On Empirical Distribution Functions Under Auxiliary Information

Erich Haeusler ()
Additional contact information
Erich Haeusler: University of Giessen, Mathematical Institute

Chapter Chapter 9 in From Statistics to Mathematical Finance, 2017, pp 157-172 from Springer

Abstract: Abstract Being the nonparametric maximum likelihood estimator, the classical empirical distribution function is the estimator of choice for a completely unknown distribution function. As shown by Qin and Lawless in (Ann Statist 22:300–325, 1994), in the presence of some nonparametric auxiliary information about the underlying distribution function the nonparametric maximum likelihood estimator is a modified empirical distribution function. It puts random masses on the observations in order to take the auxiliary information into account. Under a second moment condition Zhang in (Metrika 46:221–244, 1997) has proved a functional central limit theorem for this modified empirical distribution function. The covariance function of the centered Gaussian limit process in his result is smaller than the covariance function of the Browinan brigde limit process in Donsker’s functional central limit theorem for the classical empirical distribution function. If the auxiliary information about the underlying distribution function is knowledge of the mean, then the second moment condition in Zhang’s result requires square integrable random variables. In this note we will study integrable random variables with known mean which are not square integrable and will show that Zhang’s result is no longer true.

Date: 2017
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-50986-0_9

Ordering information: This item can be ordered from
http://www.springer.com/9783319509860

DOI: 10.1007/978-3-319-50986-0_9

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().