553_Final_Project_Bokser_Litalien

The Extreme Value of Regular Random Variables
Rory Bokser∗
Joey Litalien†
COMP/MATH 553: Algorithmic Game Theory
Presented to Assistant Professor Yang Cai
Departement of Mathematics and Statistics, McGill University
School of Computer Science, McGill University
December 10, 2014
Abstract
In this paper, we attempt to determine if the maximum of n regular
random variables X1, . . . , Xn is regular. We examine the cases when the
random variables are independently and identically distributed (i.i.d.)
and when they are independent but not necessarily identical.
∗
Email: rory.bokser@mail.mcgill.ca
†
Email: joey.litalien@mail.mcgill.ca
1

2 R. BOKSER AND J. LITALIEN
Contents
1 Introduction 3
2 Preliminaries 4
2.1 Regular and MHR Distributions . . . . . . . . . . . . . . . . 4
2.2 Fat-Tail Distributions . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Maximum of Random Variables . . . . . . . . . . . . . . . . . 5
3 Properties of Fat-Tail Distributions 6
3.1 Regular and Non-MHR Property . . . . . . . . . . . . . . . . 6
3.2 Equal-Revenue Distributions . . . . . . . . . . . . . . . . . . 7
4 Maximum of MHR Random Variables 7
4.1 I.I.D. MHR Distributions . . . . . . . . . . . . . . . . . . . . 7
5 Maximum of Regular Random Variables 9
5.1 I.I.D. Regular Ristributions . . . . . . . . . . . . . . . . . . . 9
5.2 Independent Non-Identical Regular Random Variables . . . . 10
5.3 A Diﬀerential Approach . . . . . . . . . . . . . . . . . . . . . 10
6 Conclusion 11
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6.2 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 12

THE EXTREME VALUE OF REGULAR RANDOM VARIABLES 3
1 Introduction
Montone Hazard Rate (MHR) distributions have received considerable at-
tention in the past (e.g. [BWP63]) due to their practical nature. However,
of equal importance are regular distributions, a superclass of MHR distribu-
tions. Both frequently appear in the mechanism design literature because
they represent a collection of distributions for which Myerson’s Optimal Auc-
tion is simple and natural. Indeed, regular distributions correspond to the
distributions for which Myerson’s virtual value function φ(x) is monotone
[Mye81]. These properties make the study of both classes of distributions
particularly rich and interesting.
We claim that the maximum of n i.i.d. regular random variables need not
be regular. It is known that the maximum of n i.i.d. MHR random variables
is MHR. Thus we construct a counterexample using the Pareto distribution,
which lies in the space of non-MHR distributions. Having constructed this
counterexample, we formally prove its validity.

2 Preliminaries
In this section, we present definitions and results needed for later sections.
This is in part based on Section 2 of [CD11].
2.1 Regular and MHR Distributions
Let (Ω, F, Pr) be a probability space and let X : Ω → R be a random
variable. We denote the cumulative distribution function (CDF) of X by
FX and the probability density function (PDF) of X by fX. Further define
γmin = sup{x : FX(x) = 0} and similarly, γmax = inf{x : FX(x) = 1}.
Note that we can have γmax = +∞ and we also assume that γmin 0. We
will often drop the subscript when X is clear from the context. It is also
convenient to denote F(x) = 1 − F(x).
In this paper, we will mainly focus on regular distributions as a superclass
of Monotone Hazard Rate (MHR) distributions. Hence, we proceed with a
precise definition for both classes.
Definition 1 (Regular Distribution). A one-dimensional differentiable dis-
tribution F is said to be regular if the Myerson virtual value of F
φF (x) = x −
F(x)
f(x)
(2.1)
is non-decreasing on the interval [γmin, γmax), where f is the corresponding
PDF.
Definition 2 (Monotone Hazard Rate Distribution). A one-dimensional
differentiable distribution F has a Monotone Hazard Rate if the hazard rate
function of F
HF (x) =
f(x)
F(x)
(2.2)
is non-decreasing on [γmin, γmax).
Note that the correspondence HF (x) = 1/(x − φF (x)) suggests an impor-
tant relationship between the two distributions. Indeed, it is easy to see
that all MHR distributions are regular distributions. This follows from the
fact that if HF (x) is increasing then 1/HF (x) is decreasing, in which case
φF (x) = x − 1/HF (x) is also increasing. However the converse fails to hold
in general, that is, there exist non-MHR regular distributions.
The family of MHR distributions includes the exponential, normal and
uniform distributions, to name a few.

2.2 Fat-Tail Distributions
Fat-tail distributions have the property, along with the other heavy-tailed
distributions, that they exhibit large kurtosis or skewness. Intuitively, this
means that their variance is unbounded, yielding unexpected behaviours at
the upper and lower tails. This translates to the following definition.
Definition 3 (Fat-tail Distribution). A one-dimensional differentiable dis-
tribution F is fat-tailed if F(x) ∼ x−α for some α > 0 as x → ∞. In other
words, its density function f is asymptotically f(x) ∼ x−(1+α) for some
α > 0 as x → ∞.
Fat-tail distributions include the Cauchy distribution as well as all other
stable distributions, with the exception of normal distributions as illustrated
below.
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
x
Pr(x)
Normal
Cauchy
Figure 1. The fat-tailedness behaviour of the Cauchy distribution, as
compared with the normal distribution.
2.3 Maximum of Random Variables
Let X1, . . . , Xn be independently and identically distributed (i.i.d.) random
variables, each with the same cumulative distribution function
F(x) = Pr[Xi x]. Let Xmax = max1 i n{Xi}. Then the CDF of Xmax is
Fmax(x) = Pr[Xmax x] = Pr
n
i=1
Xi x =
n
i=1
Pr[Xi x]
= F(x)n
.
(2.3)

If each Xi has density f(x) then Xmax has PDF
fmax(x) =
d
dx
F(x)n
= nF(x)n−1 d
dx
F(x)
= nF(x)n−1
f(x).
(2.4)
There is no closed-form for this expression when X1, . . . , Xn are independent
but not necessarily identical random variables. In this case, we use the formal
definitions.
Fmax =
n
i=1
Pr[Xi x] (2.5)
fmax =
d
dx
Fmax. (2.6)
3 Properties of Fat-Tail Distributions
3.1 Regular and Non-MHR Property
As previously noted, the class of fat-tail distributions is defined by the re-
lation F(x) ∼ x−α for some α > 0 as x → ∞. We verify the regularity
of these distributions and show that they do not belong to the subclass of
MHR distributions.
One particular construction of a fait-tail distribution is done by letting
the random variable X have the PDF
f(x) =
0, if x < 1
αx−(1+α), if x 1
(3.1)
for some α > 0. This is known as the Type I Pareto distribution1 with
scale 1. Computing its cumulative distribution function, we find F(x) =
1 − x−α. Thus, X has Myerson virtual value function φ(x) = x − x−(1+2α).
Considering the derivative φ (x) = x−2(1+α)(2α + 1) + 1, we see that the
virtual value function is increasing for all α > 0, and thus is regular.
However, we note that HF (x) = x−(1+2α) has derivative −(2α+1)x−2(α+1)
which is decreasing for α > 0. Hence, the fat-tailed distributions are regular
but non-MHR, as desired.
1
The interested reader is referred to [JKB94].

3.2 Equal-Revenue Distributions
One particularly interesting distribution in mechanism design is obtained by
setting α = 1 in (3.1). The equal-revenue distribution, denoted ER, with
corresponding probability functions F(x) = 1 − 1/x and f(x) = 1/x2 for
x 1, has some elegant properties. Indeed, if we price an item at any
price x ∈ [1, +∞) in a single-item auction, the buyer’s expected revenue is
Rev(ER) = xF(x) = 1. On the other hand, its expected value is unbounded,
i.e. E(ER) = +∞. The same expected revenue is obtained by oﬀering the
agent any price in the distribution’s support, hence the name.
The equal revenue distribution lies on the boundary of regularity and
irregularity since it has virtual value φ is equal to zero:
φ(x) = x −
1 − (1 − 1/x)
1/x2
= 0. (3.2)
It turns out that ER-distributions reveal an important property of the max-
imum of n regular random variables. We shall discuss this in further details
in Section 5.
4 Maximum of MHR Random Variables
4.1 I.I.D. MHR Distributions
We show that the maximum of n i.i.d. MHR random variables is also MHR.
Let X1, . . . , Xn be a ﬁnite collection of i.i.d. MHR random variables. Since
each Xi is MHR, for the common CDF F, we have that
HF (x) =
d
dx
f(x)
1 − F(x)
0, (4.1)
on all of [γmin, γmax). Equivalently,
f (x) +
f(x)2
1 − F(x)
0 (4.2)
since 1 − F(x) > 0.
By (2.3) and (2.4), the hazard rate Hmax of Xmax = max1 i n{Xi} is
Hmax(x) =
nF(x)n−1f(x)
1 − F(x)n
, (4.3)

where we use the shorthand F := Fmax and f := fmax. We wish to show
that Hmax is non-decreasing on [γmin, γmax). Differentiating with respect to
x, we find
Hmax(x) =
nf (x)F(x)n−1
1 − F(x)n
+ nf(x)2
F(x)n−2 n − 1 + F(x)n
1 − F(x)n 2 . (4.4)
Since F(x)k ∈ [0, 1) for any k ∈ Z+, it suffices to show that
ξ(x) = f (x)F(x)n−1
+ f(x)2
F(x)n−2 n − 1 + F(x)n
1 − F(x)n
0 (4.5)
for all x ∈ [γmin, γmax).
We next prove an important inequality.
Proposition 1. The density functions F and f satisfy the inequality
F(x)
1 − F(x)
n − 1 + F(x)n
1 − F(x)n
. (4.6)
Proof. Note that n−1
k=0 F(x)k n. Thus
F(x)
1 − F(x)
=
1
1 − F(x)
− 1
n
1 − F(x) 1 + F(x) + · · · + F(x)n−1
− 1
=
n
1 − F(x)n
− 1
=
n − 1 + F(x)n
1 − F(x)n
,
(4.7)
giving the desired inequality.
Multiplying (4.6) by f(x)2F(x)n−2 ( 0), the inequality is preserved. Hence,
f(x)2
F(x)n−2 n − 1 + F(x)n
1 − F(x)n
f(x)2F(x)n−1
1 − F(x)
. (4.8)
By Proposition 1, we then have
ξ(x) f (x)F(x)n−1
+
f(x)2Fn−1(x)
1 − F(x)
= Fn−1
(x) f (x) +
f(x)2
1 − F(x)
0,
(4.9)
proving (4.5). Therefore, the maximum of n i.i.d. MHR distribution is
regular, as expected.

5 Maximum of Regular Random Variables
5.1 I.I.D. Regular Ristributions
We now consider the maximum of n i.i.d. regular random variables. As
before, let X1, . . . , Xn be a ﬁnite collection of i.i.d.-ER2 random variables
and let Xmax = max1 i n{Xi}. The main result is the following.
Theorem 1. Xmax is non-regular.
Proof. By (2.3) and (2.4), Xmax has CDF and PDF
Fmax(x) = F(x)n
= 1 −
1
x
n
(5.1)
fmax(x) = nF(x)n−1
f(x) = n 1 −
1
x
n−1
x−2
(5.2)
respectively. Thus, the virtual value function of Xmax is
φmax(x) = 1 −
1 − (1 − 1/x)n
n (1 − 1/x)n−1 x2
. (5.3)
We can now see by diﬀerentiating φmax that the virtual value function is
decreasing on a subinterval of [1, +∞). Indeed,
φmax(x) =
1
n
(n − 2x + 1)xn
(x − 1)−n
+ 2x − 1 + 1 (5.4)
and so taking x → ∞ we get
lim
x→∞
φmax(x) = −1. (5.5)
Thus, for x ∈ [1, +∞) large enough, the virtual value function is decreasing.
It follows that the maximum of n ER-distributions is non-regular.
We have found the desired counterexample for i.i.d. distributions.
2
For a one-dimensional distribution F, “i.i.d.-F” refers to a family of independent
random variables each distributed according to F.

5.2 Independent Non-Identical Regular Random Variables
We now turn to the case of n independent, not necessarily identical regular
random variables. The next main result is as follows.
Theorem 2. Let X1, . . . , Xn be n independent but not necessarily identical
regular random variables. Then, Xmax = max1 i n{Xi} is generally non-
regular.
Proof. To prove this, we exhibit a counterexample when n = 2. Consider the
random variables X1 and X2 with Pareto distributions (3.1) and parameter
α = 1 and β = 1
2, respectively. Denote by Fi and fi their respective CDF
and PDF, for i = 1, 2. We have shown in Section 3.1 that each Xi is regular.
Now, let Xmax = max{X1, X2}. By (2.5) and (2.6), Xmax has distribu-
tion functions
Fmax(x) = F1(x)F2(x) = 1 −
1
x
1 −
1
√
x
=
(
√
x − 1)(x − 1)
x3/2
(5.6)
fmax(x) =
d
dx
Fmax =
2
√
x + x − 3
2x5/2
. (5.7)
Computing the virtual value function and simplifying, we obtain
φmax(x) = −
x(x + 1)
2
√
x + x − 3
(5.8)
whose derivative is
φmax(x) = −
3x2 + x5/2 − 6x3/2 + x − 3
√
x
(2
√
x + x − 3)2
√
x
. (5.9)
Hence, φmax is negative when 3x2 + x5/2 − 6x3/2 + x − 3
√
x > 0 since the
denominator is always positive for x 1. Solving, we find that φmax is
decreasing for x > 2.21606 and thus Xmax is non-regular.
5.3 A Differential Approach
Another approach to solving φmax < 0 for some interval I ⊂ R+ is to dif-
ferentiate φmax in terms of F and consider the resulting non-linear second
order differential inequality
φmax(x) =
2F (x)2 − F(x)F (x) + F (x)
F (x)2
< 0 (5.10)

subject to the condition that F is a cumulative distribution function, i.e.
lim
x→−∞
F(x) = 0, lim
x→+∞
F(x) = 1, (5.11)
and F is càdlàg3. For an arbitrary F, define JF = 1/F. Differentiating JF ,
substituting into (5.10) and rearranging, we find
φmax(F )2
= JF (1 − F)3
< 0. (5.12)
Since F(x) ∈ [0, 1), an equivalent condition for non-regularity is strict con-
cavity of JF . Thus, every CDF F whose JF is strictly concave on I solves
the inequality. In other words, every function F = 1−1/j with j > 1 càdlàg,
increasing and concave on I, solves the differential inequality.
6 Conclusion
6.1 Summary
We first proved that the maximum of MHR random variables is indeed a
MHR random variable. Our goal was then to demonstrate that the super-
class of regular random variables does not satisfy this property. Namely we
demonstrated a counterexample, the maximum of n i.i.d.-ER random vari-
ables is non-regular. Further, we showed that even in the non-i.i.d. case, the
maximum of two Pareto distributions with parameters 1 and 1
2 respectively
is non-regular. A few clear questions arise about the possible generalizations
of these results.
6.2 Open Questions
If we consider the random variables X1 and X2 with Pareto distributions
(3.1) and parameters 0 < α < β < 1 without loss of generality, we can
obtain the CDF and PDF for Xmax = max{X1, X2}. Denote by Fi and fi
the respective CDF and PDF for each Xi, i = 1, 2.
3
These correspond to right continuous with left limits. The acronym comes from the
French version “continue à droite, limite à gauche”.

By (2.5) and (2.6), Xmax has distribution functions
Fmax(x) = F1(x)F2(x) = 1 −
1
xα
1 −
1
xβ
=
(xα − 1)(xβ − 1)
xα+β
(6.1)
fmax(x) =
d
dx
Fmax =
α(xβ − 1) + β(xα − 1)
xα+β+1
. (6.2)
Computing the virtual value function and simplifying, we obtain
φmax(x) = x − x
xα + xβ − 1
α(xβ − 1) + β(xα − 1)
(6.3)
which begs the following questions:
i. Can a closed form be obtained for parameters α and β such that Xmax
is non-regular for all such pairs?
ii. Can this be extended to n random variables with Pareto distributions
and parameters 0 < α1 < α2 < · · · < αn < 1?
iii. Is there a more general criterion for which regular non-MHR random
variables will satisfy this extreme value property?
6.3 Acknowledgements
The authors thank Assistant Professor Yang Cai for providing this interest-
ing research question as well as a thorough introduction to algorithmic game
theory.

References
[BWP63] Richard E. Barlow, Albert W. Marshall, and Frank Proschan,
Properties of Probability Distributions with Monotone Hazard
Rate, Annals of Mathematical Statistics 34 (1963), 375–389.
[CD11] Yang Cai and Constantinos Daskalakis, Extreme-Value Theo-
rems for Optimal Multidimensional Pricing, Annual IEEE Sym-
posium on Foundations of Computer Science 52 (2011).
[JKB94] Norman L. Johnson, Samuel Kotz, and N. Balakrishnan, Con-
tinuous Univariate Distributions, Vol. 1, 2nd ed., Wiley Series
in Probability and Mathematical Statistics, Wiley Interscience
publication (1994), chap. 20.
[HN14] Sergiu Hart and Noam Nisan, Approximate Revenue Maximiza-
tion with Multiple Items, arXiv:1204.1846v2 (2014), 10–11.
[Mye81] Roger B. Myerson, Optimal Auction Design, Mathematics of
Operations Research 6 (1981), 58–73.

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