
−2 0 2 4
0.00.10.20.30.4
x
y
qqq q qqqq q q qqq qqq q qq qqq ...
.90 L2E Fit on Generated Normal(0,1) Data with
Contamination
−2 0 2 4
−0.050.000.050.100.15
x
y
qqq q qqqq q q qqq qqq q q...
Regression Results
We can use the L2E loss function to perform quantile
regression
Robust
The following plots have 900 poi...
Comparison of L2E and Koenker Quantile Regression
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Comparison of L2E and Koenker Quantile Regression
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Comparison of L2E and Koenker Quantile Regression
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Comparison of L2E and Koenker Quantile Regression
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Comparison of L2E and Koenker Quantile Regression
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Comparison of L2E and Koenker Quantile Regression
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Comparison of L2E and Koenker Quantile Regression
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Comparison of L2E and Koenker Quantile Regression
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Comparison of L2E and Koenker Quantile Regression
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Comparison of L2E and Koenker Quantile Regression
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Quantile Regression Results For N(0, 1) Residuals
q L2E ˆσL2E Koenker (F) ˆσKf Koenker (UC) ˆσKuc
.010 .012 .001 .011 .000...
Dealing with Unknown Sigma
Though we might be able to assume normal residuals about
the mean regression line, Assuming N(0...
Comparison of L2E and Koenker Quantile Regression
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Quantile Regression Results For N(0, σ2
) Residuals
q L2E ˆσL2E Koenker (F) ˆσKf Koenker (UC) ˆσKuc
.01 .011 .002 .011 .00...
Summary and Future Research
If parametric assumptions are valid, L2E quantile regression
provides a robust method to estim...
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Interface 2010

  1. 1. Robust Quantile Regression Using L2E Jonathan Lane and David W. Scott 1 Rice University Interface 2010 June 17, 2010 1 NSF DMS-09-07491 Grant
  2. 2. Outline Quantile Regression Koenker’s Method MLE L2E Extention L2E Theoretical Results Regression Results Dealing with Unknown Sigma Summary and Future Research
  3. 3. Quantile Regression Determining conditional quantiles. Multiple methods, classical method is the method presented by Koenker and Bassett (1978).
  4. 4. Koenker’s Quantile Regression q q q q q q qq q q q q q q q q q q qq q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q qq q q q q q q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 02468 x y
  5. 5. Koenker’s Quantile Regression on Contaminated Data q q q q q q qq q q q q q q q q q q qq q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qqq q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q qq q q q q q q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y
  6. 6. Koenker and Bassett Define the loss function for τ ∈ (0, 1): ρτ (x) = −(1 − τ)x if x < 0 τx if x ≥ 0 Then we can find that for a sample (x1, x2, . . . , xn) from X the solution to: arg min θ n i=1 ρτ (xi − θ) Is θ = the τth quantile of the sample.
  7. 7. Koenker and Bassett We can use this loss function to perform quantile regression. For simple linear regression in two dimensions with a sample ((x1, y1), (x2, y2), . . . , (xn, yn)) from (X, Y ), we can solve the minimization problem arg min β0,β1 n i=1 ρτ (yi − β0 − β1xi ) to obtain our lines for conditional quantile estimation. That is, we can estimate the τth quantile of Y for a given value of X by ˆY = β0 + β1X.
  8. 8. MLE We note that if we take a, b > 0, we can reparameterize the loss function by: ρa,b(x) = −ax if x < 0 bx if x ≥ 0 This gives an equivalent minimization to ρτ where τ = b a+b . We can also create a double exponential distribution by: fa,b(x) = c ∗ e−ρa,b(x) Where c = ab a+b so that this function integrates to 1.
  9. 9. Plot of Koenker’s Constant Values for τ = .75 −3 −2 −1 0 1 2 3 −0.50.00.51.01.5 x y −a = −.5 b = 1.5
  10. 10. Plot of Koenker’s ρ Function for τ = .75 −3 −2 −1 0 1 2 3 01234 x y −a = −.5 b = 1.5
  11. 11. Plot of Corresponding Double Exponential Distribution −3 −2 −1 0 1 2 3 0.00.10.20.30.4 x y −a = −.5 b = 1.5
  12. 12. MLE Note that minimizing Koenker’s loss function is equivalent to fitting a double exponential distribution to data using MLE. That is: arg min θ n i=1 ρa,b(xi − θ) = arg max θ n i=1 fa,b(xi − θ)
  13. 13. MLE Classical quantile regression, like L1 error, does not have an analytic solution for MLE, as the derivative of ρa,b is discontinuous A version of the Simplex method is used instead to solve this minimization efficiently In an attempt to find an analytic solution, we create a continuous version of the derivative Some possible solutions are ”S-curves”, such as the cdf of a normal distribution or the cdf of a logistic distribution
  14. 14. S-Curve with Scaling Factor c = 1 −3 −2 −1 0 1 2 3 −0.50.00.51.01.5 x y −a = −.5 b = 1.5 For this example, we use a modified version of the logistic distribution, where it is shifted and scaled such that it passes through the origin and the horizontal asymptotes occur at −a and b.
  15. 15. S-Curves with Various Scaling Factors −3 −2 −1 0 1 2 3 −0.50.00.51.01.5 x y A tuning parameter, denoted by c, adjusts the slope through the origin. The higher the value of c, the steeper the slope.
  16. 16. Smooth ρ Function with Scaling Factor c = 1 −3 −2 −1 0 1 2 3 01234 x y −a = −.5 b = 1.5
  17. 17. Smooth ρ Functions with Various Scaling Factors −3 −2 −1 0 1 2 3 01234 x y
  18. 18. Smooth Double Exponential Function with c = 1 −3 −2 −1 0 1 2 3 0.00.10.20.30.4 x y −a = −.5 b = 1.5
  19. 19. Smooth Double Exponential Functions with Various Scaling Factors −3 −2 −1 0 1 2 3 0.00.10.20.30.4 x y
  20. 20. MLE Fit on Generated Normal(0,1) Data −3 −2 −1 0 1 2 3 4 0.00.10.20.3 x y qqqq qqqqq qq qqq qq qq qq q q qqq qq q qqq qqq qq q qqq q qqqqq qqq q qqq q q qq q qq qqqqqq qq qqqqq q qq q qq qqq qqq qq qq q qq qq q q qqq q qqq q qq q qqq qq qq qqq qq qqqq qq q qqqqq qq q qq qq qq qqq qqq qqqq qq qq qqq q qq qq qq qqq qq qqq q qq qq qq q qqq qqq qqqq qq q qq qq q qq qqq qq qqq qq qq qqq qq q qq qq qqqqqq qq qq q q q qqq qq qq qqq qqq q qqqq qq qq qq qqqq qq qqq q qq qqq q qqq qqqq q q qq qqq q q qq qqqq qq qq q q qq qqqqq q qqq q qq qq qq q qq q qq qqq qq qqqq q qq qqqqq q qq qqq qq q qq qq q qq qqqq qqq qq qqq q qq q qqq q q qq qqq q qqqq qq qq qqq qqq q qq qq q qqqq q qq q q qq qq qqq qq qq qq qqqq qq qq q qq qq q qqqq qq qq qq qq q qq q qqq qq qqqq qq q q qqq q qq qqqq q qqq q qqq q qqq qqq qqqqq qq qq qq q qq qq q qq qq qq q qqqq qq q qqqqqq q qq q qq qqq q q qqq qq qqq qqq qq qq qq qq qqqq qq q qqq qq qq qqqq q q qqqq qqq qq qq qqq qqqq q qq q qq q qq qq qq q qqq qq q qqq qq qq qq qq q qqqq qqq q q qqq q qq q qq qqq qq q qqq q qq qqq q q q q qq qq q qq qqq qq qq q qq qqq qqq qq qq qq qqqqq qq q q q qq qqq q qqq q qq q q qqq qq qqq qq qqq qqqq qq q qq q qq q qq q qqq qq qqq qqq q qq q qq q qqq qq qq qq q q qq q qq qq qqq q q qq qqqqq qq q q qqq qq qq qqqq q qq qq qq q q qq qqqq q qqqq q qq qq qq qq qq q qq qqq q qqq qqq q q qq q qqq q qqq qq qqq qq q qqq q qq q qq q qq qqqqq qqqqq q qq qqq qq qqqq qqqq q qqq qq qq qq q qqq q qq qqqq qq q qqq q qq qq qqqq qqq qqqqqq qq qq q qqq qq qqqq qqqqqqq qqq qqqqq qq qqqq qq Sample Quantile Double Exponential f(x) [c =1] f(x) [c = 10] q
  21. 21. Theoretic MLE Quantiles For c = 1 and N(0, 1) data log(a,10) log(b,10) 0.001 0.01 0.025 0.05 0.1 0.25 0.333 0.4 0.444 0.5 0.556 0.6 0.667 0.75 0.9 0.95 0.975 0.99 0.999 −2 −1 0 1 2 −2−1012
  22. 22. MLE Using smooth function allows us to perform quantile estimation while allowing for analytic solutions Parametric - an explicit parametric assumption for the distribution must be made No added robustness
  23. 23. L2E L2 estimation, or L2E, was developed by Scott (2001) as a robust, parametric density estimator. To estimate a density g(x) from a sample (x1, x2, . . . , xn) by a family of distributions f (x; θ), we find the value of θ solving the equation: arg min θ f (x; θ)2 dx − 2 n n i=1 f (xi ; θ)
  24. 24. L2E Extension We can apply this method to quantile regression by trying to fit a double exponential distribution to a sampling density g(x). For given values of a and b, we can find the theoretic value of θ for the given function g(x) by taking arg min θ f (x; θ)2 dx − 2 f (x; θ)g(x)dx Which, because f (x; θ) = fa,b(x − θ), f (x; θ)2dx does not depend on θ, reduces to: arg min θ −2 f (x; θ)g(x)dx
  25. 25. L2E Extension Theoretic quantile achieved for most distributions is not b a+b (Although this is true for Unif (0, 1)) Assumption about the distribution of the residuals must be made We examine assumption that the residuals are N(0, 1)
  26. 26. Theoretic L2E Quantiles For Normal(0,1) Data log(a,10) log(b,10) 0.001 0.01 0.025 0.05 0.1 0.25 0.333 0.4 0.444 0.5 0.556 0.6 0.667 0.75 0.9 0.95 0.975 0.99 0.999 −3 −2 −1 0 1 2 −3−2−1012
  27. 27. Theoretic L2E Quantiles For Normal(0,1) Data With a + b = 2 Line log(a,10) log(b,10) 0.001 0.01 0.025 0.05 0.1 0.25 0.333 0.4 0.444 0.5 0.556 0.6 0.667 0.75 0.9 0.95 0.975 0.99 0.999 −3 −2 −1 0 1 2 −3−2−1012
  28. 28. L2E Extension log(a,10) log(b,10) 0.001 0.01 0.025 0.05 0.1 0.25 0.333 0.4 0.444 0.5 0.556 0.6 0.6 67 0.7 5 0.9 0.9 5 0.9 75 0.9 90.999 −3 −2 −1 0 1 2 −3−2−1012 q From these contours, we can determine the theoretic quantiles achieved for a N(0, 1) sample. For example, if we want the .75 quantile, we can take a = 0.382 and b = 1.618.
  29. 29. .75 L2E Fit on Generated Normal(0,1) Data −3 −2 −1 0 1 2 3 4 0.00.10.20.30.4 x y qqq q qqqq q q qqq qqq q qq qqq q q qqqq qq qq qqq qq qq q q qq qq qq qqq qqqq qqq q qqq qq q q qqq qqq qq qq q q q qq qq qqq qqq q qq q qqqqq q qq q q qqq qqq qqq qqq q qq q qq qq q q q qqq q qq qqq q qqqq qq qqq qqqq q qq qq qq q qq qq qqq qq q qq q qqq qq qq qq q qq qq q qq q qq qqq qqq qq q qq q qqqqq q qqq qq qqq qqq q qq qq qqq q qq qq qq qq q qqqq q qq qq qqq q qqq qqqq qq qqqq qq qq qq qq qq q qq qq q qqqq qqqq qqqq qq qq q qq q qqqq qqqqq qqqq q qqqqq q qq qqq qqq q qq qqqqq qqqq qq q qq q qq qqq q qq qqq q q qq qq qq q q qqqq q qq qq qq qq qq q qqqq qq qqq qqq qq qq q qq q qqqq q qqq qq qqq qq qqq qq qq qq q qqq qqq qq qq qq qq q qqq qq qqq qq qq qq qqq qqq qq qqq q qq qqqq q qqq qqqq qqqq qq qqq qq qqqq qq q qq qq q qqq qqq qq qqq qqq qqqq qq qqq qqq qqqq qq q qqq qq qq qqqq qq qq qqq q qqq q qqq qqq qq qq qqq q qqqqq qq qq qq qqqqq qqqq qqq qq q qq q qq qq q qq qq qqqq qqq qq qq qq q qqq q q qqq qqqq q qq qqq qqqq qq q q qq qq qq qq qqqq qq qqqq q qq q qq qq q qq qqqq qq qq q qqq qq qqqq qq qq qq q q qq qq q qq q q qqqqq qq q qqqq qq qq qq qq qq qq q q qqq q q q qq q qq qq q q qq q qqqq qq qq qqq qqq qq q q qq q qq qqq qq qqq q qqqqq qq q qqq qq q qqq q qqq q qqqqq qq q qqqq q qq qq qq qq qqqqqq qq qq q qq q q qqqqq qq qq q qqq qq qq qq q q qq qqq qq q qq q qq q qqq qqq qq qq qqq qq L2e = 0.6381 Sample Quantile = 0.6101 Sample Quantile L2e Value
  30. 30. .75 L2E Fit on Generated Normal(0,1) Data with Contamination −2 0 2 4 0.00.10.20.30.4 x y qqq q qqqq q q qqq qqq q qq qqq q q qqqq qq qqqqq qq qq q q qq qq qq qqq qqqq qqq q qqq qq q q qqq qqq qq qq q q q qq qq qqq qqq q qq q qqqqq q qq q q qqq qqq qqq qqq q qq q qq qq q q q qqq q qq qqq q qqqq qq qqq qqqq q qq qq qq q qq qq qqq qq q qq q qqq qq qq qq q qq qqq qq q qq qqq qqq qq q qq q qqqqq q qqq qq qqq qqq q qq qq qqq q qq qq qq qq q qqqq q qq qq qqq q qqq qqqq qq qqqq qq qq qq qq qq q qq qq q qqqq qqqq qqqq qq qq q qq q qqqq qqqqq qqqq q qqqqq q qq qqq qqq q qq qqqqq qqqq qq q qq q qq qqq q qq qqq q q qq qq qq q q qqqq q qq qq qq qq qq q qqqq qq qqq qqq qq qq q qq q qqqq q qqq qq qqq qq qqq qq qq qq q qqq qqq qq qq qq qq q qqq qq qqq qq qq qq qqq qqq qq qqq qqq qqqq q qqq qqqq qqqq qq qqq qq qqqq qq q qq qq q qqq qqq qq qqq qqq qqqq qq qqq qqq qqqq qq q qqq qq qq qqqq qq qq qqq q qqq q qqq qqq qq qq qqq q qqqqq qq qq qq qqqqq qqqq qqq qq q qq q qq qq q qq qq qqqq qqq qq qq qq q qqq q q qqq qqqq q qq qqq qqqq qq q q qq qq qq qq qqqq qq qqqq q qq q qq qq q qq qqqq qq qq q qqq qq qqqq qq qq qq q q qq qq q qq q q qqqqq qq q qqqq qq qq qq qq qq qq q q qqq q q q qq q qq qq q q qq q qqqq qq qq qqq qqq qq q q qq q qq qqq qq qqq q qqqqq qq q qqq qq q qqq q qqq q qqqqq qq q qqqq q qq qq qq qq qqqqqq qq qq q qq q q qqqqq qq qq q qqq qq qq qq q q qq qqq qq q qq q qq q qqq qqq qq qq qqq qq qqqqqqqqqqqqqq qqq qqqqqqqqqqqq qqq qqqqqqq qqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqq qqq qqqqqqq qqqqq q L2e = 0.6381 Sample Quantile = 0.6101 Sample Quantile L2e Value q
  31. 31. .90 L2E Fit on Generated Normal(0,1) Data with Contamination −2 0 2 4 −0.050.000.050.100.15 x y qqq q qqqq q q qqq qqq q qq qqq q q qqqq qq qqqqq qq qq q q qq qq qq qqq qqqq qqq q qqq qq q q qqq qqq qq qq q q q qq qq qqq qqq q qq q qqqqq q qq q q qqq qqq qqq qqq q qq q qq qq q q q qqq q qq qqq q qqqq qq qqq qqqq q qq qq qq q qq qq qqq qq q qq q qqq qq qq qq q qq qqq qq q qq qqq qqq qq q qq q qqqqq q qqq qq qqq qqq q qq qq qqq q qq qq qq qq q qqqq q qq qq qqq q qqq qqqq qq qqqq qq qq qq qq qq q qq qq q qqqq qqqq qqqq qq qq q qq q qqqq qqqqq qqqq q qqqqq q qq qqq qqq q qq qqqqq qqqq qq q qq q qq qqq q qq qqq q q qq qq qq q q qqqq q qq qq qq qq qq q qqqq qq qqq qqq qq qq q qq q qqqq q qqq qq qqq qq qqq qq qq qq q qqq qqq qq qq qq qq q qqq qq qqq qq qq qq qqq qqq qq qqq qqq qqqq q qqq qqqq qqqq qq qqq qq qqqq qq q qq qq q qqq qqq qq qqq qqq qqqq qq qqq qqq qqqq qq q qqq qq qq qqqq qq qq qqq q qqq q qqq qqq qq qq qqq q qqqqq qq qq qq qqqqq qqqq qqq qq q qq q qq qq q qq qq qqqq qqq qq qq qq q qqq q q qqq qqqq q qq qqq qqqq qq q q qq qq qq qq qqqq qq qqqq q qq q qq qq q qq qqqq qq qq q qqq qq qqqq qq qq qq q q qq qq q qq q q qqqqq qq q qqqq qq qq qq qq qq qq q q qqq q q q qq q qq qq q q qq q qqqq qq qq qqq qqq qq q q qq q qq qqq qq qqq q qqqqq qq q qqq qq q qqq q qqq q qqqqq qq q qqqq q qq qq qq qq qqqqqq qq qq q qq q q qqqqq qq qq q qqq qq qq qq q q qq qqq qq q qq q qq q qqq qqq qq qq qqq qq qqqqqqqqqqqqqq qqq qqqqqqqqqqqq qqq qqqqqqq qqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqq qqq qqqqqqq qqqqq q L2e = 1.2464 Sample Quantile = 1.2466 Sample Quantile L2e Value q
  32. 32. Regression Results We can use the L2E loss function to perform quantile regression Robust The following plots have 900 points of uncontaminated multivariate normal data where the residuals around the least squares regression line are distributed N(0, 1). 100 points of contamination are placed above the uncontaminated points L2E quantile regression and classical quantile regression are then compared
  33. 33. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full 0.01
  34. 34. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full 0.05
  35. 35. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full 0.1
  36. 36. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full 0.25
  37. 37. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full 0.5
  38. 38. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full 0.75
  39. 39. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full 0.9
  40. 40. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full 0.95
  41. 41. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full 0.99
  42. 42. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 8 0510 x y L2e Koenker Full Koenker UC
  43. 43. Quantile Regression Results For N(0, 1) Residuals q L2E ˆσL2E Koenker (F) ˆσKf Koenker (UC) ˆσKuc .010 .012 .001 .011 .000 .011 .000 .050 .056 .001 .056 .000 .050 .000 .100 .109 .001 .111 .001 .099 .000 .250 .252 .001 .278 .001 .251 .000 .500 .491 .008 .556 .001 .500 .000 .750 .761 .003 .834 .000 .751 .000 .900 .897 .001 .980 .002 .901 .000 .950 .958 .001 .993 .001 .951 .000 .990 .990 .001 .999 .001 .991 .000 Table: Quantile Results For N(0, 1) Residuals
  44. 44. Dealing with Unknown Sigma Though we might be able to assume normal residuals about the mean regression line, Assuming N(0, 1) is a stretch To obtain a robust estimate of the standard deviation of the residuals, we can use regular L2E regression Scale the data so that the residuals are N(0, 1) Perform method from before Rescale slope and intercept parameters by the standard deviation estimate to obtain parameters for original data
  45. 45. Comparison of L2E and Koenker Quantile Regression q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q −2 0 2 4 6 8 10 −505101520 x y L2e Koenker Full Koenker UC
  46. 46. Quantile Regression Results For N(0, σ2 ) Residuals q L2E ˆσL2E Koenker (F) ˆσKf Koenker (UC) ˆσKuc .01 .011 .002 .011 .001 .010 .001 .05 .054 .005 .056 .001 .050 .001 .10 .105 .007 .112 .001 .100 .001 .25 .253 .012 .278 .001 .251 .001 .50 .500 .014 .556 .001 .501 .001 .75 .750 .012 .834 .001 .751 .001 .90 .900 .006 .986 .002 .901 .001 .95 .950 .004 .999 .001 .951 .001 .99 .991 .002 1.00 .000 .991 .001 Table: Summary of Quantile Results For N(0, σ2 ) Residuals From 1000 Simulations
  47. 47. Summary and Future Research If parametric assumptions are valid, L2E quantile regression provides a robust method to estimate the conditional quantiles of data, ignoring the contamination in the data Extends to higher dimensions Smooth versions of the double exponential distribution may lead us to analytic results From here, we will examine non-linear regression, semi-parametric methods, and M-estimation methods
  48. 48. Thank you!

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