SMS scnews item created by John Ormerod at Thu 8 May 2014 1110
Type: Seminar
Distribution: World
Expiry: 17 May 2014
Calendar1: 16 May 2014 1400-1500
CalLoc1: Carslaw 173
CalTitle1: Statistics Seminar: Ngoc Tran -- Size-biased permutation for a finite i.i.d sequence
Auth: jormerod@pjormerod4.pc (assumed)

Statistics Seminar: Ngoc Tran -- Title: Size-biased permutation for a finite i.i.d sequence

Abstract: Line up n blocks of lengths X_1 ... X_n, where the X_i’s
are independent and identically distributed (i.i.d) positive random
variables. Throw a ball uniformly at random on the interval 0 to
X_1 + ... + X_n, and record length of the block X_i that the ball
falls in, and then remove this block. This is called size-biased
sampling, since the bigger blocks are more likely to be discovered
earlier. Do this recursively n times. This yields a size-biased
permutation of the finite i.i.d sequence (X_1, ... X_n).

This model is known as Kingman’s paintbox, put forward by Kingman
in 1976 to study random infinite partitions. In the infinite
version of the above model, the X_i’s are jumps of a subordinator.
In 1992, Perman, Pitman and Yor derived various distributional
properties of this infinite size-biased permutation. Their work
found applications in species sampling, oil and gas discovery,
topic modeling in Bayesian statistics, amongst many others.

In this talk, we will derive distributional properties of finite
i.i.d size-biased permutation, both for fixed and asymptotic n.
We have multiple derivations using tools from Perman-Pitman-Yor,
as well as the induced order statistics literature. Their
comparisons lead to new results, as well as simpler proofs of
existing ones. Our main contribution describes the asymptotic
distribution of the last few terms in a finite i.i.d size-biased
permutation via a Poisson coupling with its few smallest order
statistics. For example, we will answer the question: what is the
asymptotic probability that the smallest block is discovered last?

Joint work with Jim Pitman


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