[Paper Review] Size bias, sampling, the waiting time paradox, and infinite divisibility: when is the increment independent?

Richard Arratia, Larry Goldstein|arXiv (Cornell University)|Jul 22, 2010

Bayesian Methods and Mixture Models12 references20 citations

TL;DR

This paper investigates the conditions under which a random variable $X$ with a size-biased distribution $X^*$ can be expressed as $X^* = X + Y$ with $Y \geq 0$ and $X$ and $Y$ independent. It establishes that this decomposition is possible if and only if $X$ is infinitely divisible, linking size biasing to renewal theory, the waiting time paradox, and number-theoretic distributions such as Dickman’s and Buchstab’s functions.

ABSTRACT

With $X^*$ denoting a random variable with the $X$-size bias distribution, what are all distributions for $X$ such that it is possible to have $X^*=X+Y$, $Y\geq 0$, with $X$ and $Y$ {\em independent}? We give the answer, due to Steutel \cite{steutel}, and also discuss the relations of size biasing to the waiting time paradox, renewal theory, sampling, tightness and uniform integrability, compound Poisson distributions, infinite divisibility, and the lognormal distributions.

Motivation & Objective

• To determine all distributions of nonnegative random variables $X$ for which $X^* = X + Y$ holds with $Y \geq 0$ and $X$, $Y$ independent.
• To clarify the connection between size biasing and the waiting time paradox in Poisson processes.
• To examine the role of size biasing in sampling, renewal theory, and infinite divisibility.
• To characterize distributions arising from size biasing, particularly those linked to Dickman’s and Buchstab’s functions in number theory.
• To unify concepts of size biasing, compound Poisson laws, and infinite divisibility through the decomposition $X^* = X + Y$.

Proposed method

• Uses the size-biased distribution definition: $\mathbb{E}[g(X^*)] = \frac{1}{\mathbb{E}[X]} \mathbb{E}[X g(X)]$ for bounded continuous $g$.
• Applies the transformation $dF_{X^*}(x) = \frac{x}{\mathbb{E}[X]} dF(x)$ to derive the density of $X^*$ from $X$.
• Analyzes the waiting time paradox via exponential interarrival times and the paradox of memoryless vs. length-biased sampling.
• Derives differential-difference equations for densities under size biasing, particularly for $g_a(x)$ in the case of uniform $Y$ on $(0,1)$ or $(\beta,1)$.
• Establishes that $X^* = X + Y$ with independent $Y \geq 0$ holds if and only if $X$ is infinitely divisible.
• Uses convolution powers and compound Poisson constructions to link $X$ to number-theoretic functions like Dickman’s $\rho(u)$ and Buchstab’s $\omega(u)$.

Experimental results

Research questions

• RQ1 For which nonnegative random variables $X$ does there exist an independent $Y \geq 0$ such that $X^* = X + Y$?
• RQ2 How does size biasing resolve the waiting time paradox in Poisson processes?
• RQ3 What is the connection between size biasing and infinite divisibility in distributional decompositions?
• RQ4 How do the densities of size-biased $X$ relate to differential-difference equations and number-theoretic functions?
• RQ5 What role does the size-biased distribution play in sampling from populations with heterogeneous group sizes?

Key findings

• The decomposition $X^* = X + Y$ with $X$ and $Y$ independent and $Y \geq 0$ holds if and only if $X$ is infinitely divisible.
• The waiting time paradox is resolved by recognizing that an arrival time uniformly distributed in an interval leads to a length-biased distribution of interarrival times, not memoryless expectation.
• For $X$ with $\mathbb{E}[X] = 1$, the size-biased version $X^*$ satisfies $\mathbb{P}(X^* = k) = k \mathbb{P}(X = k)$, which explains sampling bias in group size surveys.
• Dickman’s function $\rho(u)$ arises as the density of $X$ when $Y \sim \text{Uniform}(0,1)$, and satisfies $\rho'(x) = -\rho(x-1)/x$ for $x > 0$, with $\rho(x) = 1$ on $[0,1]$.
• Buchstab’s function $\omega(u)$ governs the distribution of the smallest prime factor of random integers, with $\mathbb{P}(a < X < b) = \int_a^b \omega(x/\beta) \, dx$ for $X$ compound Poisson with $\mathbb{P}(X=0) = \beta^t$.
• The size-biased distribution of $X$ leads to a defective density when $\mathbb{P}(X=0) > 0$, as in the case $\beta > 0$, and connects to the asymptotic proportion of integers with no small prime factors.