Sojourning With the Homogeneous Poisson Process

Liu, Piaomu; Peña, Edsel A.

doi:10.1080/00031305.2016.1200484

Cited by 5 publications

(4 citation statements)

References 18 publications

(20 reference statements)

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“…On the other hand, in many well-known examples, Inequality (1) is strict for t > 0 and so is (3); e.g., this is the case for a Poisson process with exponentially distributed interarrival times. The same holds true in connection with random inspection times; we refer to Liu and Peña [13] who discussed the choice of an exponentially distributed random inspection time.…”

Section: The Inspection Paradox With a Random Inspection Timementioning

confidence: 76%

“…Herff et al [11] derived an inequality for the length of the inspection interval with a random time and Rauwolf and Kamps [12] gave a general representation for the expected inspection interval length, which served as the basis for the main results in this work. Several explicit examples with random time and applications to earthquake and geyser data can be found in the literature (see, e.g., Liu and Peña [13], Rauwolf and Kamps [12]).…”

Section: Introductionmentioning

confidence: 99%

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On Non-Occurrence of the Inspection Paradox

Rauwolf,

Kamps

2024

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The well-known inspection paradox or waiting time paradox states that, in a renewal process, the inspection interval is stochastically larger than a common interarrival time having a distribution function F, where the inspection interval is given by the particular interarrival time containing the specified time point of process inspection. The inspection paradox may also be expressed in terms of expectations, where the order is strict, in general. A renewal process can be utilized to describe the arrivals of vehicles, customers, or claims, for example. As the inspection time may also be considered a random variable T with a left-continuous distribution function G independent of the renewal process, the question arises as to whether the inspection paradox inevitably occurs in this general situation, apart from in some marginal cases with respect to F and G. For a random inspection time T, it is seen that non-trivial choices lead to non-occurrence of the paradox. In this paper, a complete characterization of the non-occurrence of the inspection paradox is given with respect to G. Several examples and related assertions are shown, including the deterministic time situation.

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Section: The Inspection Paradox With a Random Inspection Timementioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

On Non-Occurrence of the Inspection Paradox

Rauwolf,

Kamps

2024

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“…On the other hand, if

λ

depends on

t

, then it is called a non‐homogeneous Poisson process (NHPP). These processes possess some very interesting properties; see, for instance, Resnick (1992) and Liu and Peña (2016).…”

Section: Counting Variables and Poisson Modelsmentioning

confidence: 99%

Prediction intervals for Poisson‐based regression models

Kim

Lieberman

Luta

et al. 2021

WIREs Computational Stats

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This paper provides a review of the literature regarding methods for constructing prediction intervals for counting variables, with particular focus on those whose distributions are Poisson or derived from Poisson and with an over‐dispersion property. Independent and identically distributed models and regression models are both considered. The motivating problem for this review is that of predicting the number of daily and cumulative cases or deaths attributable to COVID‐19 at a future date. This article is categorized under: Applications of Computational Statistics > Clinical Trials Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical Models > Generalized Linear Models

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“…In addition, if such information is to be used in the prediction model, then we may not have their realized values at the future date on which the prediction region is desired. We point out that even though we are employing probabilistic models in the form of the Poisson or an over-dispersed Poisson model, which are derivable from intuitive conditions when dealing with rare events (cf., [23,16]), our prediction method is still purely data-driven being only reliant on the observed data.…”

Section: Introductionmentioning

confidence: 99%

Prediction Regions for Poisson and Over-Dispersed Poisson Regression Models with Applications to Forecasting Number of Deaths during the COVID-19 Pandemic

KIm¹,

Lieberman²,

Luta³

et al. 2020

Preprint

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Motivated by the current Coronavirus Disease (COVID-19) pandemic, which is due to the SARS-CoV-2 virus, and the important problem of forecasting daily deaths and cumulative deaths, this paper examines the construction of prediction regions or intervals under the Poisson regression model and for an over-dispersed Poisson regression model. For the Poisson regression model, several prediction regions are developed and their performance are compared through simulation studies. The methods are applied to the problem of forecasting daily and cumulative deaths in the United States (US) due to COVID-19. To examine their performance relative to what actually happened, daily deaths data until May 15th were used to forecast cumulative deaths by June 1st. It was observed that there is over-dispersion in the observed data relative to the Poisson regression model. An over-dispersed Poisson regression model is therefore proposed.This new model builds on frailty ideas in Survival Analysis and over-dispersion is quantified through an additional parameter. The Poisson regression model is a hidden model in this over-dispersed Poisson regression model and obtains as a limiting case when the over-dispersion parameter increases to infinity. A prediction region for the cumulative number of US deaths due to COVID-19 by July 16th, given the data until July 2nd, is presented. Finally, the paper discusses limitations of proposed procedures and mentions open research problems, as well as the dangers and pitfalls when forecasting on a long horizon, with focus on this pandemic where events, both foreseen and unforeseen, could have huge impacts on point predictions and prediction regions.

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Sojourning With the Homogeneous Poisson Process

Cited by 5 publications

References 18 publications

On Non-Occurrence of the Inspection Paradox

On Non-Occurrence of the Inspection Paradox

Prediction intervals for Poisson‐based regression models

Prediction Regions for Poisson and Over-Dispersed Poisson Regression Models with Applications to Forecasting Number of Deaths during the COVID-19 Pandemic

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