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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 for k>2.
So and
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 !