Skip to main content Accessibility help
Home
Hostname: page-component-564cf476b6-lwxm7 Total loading time: 0.226 Render date: 2021-06-18T19:20:13.400Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Uniform order statistics property and ℓ∞-spherical densities

Published online by Cambridge University Press:  18 December 2007

Corresponding
E-mail address:

Abstract

Type
Errata
Information
Copyright
Copyright © Cambridge University Press 2008

(This article appeared in Volume 18, Number 3, 2004 pages 275–297)

Professor Taizhong Hu has pointed out to us that the interpretation that follows the definition of the UOSP(≤) property in page 287 in the above paper is incorrect. As a consequence, Theorem 4.7 and Remark 4.8 in the above paper need to be modified.

In order to do that, we replace the definition of the UOSP(≤) property in the above paper by two definitions that are given below. Let the discrete random variables X 1, X 2, … be such that P{0 ≤ X 1 ≤ X 2 ≤ ···} = 1. We say that these X i's have the UOSP1(≤) property if for discrete 0 ≤ x 1 ≤ x 2 ≤ ··· ≤ x k ≤ t we have

where, for l ∈ {0, 1, … , t}, j l is the number of values in {x 1, x 2, … , x k} that are equal to l. That is, conditional on X k ≤ t and X k+1 > t, the random variables X 1 ≤ X 2 ≤ ··· ≤ X k are distributed as order statistics of a sample of size k drawn from the set {0, 1, … , t} with replacement. On the other hand, we say that these X i's have the UOSP2(≤) property if for discrete 0 ≤ x 1 ≤ x 2 ≤ ··· ≤ x k ≤ t we have

this is the definition of the UOSP(≤) property given in the original paper. The meaning of this definition is that conditional on X k ≤ t and X k+1 > t, the random variables X 1 ≤ X 2 ≤ ··· ≤ X k are distributed as order statistics of a sample of size k drawn from the set {0, 1, … , t} with double replacement; see (23) in page 184 of de Finetti (Reference de Finetti1975) and see also Exercise 1.62 in page 41 of Spizzichino (Reference Spizzichino2001).

With these definitions we first note that Proposition 4.2 in the original paper remains correct if UOSP(≤) is understood to mean UOSP2(≤).

Next, let {B(t), t = 0, 1, … } be a nondecreasing discrete-time discrete-state random process as described in page 291 of the original paper, and let T 1, T 2, … be the corresponding “unit jump” times, again, as described in page 291 of the original paper. We say that the process {B(t), t = 0, 1, …} has the UOSP1(≤) [respectively, UOSP2(≤)] property if T 1, T 2, … have the UOSP1(≤) [respectively, UOSP2(≤)] property. Then

• item (i) in Theorem 4.7 of the original paper, with UOSP1(≤) instead of UOSP(≤), is equivalent to item (ii) of that theorem, and

• item (i) in Theorem 4.7 of the original paper, with UOSP2(≤) instead of UOSP(≤), is equivalent to each of the items (iii), (iv), and (v) of that theorem.

Finally there are a couple of minor corrections in Remark 4.8. In lines 1 and 8 on page 295, UOSP2(≤) should replace UOSP(≤). The claim that the processes from Theorem 4.5 satisfy “a version of the statement in Theorem 4.7(ii)” is not true.

References

1.de Finetti, B. (1975). Theory of Probability, Volume 2. John Wiley & Sons, London.Google Scholar
2.Spizzichino, F. (2001). Subjective Probability Models for Lifetimes. Chapman & Hall/CRC, Boca Raton.CrossRefGoogle Scholar
You have Access
1
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Uniform order statistics property and ℓ-spherical densities
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Uniform order statistics property and ℓ-spherical densities
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Uniform order statistics property and ℓ-spherical densities
Available formats
×
×

Reply to:Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *