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Using elements to represent exchangeable sequences

 

Consider a second-order exchangeable sequence tex2html_wrap_inline34798 . This has the property (see [37]) that each tex2html_wrap_inline34834 may be decomposed as tex2html_wrap_inline34800 , where

displaymath34796

are uncorrelated; and where the tex2html_wrap_inline34808 have expectation zero and common variance tex2html_wrap_inline34810 . For such sequences it is typically possible to learn only about the common mean quantity M. Therefore, the sequence can be summarised by knowledge of tex2html_wrap_inline34812 , tex2html_wrap_inline34814 , tex2html_wrap_inline34810 , and the notion of repetition.

In [B/D] we choose to represent an exchangeable sequence by a solitary element with expectation tex2html_wrap_inline34812 being the expectation common to each tex2html_wrap_inline34834 . For the variance quantities tex2html_wrap_inline34814 and tex2html_wrap_inline34810 (the sum tex2html_wrap_inline34824 is the common variance tex2html_wrap_inline34826 ) we associate with the element two different variance storage areas. For example, to summarise this exchangeable sequence we could define an element called X with associated expectation tex2html_wrap_inline34828 , and specify two kinds of variances: tex2html_wrap_inline34830 and tex2html_wrap_inline34832 . (it is usually more covenient to store the overall variance, as the residual variance can always be found by subtraction if necessary).



David Wooff
Wed Oct 21 15:14:31 BST 1998