Barry C. Arnold University of California, Riverside

1. An investigation of certain characteristic properties of the exponential distribution based on maxima in small samples Barry C. Arnold University of California, Riverside

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

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

4. As a change of pace, instead of looking at maxima of large samples. Let’s look at smaller samples.

5. As a change of pace, instead of looking at maxima of large samples. Let’s look at smaller samples. Really small samples !!

6. 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.

16. 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.

17. An obvious result

26. v So, we have

28. 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.

29. Power function distributions

30. Power function distributions • In this case we find:

31. Pareto (II) distributions

32. Pareto (II) distributions • Here the covariance is always positive

33. Open question Does reciprocation always reverse the sign of the covariance ?

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

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

36. 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.

37. Pareto (IV) or Burr distributions

38. Pareto (IV) or Burr distributions So that

39. Pareto (IV) or Burr distributions

40. Can you find a “nicer” example ?

41. Extensions for n>2 Some negative results extend readily:

42. Back to Property (B)

43. 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.

44. 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.

45. 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.

46. 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 !!

47. 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.

48. 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.

49. 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

50. The 10 characteristic properties