The Poisson Approximation to the Poisson Binomial Distribution

Hodges, J. L.; Cam, Lucien Le

doi:10.1214/aoms/1177705799

Cited by 151 publications

(69 citation statements)

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“…Prohorov (1953), Hodges & Le Cam (1960) [also known as the Hodges-Le Cam theorem] and Chen (1975) can be regarded as the milestones for quantifying the Poisson limit theorem. However, Barbour and Hall (1984) proved that the accuracy of Poisson approximation in total variation is determined by the rarity of the events and its order does not improve when the sample size increases.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

Xia

Zhang

2009

J Theor Probab

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The polynomial birth-death distribution (abbr. as PBD) on I = {0, 1, 2, ...} or I = {0, 1, 2, ..., m} for some finite m introduced in Brown & Xia (2001) is the equilibrium distribution of the birth-death process with birth rates {α i } and death rates {β i }, where α i ≥ 0 and β i ≥ 0 are polynomial functions of i ∈ I. The family includes Poisson, negative binomial, binomial and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein's factors for the PBD approximation with α i = a and β i = i + bi(i − 1) in terms of the Wasserstein distance. The paper complements the work of Brown & Xia (2001) and generalizes the work of Barbour & Xia (2006) where Poisson approximation (b = 0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

Xia

Zhang

2009

J Theor Probab

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“…The probability mass function of (18) is easily seen to be a Poisson binomial distribution [16] with parameters {N, p 1 , p 2 , . .…”

Section: 1 Single Stage Performancementioning

confidence: 99%

“…For large N the number of terms in the summation of (18) becomes very large. However, the Poisson binomial distribution is accurately approximated, with known total variation, by either the binomial [17] or Poisson distributions [16], depending on the distribution of p 1 , . .…”

Section: 1 Single Stage Performancementioning

confidence: 99%

Lattice Codes and Generalized Minimum-Distance Decoding for OFDM Systems

Clark

Taylor

2007

IEEE Trans. Commun.

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Lattice coding of orthogonal frequency division multiplexing (OFDM) systems is considered.Mapping of multilevel construction lattices to OFDM blocks is shown, and a methodology for probabilistic analysis of multistage generalized minimum distance (GMD) decoding of the received OFDM blocks is derived. As a case study transmission of points from a 128-dimensional Barnes-Wall lattice is considered. Tight approximations to the system error rate are obtained and verified by simulation. It appears that GMD decoding of lattice encoded OFDM provides high coding gain at low complexity.2

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“…This generalized distribution is referred to as the Poisson-Binomial distribution (Hodges and Le Cam, 1960), and is presented in equation (53). In equation From equation (53), the entire string of drought probabilities is used to estimate the sum of the probabilities of each combination of drought events occurring, up to the observed 66 number of droughts.…”

Section: Exact Solutionmentioning

confidence: 99%

Quantifying the Impacts of Initial Condition and Model Uncertainty on Hydrological Forecasts

DeChant¹

2000

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Forecasts of hydrological information are vital for many of society's functions.Availability of water is a requirement for any civilization, and this necessitates quantitative estimates of water for effective resource management. The research in this dissertation will focus on the forecasting of hydrological quantities, with emphasis on times of anomalously low water availability, commonly referred to as droughts. Of particular focus is the quantification of uncertainty in hydrological forecasts, and the factors that affect that uncertainty. With this focus, Bayesian methods, including ensemble data assimilation and multi-model combinations, are utilized to develop a probabilistic forecasting system. This system is applied to the upper Colorado River Basin for water supply and drought forecast analysis.This dissertation examines further advancements related to the identification of drought intensity. Due to the reliance of drought forecasting on measures of the magnitude of a drought event, it is imperative that these measures be highly accurate. In order to quantify drought intensity, hydrologists typically use statistical indices, which place observed hydrological deficiencies within the context of historical climate.Although such indices are a convenient framework for understanding the intensity of a drought event, they have obstacles related to non-stationary climate, and non-uniformly distributed input variables. This dissertation discusses these shortcomings, demonstrates some errors that conventional indices may lead to, and then proposes a movement towards physically-based indices to overcome these issues.ii A final advancement in this dissertation is an examination of the sensitivity of hydrological forecasts to initial conditions. Although this has been performed in many recent studies, the experiment here takes a more detailed approach. Rather than determining the lead time at which meteorological forcing becomes dominant with respect to initial conditions, this study quantifies the lead time at which the forecast becomes entirely insensitive to initial conditions, and estimating the rate at which the forecast loses sensitivity to initial conditions. A primary goal with this study is to examine the recovery of drought, which is related to the loss of sensitivity to below average initial moisture conditions over time. Through this analysis, it is found that forecasts are sensitive to initial conditions at greater lead times than previously thought, which has repercussions for development of forecast systems.iii Acknowledgements

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The Poisson Approximation to the Poisson Binomial Distribution

Cited by 151 publications

References 0 publications

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

Lattice Codes and Generalized Minimum-Distance Decoding for OFDM Systems

Quantifying the Impacts of Initial Condition and Model Uncertainty on Hydrological Forecasts

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