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