Interface 2010

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

Outline
Quantile Regression
Koenker’s Method
MLE
L2E Extention
L2E 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
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0 2 4 6 8
02468
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Koenker’s Quantile Regression on Contaminated Data
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0 2 4 6 8
0510
x
y

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.

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.

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.

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 MLE. That
is:
arg min
θ
n
i=1
ρa,b(xi − θ) = arg max
θ
n
i=1
fa,b(xi − θ)

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
−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.

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.

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
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Sample Quantile
Double Exponential
f(x) [c =1]
f(x) [c = 10]
q

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

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

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)

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

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)

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

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

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.

.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

.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
L2e Value
q

.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
L2e Value
q

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

Comparison of L2E and Koenker Quantile Regression
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0 2 4 6 8
0510
x
y L2e
Koenker Full 0.01

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0 2 4 6 8
0510
x
y L2e
Koenker Full 0.05

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0 2 4 6 8
0510
x
y L2e
Koenker Full 0.1

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0 2 4 6 8
0510
x
y L2e
Koenker Full 0.25

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0 2 4 6 8
0510
x
y L2e
Koenker Full 0.5

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0 2 4 6 8
0510
x
y L2e
Koenker Full 0.75

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0 2 4 6 8
0510
x
y L2e
Koenker Full 0.9

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0 2 4 6 8
0510
x
y L2e
Koenker Full 0.95

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0 2 4 6 8
0510
x
y L2e
Koenker Full 0.99

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0 2 4 6 8
0510
x
y L2e
Koenker Full
Koenker UC

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

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

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−2 0 2 4 6 8 10
−505101520
x
y
L2e
Koenker Full
Koenker UC

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

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

Interface 2010

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