An investigation of certain
characteristic properties of the
exponential distribution based on
maxima in small samples
Barry C. Arnold
University of California, Riverside
Joint work with
Jose A. Villasenor
Colegio de Postgraduados
Montecillo, Mexico
As a change of pace, instead of looking at
maxima of large samples.
As a change of pace, instead of looking at
maxima of large samples.
Let’s look at smaller samples.
As a change of pace, instead of looking at
maxima of large samples.
Let’s look at smaller samples.
Really small samples !!
As a change of pace, instead of looking at
maxima of large samples.
Let’s look at smaller samples.
Really small samples !!
In fact n=2.
First, we note that neither (A*) nor (A**) is
sufficient to guarantee that the X’s are
exponential r.v.s.
For geometrically distributed X’s, (A*) and
(A**) both hold, since the corresponding
spacings are independent.
An obvious result
v
So, we have
Weibull distributions provide examples in
which the covariance between the first two
spacings is positive, negative or zero (in the
exponential case).
But we seek an example in which we have
zero covariance for a non-exponential
distribution.
It’s not completely trivial to achieve this.
Power function distributions
Power function distributions
In this case we find:
Pareto (II) distributions
Pareto (II) distributions
Here the covariance is always positive
Open question
Does reciprocation always reverse the sign
of the covariance ?
The hunt for a non-exponential example
with zero covariance continues.
The hunt for a non-exponential example
with zero covariance continues.
What would you try ?
The hunt for a non-exponential example
with zero covariance continues.
What would you try ?
Success is just around the corner, or rather
on the next slide.
Pareto (IV) or Burr distributions
Pareto (IV) or Burr distributions
So that
Pareto (IV) or Burr distributions
Can you find a “nicer” example ?
Extensions for n>2
Some negative results extend readily:
Back to Property (B)
Back to Property (B)
Recall:
This holds if the X’s are i.i.d. exponential
r.v.’s. It is unlikely to hold for other parent
distributions. More on this later.
Another exponential property
If a r.v. has a standard exponential
distribution (with mean 1) then its density
and its survival function are identical, thus
And it is well-known that property (C) only
holds for the standard exponential
distribution.
Combining (B) and (C).
By taking various combinations of (B) and
(C) we can produce a long list of unusual
distributional properties, that do hold for
exponential variables and are unlikely to
hold for other distributions.
Combining (B) and (C).
By taking various combinations of (B) and
(C) we can produce a long list of unusual
distributional properties, that do hold for
exponential variables and are unlikely to
hold for other distributions.
In fact we’ll list 10 of them !!
Combining (B) and (C).
By taking various combinations of (B) and
(C) we can produce a long list of unusual
distributional properties, that do hold for
exponential variables and are unlikely to
hold for other distributions.
In fact we’ll list 10 of them !!
Each one will yield an exponential
characterization.
Combining (B) and (C).
By taking various combinations of (B) and
(C) we can produce a long list of unusual
distributional properties, that do hold for
exponential variables and are unlikely to
hold for other distributions.
In fact we’ll list 10 of them !!
Each one will yield an exponential
characterization.
They appear to be closely related, but no
one of them implies any other one.
Combining (B) and (C).
The good news is that I don’t plan to prove
or even sketch the proofs of all 10.
We’ll just consider a sample of them
The 10 characteristic properties
The 10 characteristic properties
The following 10 properties all hold if the X’s
are standard exponential r.v.’s.
The 10 characteristic properties
The 10 characteristic properties
Property (2)
Property (2)
PROOF:
Define
then
Property (2)
PROOF continued:
and we conclude that
Property (5)
Property (5)
PROOF:
From (5)
Property (5)
PROOF continued:
As before define
It follows that
We can write
and
Property (5)
PROOF continued:
which implies that
which implies that
Property (5)
PROOF continued:
For k>2 we have
which via induction yields
So
and
for k>2.
Property (10)
Property (10)
PROOF: Since
we have
and so
Property (10)
PROOF continued:
also
Property (10)
PROOF continued:
But
so we have
for every x,
i.e., a constant failure rate =1, corresponding
to a standard exponential distribution.
Since we have lots of time, we can also go
through the remaining 7 proofs.
Since we have lots of time, we can also go
through the remaining 7 proofs.
HE CAN’T BE SERIOUS !!!
Thank you for your attention
Thank you for your attention
and for suffering through 3 of the 10 proofs !