On the Sums of Compound Negative Binomial and Gamma Random Variables

Vellaisamy, P.; Upadhye, N. S.

doi:10.1239/jap/1238592129

Cited by 21 publications

(21 citation statements)

References 12 publications

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“…However, when any two of β i s are different, there exists no exact expression for the n -fold convolution of the gamma distribution. A number of independent studies have investigated this problem (Thom 1968, Mathai 1982, Moschopoulos 1985, Sim 1992, Akkouchi 2005, Stewart et al 2007, Vellaisamy and Upadhye 2009). For computing efficiency, the expression derived by Sim (1992) is used in this study when the scale parameters are close to each other.…”

Section: Simulation Studiesmentioning

confidence: 99%

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

et al. 2011

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Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith-Martin, and a linear birth-death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.

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Section: Simulation Studiesmentioning

confidence: 99%

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

et al. 2011

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“…D(v)=∑b=1BXbPfalse(v∣false〈Nfalse〉,false(S-1false)false〈vfalse〉b,false(S-1false)false〈Nfalse〉false)≈∑b=1BXbNBfalse(v∣1+false(S-1false)false〈vfalse〉b,1Sfalse), where B is the number of bins, X b is the number of variants in each bin, and 〈 N 〉 is the average sequencing depth across S samples. It has been shown that the sum of negative binomial distributions with equal success probabilities is also a negative binomial distribution, though with a random parameter [7, 50]. Thus, the approximation of D ( v ) in equations (3) and (4) with sums of negative binomial distributions that have success probability of 1S, suggests empirical observations [41].…”

Section: Methodsmentioning

confidence: 99%

On Statistical Modeling of Sequencing Noise in High Depth Data to Assess Tumor Evolution

et al. 2017

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One cause of cancer mortality is tumor evolution to therapy-resistant disease. First line therapy often targets the dominant clone, and drug resistance can emerge from preexisting clones that gain fitness through therapy-induced natural selection. Such mutations may be identified using targeted sequencing assays by analysis of noise in high-depth data. Here, we develop a comprehensive, unbiased model for sequencing error background. We find that noise in sufficiently deep DNA sequencing data can be approximated by aggregating negative binomial distributions. Mutations with frequencies above noise may have prognostic value. We evaluate our model with simulated exponentially expanded populations as well as data from cell line and patient sample dilution experiments, demonstrating its utility in prognosticating tumor progression. Our results may have the potential to identify significant mutations that can cause recurrence. These results are relevant in the pre-treatment clinical setting to determine appropriate therapy and prepare for potential recurrence pretreatment.

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“…Convolutions of gamma distributions commonly arise in statistics [Vellaisamy & Upadhye (2009), Covo & Elalouf (2014)]. The next corollary quantifies gamma distribution approximation to such convolutions.…”

Section: Around Gamma Distribution Approximationmentioning

confidence: 97%

Geometric sums, size biasing and zero biasing

Li¹,

Xia²

2021

Preprint

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The geometric sum plays a significant role in risk theory and reliability theory [Kalashnikov (1997)] and a prototypical example of the geometric sum is Rényi's theorem [Rényi (1956)] saying a sequence of suitably parameterised geometric sums converges to the exponential distribution. There is extensive study of the accuracy of exponential distribution approximation to the geometric sum [Sugakova (1995), Kalashnikov (1997), Peköz & Röllin (2011)] but there is little study on its natural counterpart of gamma distribution approximation to negative binomial sums. In this note, we show that a nonnegative random variable follows a gamma distribution if and only if its size biasing equals its zero biasing. We combine this characterisation with Stein's method to establish simple bounds for gamma distribution approximation to the sum of nonnegative independent random variables, a class of compound Poisson distributions and the negative binomial sum of random variables.

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On the Sums of Compound Negative Binomial and Gamma Random Variables

Cited by 21 publications

References 12 publications

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

On Statistical Modeling of Sequencing Noise in High Depth Data to Assess Tumor Evolution

Geometric sums, size biasing and zero biasing

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