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    Interface 2010 Interface 2010 Presentation Transcript

    • Robust Quantile Regression Using L2 E Jonathan Lane and David W. Scott 1 Rice University Interface 2010 June 17, 2010 1 NSF DMS-09-07491 Grant
    • Outline Quantile Regression Koenker’s Method MLE L2 E Extention L2 E Theoretical Results Regression Results Dealing with Unknown Sigma Summary and Future Research
    • Quantile Regression Determining conditional quantiles. Multiple methods, classical method is the method presented by Koenker and Bassett (1978).
    • Koenker’s Quantile Regression q q 8 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 qq qq 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 qq q q q q q q qq q q qq 6 qq q qq q qq q q q qqq q q q q q q q qq q q q q q q q q qqq q q q q qq q q q q qq q q qqqq q q q q qqq q qq q q q qq q qq q qqqq q q q q q qq q q q q q q q qq q q qqq q qqq q qq qq q qq q qq q q q q q qq q q qq q q q q q qq qq qq q q qq qq q q q q q q q q qq q q q q q q q qq q qqqq q q q qq qq q q qq q q q q q q q qq q q q q qq qq q q qq qq q q q q qq qq q q q q q q q qqq qq qq q q qq qq qqq q qq q q qqq q q qq q q q qq q q q q qq qq q q q q qq q q q q qq q q q qq q q qqqqq q q q q q q q q q q qqq qq qq q qq q q q qq q qqq qq qq q qqq q qq q q q q q qq q q 4 q q q q q q qq q q q q q q q q qqq q qqq q q qqq q q q q q q y qq q qq q q q qq q q qq q q q q q q qqq q q qqq q q qqqq q q q q q q q qq q q q q q q q q q q q q qqqq qq qqq qq qq q qqq qq qq q q q q q qq q q qqq q q q q qq q q q qq q qq q q qqqq qqqqqq q qqqq qq q q q q q q q q q q q qq qq q qqq q q q q qq q qqq q qq q qq q qq qqq qqqqq q q q q q qq q q qq q q q q q q q qq q q qq q q q q qq q q qqqq q q qq q qq q q q q q q q qq q q qq qq q q q q q q q q q q qq q q q qqq qq q q q q q qqq q qq q qq qq qq qq q qq q q q q q q q q qq q q q q q q qqqq q q q q q qq q q q q q qq q q qq q q q q qq q q q q q q q qq q q q q q 2 q q q q q q qq 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 qq q q q q qq qq q q q q q q q q q q q q q q q 0 q 0 2 4 6 8 x
    • Koenker’s Quantile Regression on Contaminated Data q q q q q qq q q qq q q q q qq q qqq q q q q q q qq q q q q q qq q q q q q q q q q q q qq q q q q q q qq q q q qqq q q q q q q q q q q q q qq q qq q q q q qq q qq q q q q 10 q q q q q q q q q q q q q q q q q qq q q qq q q q y q q q q q q q q q q q qq q qq qq q q q q q qq qq qq q q q q q q qq q q q q q q qq qqqq qq qq q q qq q q q qq q q q qqq q q qq qq qq q q q q qq q qq qq q q q q q qqq qq q qq q q qqqq q qqqq q q q q q qq q q qq q q q qq q q q q q qq q q q q qq q qqq qq qqq qq q q q q q q q q qq qqqqqqqqq qq q q qq qq q q qq qq q qqq q q qqq 5 q q qq q q qq q q qq qqqqq qqqqqq qq qqq qq q q q qqqqq q qqq q q q q q q q q qq qq q q q q q qqq q qq q q qq qqqqqq q qq q q q q q q qq q q qqqq qq q q q q q q q q qq qqqqqq q qq q qq qq q qqqq q q q q q qq q q qqqqq qqqqqqqq qqq qqq q qq qq q q q q q qq q q q qq qq q qqq q q q q q q q q q q q q q q q q qqq qq q q q qq q q q qq q q q q q q q qq q q qqq q q qqq q q qqqqqq qqqq qq q qq q q q q q qq q qqqq q q q q qq q qqq qq qq q q q qqq qqqq qq q qq q qq q q q qq qqq qqq qqqqqq qqq q qq q qq qq qq q q q q q q q q q q q qqq q q q q q q q qqqqqqq qqq qqqqq q q q q q q qq qq q q q qq q q q qqqq q qq q q q q qqqq q q qq q q qq q q q qqqqq q q qq qq q q q q q qq q q qq q qq qqqqqqq qq qq q q q q qq q q qq qqqqq q qq q qqqq q q q q q q q qq qq q qq q q q q q q qq q q q q q q q q q q q q q qqq q q q q q q q q q q qq q q q qqqqq qq q q qq q q qq q qq qq q q q q q q q q q q qq qq 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 0 q 0 2 4 6 8 x
    • Koenker and Bassett Define the loss function for τ ∈ (0, 1): −(1 − τ )x if x < 0 ρτ (x) = τx if x ≥ 0 Then we can find that for a sample (x1 , x2 , . . . , xn ) from X the solution to: n arg min ρτ (xi − θ) θ i=1 Is θ = the τ th quantile of the sample.
    • 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 n arg min ρτ (yi − β0 − β1 xi ) β0 ,β1 i=1 to obtain our lines for conditional quantile estimation. That is, we can estimate the τ th quantile of Y for a given value of X by ˆ Y = β0 + β1 X .
    • MLE We note that if we take a, b > 0, we can reparameterize the loss function by: −ax if x < 0 ρa,b (x) = bx if x ≥ 0 b This gives an equivalent minimization to ρτ where τ = a+b . We can also create a double exponential distribution by: fa,b (x) = c ∗ e −ρa,b (x) ab Where c = a+b so that this function integrates to 1.
    • Plot of Koenker’s Constant Values for τ = .75 1.5 b = 1.5 1.0 0.5 y 0.0 −a = −.5 −0.5 −3 −2 −1 0 1 2 3 x
    • Plot of Koenker’s ρ Function for τ = .75 4 3 y 2 −a = −.5 b = 1.5 1 0 −3 −2 −1 0 1 2 3 x
    • Plot of Corresponding Double Exponential Distribution 0.4 0.3 0.2 y −a = −.5 b = 1.5 0.1 0.0 −3 −2 −1 0 1 2 3 x
    • MLE Note that minimizing Koenker’s loss function is equivalent to fitting a double exponential distribution to data using MLE. That is: n n arg min ρa,b (xi − θ) = arg max fa,b (xi − θ) θ i=1 θ i=1
    • 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
    • S-Curve with Scaling Factor c = 1 1.5 b = 1.5 1.0 0.5 y 0.0 −a = −.5 −0.5 −3 −2 −1 0 1 2 3 x 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.
    • S-Curves with Various Scaling Factors 1.5 1.0 0.5 y 0.0 −0.5 −3 −2 −1 0 1 2 3 x A tuning parameter, denoted by c, adjusts the slope through the origin. The higher the value of c, the steeper the slope.
    • Smooth ρ Function with Scaling Factor c = 1 4 3 y 2 −a = −.5 b = 1.5 1 0 −3 −2 −1 0 1 2 3 x
    • Smooth ρ Functions with Various Scaling Factors 4 3 y 2 1 0 −3 −2 −1 0 1 2 3 x
    • Smooth Double Exponential Function with c = 1 0.4 0.3 0.2 y −a = −.5 b = 1.5 0.1 0.0 −3 −2 −1 0 1 2 3 x
    • Smooth Double Exponential Functions with Various Scaling Factors 0.4 0.3 0.2 y 0.1 0.0 −3 −2 −1 0 1 2 3 x
    • MLE Fit on Generated Normal(0,1) Data 0.3 Sample Quantile Double Exponential f(x) [c =1] f(x) [c = 10] 0.2 y 0.1 0.0 q qq q qqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qq qq q q q qq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qq q q q qq qqqqqqq qqqqqqqqqqqqqqqqqqqqqqq qq qqqq qq q q q q q q qq qqq qq qqqqqqqqqqqq qqqqqqqqqqq qqqq qq q q q q qq qqqqq qqqq qqqqq qq qqqqqq qq qq q qq qqqqqqq qq qqqq qq qqq q q qq qqqq q q q qq qq q q qq q −3 −2 −1 0 1 2 3 4 x
    • Theoretic MLE Quantiles For c = 1 and N(0, 1) data 2 0.5 9 99 0. 1 0.444 0.4 log(b,10) 5 0.333 97 99 0. 0. 0.25 0 6 55 7 0. 66 0. 0.1 0.05 95 0.025 0. 0.01 9 0. −1 75 0. 6 0. 01 0.0 −2 −2 −1 0 1 2 log(a,10)
    • 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
    • L2 E L2 estimation, or L2 E , 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: n 2 arg min f (x; θ)2 dx − f (xi ; θ) θ n i=1
    • L2 E 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 f (x; θ)2 dx − 2 f (x; θ)g (x)dx θ Which, because f (x; θ) = fa,b (x − θ), f (x; θ)2 dx does not depend on θ, reduces to: arg min −2 f (x; θ)g (x)dx θ
    • L2 E Extension b Theoretic quantile achieved for most distributions is not 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)
    • Theoretic L2 E Quantiles For Normal(0,1) Data 2 0.5 1 0.444 99 0.4 0.9 0.333 0 0.25 log(b,10) 667 5 0. 0.1 97 0. 99 6 0. 0. 0.05 −1 0.025 0.01 95 0. −2 9 0. 75 0. 0.001 5 6 0.5 −3 −3 −2 −1 0 1 2 log(a,10)
    • Theoretic L2 E Quantiles For Normal(0,1) Data With a + b = 2 Line 2 0.5 1 0.444 99 0.4 0.9 0.333 0 0.25 log(b,10) 6 67 5 0. 0.1 97 0. 9 6 0. 9 0. 0.05 −1 0.025 0.01 95 0. −2 9 0. 75 0. 0.001 56 0.5 −3 −3 −2 −1 0 1 2 log(a,10)
    • L2 E Extension 2 0.5 1 0.444 99 0.4 0.9 q 0.333 0 0.25 log(b,10) 67 0.6 0.1 75 0.9 9 0.6 0.05 0.9 −1 0.025 5 0.01 0.9 −2 0.9 5 0.7 0.001 56 0.5 −3 −3 −2 −1 0 1 2 log(a,10) 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.
    • .75 L2 E Fit on Generated Normal(0,1) Data 0.4 Sample Quantile L2e Value 0.3 0.2 y 0.1 L2e = 0.6381 Sample Quantile = 0.6101 0.0 q q qq qq qq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq q q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqq qq qq q q q qqqqqqq qqqqqqqqqqqqqqqqqq qqqq qq qqqqqqqq qqq q q qqqqqqqqqqqqqqqqqqqqqq qqqqq qq qqqqqqq qqq qqqqqqq qqqqq qqq qqqqq qq q qqq q qq qq qqqqq qqq qqqq qq qq q qqqqq qq qqq qq qq qqq q qq q −3 −2 −1 0 1 2 3 4 x
    • .75 L2 E Fit on Generated Normal(0,1) Data with Contamination 0.4 Sample Quantile L2e Value 0.3 0.2 y 0.1 L2e = 0.6381 Sample Quantile = 0.6101 0.0 q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q qq q qqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qq qq q qqqqqqqqqqqqqqqqqqq qqqqqq qq q qqqq qqqqqqqqqqqqq q qqqq qqqqq q qqqqq qqq q qq qqqq q qq qqqqqq qq qq qqq qq q q q qqqqq qqqq qqqq qqq qqq qqq qqq qq qq qq q q −2 0 2 4 x
    • .90 L2 E Fit on Generated Normal(0,1) Data with Contamination 0.15 0.10 Sample Quantile 0.05 L2e Value y L2e = 1.2464 Sample Quantile = 1.2466 0.00 q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q qq q qqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qq qq q qqqqqqqqqqqqqqqqqqq qqqqqq qq q qqqq qqqqqqqqqqqqq q qqqq qqqqq q qqqqq qqq q qq qqqq q qq qqqqqq qq qq qqq qq q q q qqqqq qqqq qqqq qqq qqq qqq qqq qq qq qq q q −0.05 −2 0 2 4 x
    • Regression Results We can use the L2 E 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 L2 E quantile regression and classical quantile regression are then compared
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q L2e 0.01 q q q q q q q q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q q qq qq q q q q q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q L2e 0.05 q q q q q q q q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q q qq qq q q q q q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q L2e 0.1 q q q q q q q q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q q qq qq q q q q q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q L2e 0.25 q q q q q q q q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q q qq qq q q q q q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q L2e 0.5 q q q q q q q q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q q qq qq q q q q q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q L2e 0.75 q q q q q q q q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q q qq qq q q q q q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q L2e 0.9 q q q q q q q q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q q qq qq q q q q q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q L2e 0.95 q q q q q q q q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q q qq qq q q q q q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q L2e 0.99 q q q q q q q q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q q qq qq q q q q q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q q q q q q q q L2e q q q q qq q q qq q qq q q q qq q q q qq qq q q q Koenker Full qq q q q q q q q q qq q q q q qq q q q q qq q Koenker UC q q qq 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 10 q q q q q q q q q q q q q q q q q qq q q q y q q q q q q q qq q qq q q q q q q q q q q q qq q qq qq q q q q q qq q qq qq 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 qq q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq q q q q q q q q qq q q q q q qq q qq qqq 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 qq qq qqq q q qq qq q q q q q 5 q q q q q q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q q q qqq qq q qq q q q q qqq q q qq qq q q qq q q q q qqq q q qq qqq 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 qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q qq qq q qq q q q qq q q q q q q qq q q q q qqq q q q q qq q q q q q q q q qq q q qq q q q q qq qq q q q q q q qq q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q qq qqq 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 qq q q q q q qqq q q qq qqq q q q q q q q qq q qq q q q qq q q q qq qq q q q q q q q qq qqqq q q q q q q q qq q q q qq q qq qq q q qqq qqq q q q q q qq q q q q q q qq q qq q q q q q q qq q q q q qq q qq q qq q q q qq q q q q q qq q q qq q qqq qq q q q q q qq qq q q q qq qqq q q q qqq q q q q q q q q q qq q q q qq q qq q q q q q q qq q q q qq qqqq q q q q q qq q qq q q q q q q q q qq qq q qqq qq q qq q q q q q q qq q q q qqq qq q q q q q q q q q qq q q q q qq q q q q q qq q q q q q q q q q q qq q q q q q qq q q q qq q q qq q q q q q qq q qqq q q q q q q q qq qq q qq q q q qq q q q q q qq q q q qq qq qq q q q q qq q q q q q q q q q qq 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 0 q q 0 2 4 6 8 x
    • Quantile Regression Results For N(0, 1) Residuals q L2 E σL2 E ˆ Koenker (F) σKf ˆ Koenker (UC) σKuc ˆ .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
    • 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 L2 E 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
    • Comparison of L2 E and Koenker Quantile Regression q q q q q q q q q q q q q 20 q q q q q q q q q q q qq q q q q qq q qq q q q qq q q q q qq q q q q q qq q q q q q q q qq q qqq q q qq q qq 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 15 q L2e q q q q q Koenker Full q q q q q q q q q Koenker UC q 10 q q q q q q q q q q q q qq q q q qq q q q q qq q q q q q q q q q q q q qq q q q q q qq q q q q qq q q q q q q qq q q q q q qq q q q q q q q q y q q q q qq q q q q q q qq q q q q q q q qq q qq q qq q q q q q qq q q qq q q q q qq 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 qq q q q q q q q q q q q qq q q q qq q qq qq q q qqq q q q q q qq q q q q q q q qq q q q q q q q qq qq q q q q q q q qq q q q q q qqq q q qq q q q q q q qq q q q q q q q qq q q q q q qq q q q q qqq q qq q q qq q q q qq q q q qqq q q qq q q q q q q q q q q q q qq q q qq q q 5 q qq q qq q q q qq q q q q q q q q qq q qq 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 qqqq qq qq 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 qqq q qq qq q qq q q q q q q q q q q q qq q q q qq qq q q q q q q qq qq q q q q q q q q q q qq qq q q q q q q q qq q qq qqq q q qq q q q q q q q qq q q q q q q qq q qq q q q q qq q q q q q q qq q q qq q qq q q q q q q q qq q q q qq q q q qq q q qq q q q qq q qqq q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q qq q q q qq q q q qq q q q q q q q q q q q q qq q q q q qq q q q q q qq q q qq q q q q q q q qq q q q q qq q q q q q q q q q q qq q q q q q q qq q q qq q q q q q q q qq q q qq q q q q q q qq q qq qq q q q q q q q q qq q q q q q q qq q q q q q q qq q q q q qq q q q qq q q qqq q qq q q q qq 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 qq qqq 0 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 qq 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 −5 q q q −2 0 2 4 6 8 10 x
    • Quantile Regression Results For N(0, σ 2 ) Residuals q L2 E σL2 E ˆ Koenker (F) σKf ˆ Koenker (UC) σKuc ˆ .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, σ 2 ) Residuals From 1000 Simulations
    • Summary and Future Research If parametric assumptions are valid, L2 E 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
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