An investigation of certain characteristic properties of the exponential distribution based on maxima in small samples

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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

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• Joint work with
• Jose A. Villasenor
• Colegio de Postgraduados
• Montecillo, Mexico

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• As a change of pace, instead of looking at maxima
  of large samples.

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• As a change of pace, instead of looking at maxima
  of large samples.
• Lets look at smaller samples.

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• As a change of pace, instead of looking at maxima
  of large samples.
• Lets look at smaller samples.
• Really small samples !!

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• As a change of pace, instead of looking at maxima
  of large samples.
• Lets look at smaller samples.
• Really small samples !!
• In fact n2.

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• First, we note that neither (A) nor (A) is
  sufficient to guarantee that the Xs are
  exponential r.v.s.
• For geometrically distributed Xs, (A) and (A)
  both hold, since the corresponding spacings are
  independent.

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An obvious result
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• So, we have

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• 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.
• Its not completely trivial to achieve this.

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Power function distributions
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Power function distributions

• In this case we find

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Pareto (II) distributions
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Pareto (II) distributions

• Here the covariance is always positive

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Open question

• Does reciprocation always reverse the sign of the
  covariance ?

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• The hunt for a non-exponential example with zero
  covariance continues.

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• The hunt for a non-exponential example with zero
  covariance continues.
• What would you try ?

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• 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.

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Pareto (IV) or Burr distributions
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Pareto (IV) or Burr distributions

• So that

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Pareto (IV) or Burr distributions
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• Can you find a nicer example ?

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Extensions for ngt2

• Some negative results extend readily

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Back to Property (B)
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Back to Property (B)

• Recall
• This holds if the Xs are i.i.d. exponential
  r.v.s. It is unlikely to hold for other parent
  distributions. More on this later.

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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.

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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.

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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 well list 10 of them !!

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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 well list 10 of them !!
• Each one will yield an exponential
  characterization.

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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 well 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.

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Combining (B) and (C).

• The good news is that I dont plan to prove
• or even sketch the proofs of all 10.
• Well just consider a sample of them

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The 10 characteristic properties
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The 10 characteristic properties

• The following 10 properties all hold if the Xs
  are standard exponential r.v.s.

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The 10 characteristic properties
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The 10 characteristic properties
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Property (2)
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Property (2)

• PROOF
• Define
• then

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Property (2)

• PROOF continued
• and we conclude that

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Property (5)
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Property (5)

• PROOF
• From (5)

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Property (5)

• PROOF continued
• As before define
• It follows that
• We can write
• and

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Property (5)

• PROOF continued
• which implies that
• which implies that

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Property (5)

• PROOF continued
• For kgt2 we have
• which via induction yields for kgt2.
• So and

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Property (10)
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Property (10)

• PROOF Since
• we have
• and so

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Property (10)

• PROOF continued
• also

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Property (10)

• PROOF continued
• But
• so we have for every
  x,
• i.e., a constant failure rate 1, corresponding
  to a standard exponential distribution.

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• Since we have lots of time, we can also go
  through the remaining 7 proofs.

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• Since we have lots of time, we can also go
  through the remaining 7 proofs.
• HE CANT BE SERIOUS !!!

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Thank you for your attention
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Thank you for your attention

• and for suffering through 3 of the 10 proofs !