1

Estimation with Asymmetric Loss

Suppose you are interested in estimating a value Q, which is well-defined and can be measured 

at some later point. Your estimator is 

Q

ˆ

. Your loss is a function of your estimation error, x = Q

– Q

ˆ

; denote the loss function as L(x). The estimation problem is to find the estimator 

Q

ˆ

that 

minimizes the expected loss, E[L(x)], where the expectation is over the available information.

By far the most common loss function used is quadratic, L(x) = x

2

, and minimizing the quadratic 

loss yields the estimator, 

ˆ

Qquadratic

= E(Q), the mean of Q. In some circumstances, however, loss 

is asymmetric. Suppose, for example, that a greater loss is incurred by incorrectly choosing an 

estimate 

Q

ˆ

which is bigger than Q, than by choosing an estimate which is less than Q. Because 

x = Q – Q

ˆ

, this means that the loss function is asymmetric, and steeper on the left than the right 

of zero.

Two convenient asymmetric loss functions are the Linex and Lin-Lin loss functions. The Linex 

loss function is,

Linex: L(x) = b[e

ax

– ax – 1] (1)

where a and b are constants. The Linex loss function is approximately exponential on one side 

and approximately linear on the other; when a < 0, the loss is greater when x is negative. Near 

zero, it is locally quadratic. The Lin-Lin loss function is,

Lin-Lin: L(x) = 

1

2

| |, 0

| |, 0

c x x

c x x

 



 

, (2)

c1 and c2 are positive constants. The Lin-Lin loss function is linear on both sides, but with 

different slopes. When c1 > c2, the loss is greater when x is negative.

Figure 1 shows the Linex, Lin-Lin, and quadratic loss functions, for the parameterization in the 

numerical exercise below.
2

Figure 1. The Linex, Lin-Lin, and quadratic loss functions.

All three loss functions have a slope of -2 at x = -200; 

the Linex and Lin-Lin loss functions have a slope of 0.4 at x = 200.

The Linex and Lin-Lin loss functions are convenient because they have closed-form solutions for 

the optimal estimator. For the Linex loss function, if Q has a normal distribution, then this

closed form is,

ˆ

QLinex

= E(Q) + 1⁄2a

2

, (3)

where 

2

is the variance of the distribution of q. For example, if a = -1, then the Linex estimator 

is the mean of Q (which is the quadratic-loss estimator), minus 1⁄2

2

.

For the Lin-Lin loss function, the optimal estimator is the c2/(c1 + c2)-th percentile of the 

distribution of Q:

ˆ

QLin Lin  = 

1 2

1 2

c

F

c c



      

, (4)

where F is the cumulative distribution function of Q given currently available information. For 

example, if c1 = 3 and c2 = 1, then 

ˆ

QLin Lin 

is the 25th percentile of the distribution of Q. Note that 

if c1 = c2, the Lin-Lin estimator is the median. The Lin-Lin loss function is the loss function that 

underlies quantile regression.
3

For both the Linex and Lin-Lin loss functions, if the loss is greater when 

Q

ˆ

exceeds Q than 

when 

Q

ˆ

is less than Q, then the optimal 

Q

ˆ

is less than a measure of the center of the 

distribution, in the case of Linex, it is less than the mean, in the case of Lin-Lin, it is less than the 

median.

Numerical example

Implementation of these formulas requires calibration of the loss function parameters, plus 

knowledge of 

2

(for Linex) and of F (for Lin-Lin). Here we work through an example.

To calibrate the loss function, let (x0) = |L(-x0)/ L(x0)|, which is the ratio of the absolute slopes 

of the loss functions evaluated at x0 for some x0 > 0. This is a measure of the marginal loss 

from overpredicting to underpredicting, as a size of the overprediction error. For a symmetric 

loss function, (x0) = 1. For the Linex and Lin-Lin loss functions, 

Linex: (x0) = 

0

0

1

1

ax

ax

e

e







(5)

Lin-Lin: (x0) = 

1

2

c

c

(6)

For Lin-Lin, the ratio is just the ratio of slope coefficients. For Linex, the ratio depends on the 

point of evaluation, and given x0, a must be obtained by solving (5) nonlinearly.

We consider three values of : 5, 10, and 20. For Linex, we evaluate the derivative at x0 = 200, 

which yields the following parameters:

 c1 c2 a

5 .833 .167 -.00805

10 .909 .091 -.0115

20 .952 .048 -.0150

The Linex and Lin-Lin loss functions in Figure 1 are for  = 5, with coefficients scaled so that 

the slope at x0 = -200 is 2 (the increase in the loss when going from x = 200 to x = 201 is 2 units).
4

The implied optimal additive terms (under the assumption of normality) are given in the 

following table:

 (std. dev. of Q) 100 100 100 200 200 200

 5 10 20 5 10 20

Linex bias term -40 -58 -75 -161 -230 -300

Lin-Lin bias term -97 -134 -167 -193 -267 -334

Lin-Lin percentile .167 .091 .048 .167 .091 .048

For example, for  = 5 and  = 200, the optimal Linex bias is -161, so that the optimal Linex 

estimator would be the mean, minus 161. For Lin-Lin, the optimal bias term is -193, so that the 

optimal estimator would be the median (which for normality is the mean), minus 193.

The final row gives the percentile of the distribution of Q which would be the optimal Lin-Lin 

estimator. This row is included because it is valid under a general distribution, not just normality 

(the previous two bias rows were derived under normality). For example, for  = 5, the optimal 

Lin-Lin estimator would be the 16.7% percentile of the distribution of Q.

Reference

Christoffersen, P.F. and F.X. Diebold, 1997. “Optimal Prediction under Asymmetric Loss,” 

Econometric Theory 13, 808-817.