Independent Poisson Random Variables Research Articles

Tail bounds for weighted sums of independent Poisson random variables

Tail bounds for weighted sums of independent Poisson random variables

Bayesian spatio‐temporal survival analysis for all types of censoring with application to a wildlife disease study

AbstractIn this article, we consider modeling arbitrarily censored survival data with spatio‐temporal covariates. We demonstrate that under the piecewise constant hazard function, the likelihood for uncensored or right‐censored subjects is proportional to the likelihood of multiple conditionally independent Poisson random variables. To address left‐ or interval‐censored subjects, we propose to impute the exact event times and convert them into uncensored subjects, enabling the application of the integrated nested Laplace approximation to update model parameters using the imputed data. We introduce an iterative algorithm that alternates between imputing event times for left‐ and interval‐censored subjects and re‐estimating model parameters. The proposed method is assessed through a simulation study and applied to analyze a spatio‐temporal survival dataset in a wildlife disease study investigating bovine tuberculosis in white‐tailed deer in Michigan.

Cycles in Mallows random permutations

AbstractWe study cycle counts in permutations of drawn at random according to the Mallows distribution. Under this distribution, each permutation is selected with probability proportional to , where is a parameter and denotes the number of inversions of . For fixed, we study the vector where denotes the number of cycles of length in and is sampled according to the Mallows distribution. When the Mallows distribution simply samples a permutation of uniformly at random. A classical result going back to Kolchin and Goncharoff states that in this case, the vector of cycle counts tends in distribution to a vector of independent Poisson random variables, with means . Here we show that if is fixed and then there are positive constants such that each has mean and the vector of cycle counts can be suitably rescaled to tend to a joint Gaussian distribution. Our results also show that when there is a striking difference between the behavior of the even and the odd cycles. The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution. For the odd cycle counts on the other hand, the limiting behavior depends on the parity of when . Both and have discrete limiting distributions—they do not need to be renormalized—but the two limiting distributions are distinct for all . We describe these limiting distributions in terms of Gnedin and Olshanski's bi‐infinite extension of the Mallows model. We investigate these limiting distributions further, and study the behavior of the constants involved in the Gaussian limit laws. We for example show that as the expected number of 1‐cycles tends to —which, curiously, differs from the value corresponding to . In addition we exhibit an interesting “oscillating” behavior in the limiting probability measures for and odd versus even.

Behavioral event detection and rate estimation for autonomous vehicle evaluation

AbstractAutonomous vehicles are continually increasing their presence on public roads. However, before any new autonomous driving software can be approved, it must first undergo a rigorous assessment of driving quality. These quality evaluations typically focus on estimating the frequency of (undesirable) behavioral events. While rate estimation would be straight‐forward with complete data, in the autonomous driving setting this estimation is greatly complicated by the fact that detecting these events within large driving logs is a non‐trivial task that often involves human reviewers. In this article, we outline a streaming partial tiered event review configuration that ensures both high recall and high precision on the events of interest. In addition, the framework allows for valid streaming estimates at any phase of the data collection process, even when labels are incomplete, for which we develop the maximum likelihood estimate and show it is unbiased. Constructing honest and effective confidence intervals (CI) for these rate estimates, particularly for rare safety‐critical events, is a novel and challenging statistical problem due to the complexity of the data likelihood. We develop and compare several CI approximations, including a novel gamma CI method that approximates the exact but intractable distribution with a weighted sum of independent Poisson random variables. There is a clear trade‐off between statistical coverage and interval width across the different CI methods, and the extent of this trade‐off varies depending on the specific application settings (e.g., rare vs. common events). In particular, we argue that our proposed CI method is the best‐suited when estimating the rate of safety‐critical events where guaranteed coverage of the true parameter value is a prerequisite to safely launching a new ADS on public roads.

The Simplest Unlimited-Player Game of Skill

The Least Unique Positive Integer game (LUPI) is among the simplest games that can be played by any number of players, N > 2, and has a nontrivial strategic component. In LUPI, players try to pick the smallest positive integer nobody else picks. Despite its simplicity, the game was not widely known until fairly recently. It was actually offered as a state run lottery game in Sweden in 2007, but players collaborated, and the game was quickly stopped. LUPI has also been proposed as the basis for a reverse auction system, and here it is proposed as a “party game”. The Nash equilibrium for the game has been previously worked out in the case where the numbers of players that make each choice are independent Poisson random variables, an assumption that can often be justified when N is large. Here we summarize previous work and derive a number of interesting new results on Nash equilibria when N is small, and when N is large using the Poisson assumption. We also investigate whether the Nash equilibrium strategies for LUPI games are evolutionary stable strategies. Finally, we look at cheating strategies for LUPI and devise ways to make it harder to cheat.

Sparse matrices: convergence of the characteristic polynomial seen from infinity

We prove that the reverse characteristic polynomial det(In−zAn) of a random n×n matrix An with iid Bernoulli(d∕n) entries converges in distribution towards the random infinite product ∏ℓ=1∞(1−zℓ)Yℓ where Yℓ are independent Poisson(dℓ∕ℓ) random variables. We show that this random function is a Poisson analog of more classical Gaussian objects such as the Gaussian holomorphic chaos. As a byproduct, we obtain new simple proofs of previous results on the asymptotic behaviour of extremal eigenvalues of sparse Erdős-Rényi digraphs: for every d>1, the greatest eigenvalue of An is close to d and the second greatest is smaller than d, a Ramanujan-like property for irregular digraphs. For d<1, the only non-zero eigenvalues of An converge to a Poisson multipoint process on the unit circle. Our results also extend to the semi-sparse regime where d is allowed to grow to ∞ with n, slower than no(1). We show that the reverse characteristic polynomial converges towards a more classical object written in terms of the exponential of a log-correlated real Gaussian field. In the semi-sparse regime, the empirical spectral distribution of An∕ dn converges to the circle distribution; as a consequence of our results, the second eigenvalue sticks to the edge of the circle.

Cycle structure of random parking functions

We initiate the study of the cycle structure of uniformly random parking functions. Using the combinatorics of parking completions, we compute the asymptotic expected value of the number of cycles of any fixed length. We obtain an upper bound on the total variation distance between the joint distribution of cycle counts and independent Poisson random variables using a multivariate version of Stein's method via exchangeable pairs. Under a mild condition, the process of cycle counts converges in distribution to a process of independent Poisson random variables.

Predicting Parallelism and Quantifying Divergence in Microbial Evolution Experiments.

ABSTRACTThe degree to which independent populations subjected to identical environmental conditions evolve in similar ways is a fundamental question in evolution. To address this question, microbial populations are often experimentally passaged in a given environment and sequenced to examine the tendency for similar mutations to repeatedly arise. However, there remains the need to develop an appropriate statistical framework to identify genes that acquired more mutations in one environment than in another (i.e., divergent evolution), genes that serve as genetic candidates of adaptation. Here, we develop a mathematical model to evaluate evolutionary outcomes among replicate populations in the same environment (i.e., parallel evolution), which can then be used to identify genes that contribute to divergent evolution. Applying this approach to data sets from evolve-and-resequence experiments, we found that the distribution of mutation counts among genes can be predicted as an ensemble of independent Poisson random variables with zero free parameters. Building on this result, we propose that the degree of divergent evolution at a given gene between populations from two different environments can be modeled as the difference between two Poisson random variables, known as the Skellam distribution. We then propose and apply a statistical test to identify specific genes that contribute to divergent evolution. By focusing on predicting patterns among replicate populations in a given environment, we are able to identify an appropriate test for divergence between environments that is grounded in first principles.IMPORTANCE There is currently no universally accepted framework for identifying genes that contribute to molecular divergence between microbial populations in different environments. To address this absence, we developed a null model to describe the distribution of mutation counts among genes. We find that divergent evolution within a given gene can be modeled as the absolute difference in the total number of mutations observed between two environments. This quantity is effectively captured by a probability distribution known as the Skellam distribution, providing an appropriate statistical test for researchers seeking to identify the set of genes that contribute to divergent evolution in microbial evolution experiments.

On infinitely divisible multivariate gamma distributions

We provide an explicit construction of variables from a bivariate gamma distribution which is infinitely divisible. These distributions are currently only known through their (complicated) density functions or moment generating functions. The sampling construction uses independent gamma and Poisson random variables and sheds further light on these distributions and their infinitely divisible property. The extension to multivariate gamma infinitely divisible variables is also provided, as well as a negative binomial case.

Asymptotic distribution of Bernoulli quadratic forms

Consider the random quadratic form Tn=∑ 1≤u<v≤nauvXuXv, where ((auv))1≤u,v≤n is a {0,1}-valued symmetric matrix with zeros on the diagonal, and X1,X2,…,Xn are i.i.d. Ber(pn), with pn∈(0,1). In this paper, we prove various characterization theorems about the limiting distribution of Tn, in the sparse regime, where pn→0 such that E(Tn)=O(1). The main result is a decomposition theorem showing that distributional limits of Tn is the sum of three components: a mixture which consists of a quadratic function of independent Poisson variables; a linear Poisson mixture, where the mean of the mixture is itself a (possibly infinite) linear combination of independent Poisson random variables; and another independent Poisson component. This is accompanied with a universality result which allows us to replace the Bernoulli distribution with a large class of other discrete distributions. Another consequence of the general theorem is a necessary and sufficient condition for Poisson convergence, where an interesting second moment phenomenon emerges.

Joint Poisson distribution of prime factors in sets

AbstarctGiven disjoint subsets T1, …, Tm of “not too large” primes up to x, we establish that for a random integer n drawn from [1, x], the m-dimensional vector enumerating the number of prime factors of n from T1, …, Tm converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T1, …, Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.

Some properties of degenerate complete and partial Bell polynomials

In this paper, we study degenerate complete and partial Bell polynomials and establish some new identities for those polynomials. In addition, we investigate the connections between modified degenerate complete and partial Bell polynomials, which are closely related to the degenerate complete and partial Bell polynomials, and the joint distribution of weighted sums of independent degenerate Poisson random variables.

A probabilistic interpretation of the Bell polynomials

In this paper, we obtain a probabilistic relationship between the exponential Bell polynomials and the weighted sums of independent Poisson random variables. A recently established probabilistic connection between the Adomian polynomials and independent Poisson random variables can be derived from the obtained relationship. This result has importance because any known identity for the exponential Bell polynomials will generate a new identity for the Poisson random variables. We use the obtained relationship to derive several new identities for the joint distribution of weighted sums of independent Poisson random variables. Few examples are provided that substantiate the obtained identities.

The binomial-match, outcome uncertainty, and the case of netball

We introduce the binomial-match as a model for the bivariate score in a paired-contest. This model is naturally associated with sports in which the restart alternates following a goal. The model is a challenger to the Poisson-match, a pair of independent Poisson random variables whose means are related to the strengths of the competing teams. We use the binomial-match primarily to study the relationship between outcome uncertainty and scoring-rate, particularly for high values of the scoring-rate. Netball has a high scoring-rate and motivates our model development. In the binomial-match framework, we also evaluate rule-variations, and study tactical play in netball. Our analysis suggests that the binomial-match is not a better forecaster than the Poisson-match, but it is better for representing outcome uncertainty and evaluating rule-variations and tactics. In general, we find that the binomial-match implies greater outcome uncertainty than the Poisson match, for a given scoring-rate, and that an alternating-restart is a good rule for reducing the frequency of tied outcomes. For netball in particular, we show that starting the final quarter with possession in a close, balanced match may confer a significant advantage.

Precise asymptotics of longest cycles in random permutations without macroscopic cycles

We consider Ewens random permutations of length n conditioned to have no cycle longer than nβ with 0<β<1 and study the asymptotic behaviour as n→∞. We obtain very precise information on the joint distribution of the lengths of the longest cycles; in particular we prove a functional limit theorem where the cumulative number of long cycles converges to a Poisson process in the suitable scaling. Furthermore, we prove convergence of the total variation distance between joint cycle counts and suitable independent Poisson random variables up to a significantly larger maximal cycle length than previously known. Finally, we remove a superfluous assumption from a central limit theorem for the total number of cycles proved in an earlier paper.

A flexible multivariate model for high-dimensional correlated count data

We propose a flexible multivariate stochastic model for over-dispersed count data. Our methodology is built upon mixed Poisson random vectors (Y1,…,Yd), where the {Yi} are conditionally independent Poisson random variables. The stochastic rates of the {Yi} are multivariate distributions with arbitrary non-negative margins linked by a copula function. We present basic properties of these mixed Poisson multivariate distributions and provide several examples. A particular case with geometric and negative binomial marginal distributions is studied in detail. We illustrate an application of our model by conducting a high-dimensional simulation motivated by RNA-sequencing data.

On a tail bound for analyzing random trees

We re-examine a lower-tail upper bound for the random variable X=∏i=1∞min∑k=1iEk,1,where E1,E2,…∼iidExp(1). This bound has found use in root-finding and seed-finding algorithms for randomly growing trees, and was initially established in the context of the uniform attachment tree model. We first show that X has a useful representation as a compound product of uniform random variables that allows us to determine its moments and refine the existing nonasymptotic bound. Next we demonstrate that the lower-tail probability for X can equivalently be written as a probability involving two independent Poisson random variables, an equivalence that yields a novel general result regarding independent Poissons and that also enables us to obtain tight asymptotic bounds on the tail probability of interest.

Roundabout model with on-ramp queues: Exact results and scaling approximations.

This paper introduces a general model of a single-lane roundabout, represented as a circular lattice that consists of L cells, with Markovian traffic dynamics. Vehicles enter the roundabout via on-ramp queues that have stochastic arrival processes, remain on the roundabout a random number of cells, and depart via off-ramps. Importantly, the model does not oversimplify the dynamics of traffic on roundabouts, while various performance-related quantities (such as delay and queue length) allow an analytical characterization. In particular, we present an explicit expression for the marginal stationary distribution of each cell on the lattice. Moreover, we derive results that give insight on the dependencies between parts of the roundabout, and on the queue distribution. Finally, we find scaling limits that allow, for every partition of the roundabout in segments, to approximate (i) the joint distribution of the occupation of these segments by a multivariate Gaussian distribution, and (ii) the joint distribution of their total queue lengths by a collection of independent Poisson random variables. To verify the scaling limit statements, we develop a way to empirically assess convergence in distribution of random variables.

Feller coupling of cycles of permutations and Poisson spacings in inhomogeneous Bernoulli trials

Feller (1945) provided a coupling between the counts of cycles of various sizes in a uniform random permutation of $[n]$ and the spacings between successes in a sequence of $n$ independent Bernoulli trials with success probability $1/n$ at the $n$th trial. Arratia, Barbour and Tavaré (1992) extended Feller’s coupling, to associate cycles of random permutations governed by the Ewens $(\theta )$ distribution with spacings derived from independent Bernoulli trials with success probability $\theta /(n-1+\theta )$ at the $n$th trial, and to conclude that in an infinite sequence of such trials, the numbers of spacings of length $\ell $ are independent Poisson variables with means $\theta /\ell $. Ignatov (1978) first discovered this remarkable result in the uniform case $\theta = 1$, by constructing Bernoulli $(1/n)$ trials as the indicators of record values in a sequence of i.i.d. uniform $[0,1]$ variables. In the present article, the Poisson property of inhomogeneous Bernoulli spacings is explained by a variation of Ignatov’s approach for a general $\theta >0$. Moreover, our approach naturally provides random permutations of infinite sets whose cycle counts are exactly given by independent Poisson random variables.

The Second-Moment Phenomenon for Monochromatic Subgraphs

What is the chance that among a group of $n$ friends, there are $s$ friends all of whom have the same birthday? This is the celebrated birthday problem which can be formulated as the existence of a monochromatic $s$-clique $K_s$ ($s$-matching birthdays) in the complete graph $K_n$, where every vertex of $K_n$ is uniformly colored with 365 colors (corresponding to birthdays). More generally, for a general connected graph $H$, let $T(H, G_n)$ be the number of monochromatic copies of $H$ in a uniformly random coloring of the vertices of the graph $G_n$ with $c_n$ colors. In this paper we show that $T(H, G_n)$ converges to ${Pois}(\lambda)$ whenever $\mathbb{E} T(H, G_n) \rightarrow \lambda$ and ${Var} T(H, G_n) \rightarrow \lambda$, that is, the asymptotic Poisson distribution of $T(H, G_n)$ is determined just by the convergence of its mean and variance. Moreover, this condition is necessary if and only if $H$ is a star-graph. In fact, the second-moment phenomenon is a consequence of a more general theorem about the convergence of $T(H,G_n)$ to a finite linear combination of independent Poisson random variables. As an application, we derive the limiting distribution of $T(H, G_n)$, when $G_n\sim G(n, p)$ is the Erdös--Rényi random graph. Multiple phase transitions emerge as $p$ varies from 0 to 1, depending on whether the graph $H$ is balanced or unbalanced.