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MS&E 223
Lecture Notes #9
Simulation
Quantile Estimation
Brad Null
Spring Quarter 200809
Page 1 of 9
Quantile Estimation
1.
Quantiles (Definition and Simple Point Estimate)
Quantiles form an important class of performance measures.
The following examples show the utility of
quantile estimation:
Example 1
:
Let X be the return on an investment.
One possible measure of the downside risk (also
called Value at Risk) of the investment is the determination of that value q such that the likelihood that X
takes on a smaller value than q is some prescribed value, say 0.01. This is one simple way of quantifying
a “worstcase scenario” when assessing the riskiness of an investment.
0
f
X
(x)
99%
q
1%
The value q is known as the 0.01quantile of X.
In general, we may be interested in computing the pth
quantile of X (0<p<1). Such a quantile
p
q is defined by
1
p
X
q
F (p)

=
When
X
F is continuous and increasing, then
p
q is simply the unique root of the equation
F
X
(q
p
)
=
p.
When
X
F has jumps or piecewiseconstant sections, we define
p
q as
1
X
F (p)

, where we use our general
definition of the inverse of a distribution (see our earlier discussion of the inversion method for random
variate generation). This leads to the general definition
{ }
p
X
q
min q :F (q)
p
=
≥
.
Example 2
: A measure of spread (or dispersion) in a distribution that is sometimes used in preference to
the variance is the
interquartile range
, given by
0.75
0.25
q
q
α =

,
where
0.75
q
and
0.25
q
are the 0.25th and 0.75th quantiles, respectively, of the distribution under
consideration.
The interquartile range is much less sensitive than the variance to “outlier” observations.
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View Full DocumentMS&E 223
Lecture Notes #9
Simulation
Quantile Estimation
Brad Null
Spring Quarter 200809
Page 2 of 9
However, the estimation of quantiles is very different from the estimation of functions (both linear and
nonlinear) of means.
In other words, the above estimation problems involve methodologies that have not
previously been addressed in this course.
These procedures are discussed below.
The natural estimator for
p
q is, of course, the pth sample quantile.
Specifically, set
1
n
n
Q
F (p)

=
where
n
F is the
empirical
distribution function, defined by
n
n
j
j 1
1
F (x)
I(X
x)
n
=
=
≤
∑
.
Here,
1
n
X ,
,X
K
are i.i.d. replicates of the random variable X. (The function
n
F is piecewise constant,
with jumps at the values
1
n
X ,
,X
K
. The height of a jump at the point
i
x
X
=
is k / n , where k is the
number of data points with a value equal to
i
X .) An equivalent way of defining
n
Q is to let
(i)
X
be the
“ith order statistic” of the sample
1
n
X ,
,X
K
, so that
(1)
(2)
(n)
X
X
X
≤
≤
≤
L
. Then
(
29
n
np
Q
X
=
That is,
n
Q is the
np th smallest observation in the sample
1
n
X ,
,X
K
.
So, developing a point
estimator for
p
q is easy. (There are many other quantile estimators that are used in practice, typically
involving some sort of interpolation between
(
29
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