On the Sums of Compound Negative Binomial and Gamma Random Variables

Vellaisamy, P.; Upadhye, N. S.

doi:10.1017/s0021900200005350

Cited by 12 publications

(7 citation statements)

References 9 publications

(10 reference statements)

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“…Also, a kind of additivity property for the mixed Poisson distributions is established and using this result it is shown that a gamma model can arise as the distribution of Y 1 +Y 2 , where Y 1 is exponential, while Y 2 is a non-gamma rv. As another interesting application of Theorem 5, Vellaisamy and Upadhye (2009) recently obtained a random parameter representation for the convolution of gamma G(α j , p j ) distributions. Finally, we hope that the results in this paper would lead to the consideration of generalized gamma models, where Y k strongly depends on S k−1 , for k ≥ 2, that could be more flexible and useful for practical situations.…”

Section: Discussionmentioning

confidence: 99%

Some Intrinsic Properties of the Gamma Distribution

Vellaisamy

Sreehari²

2010

JJSS

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Let {Yn} be a sequence of nonnegative random variables (rvs), and Sn = n j=1 Yj, n ≥ 1. It is first shown that independence of S k−1 and Y k , for all 2 ≤ k ≤ n, does not imply the independence of Y1, Y2, . . . , Yn. When Yj's are identically distributed exponential Exp(α) variables, we show that the independence of S k−1 andIt is shown by a counterexample that the converse is not true. We show that if X is a non-negative integer valued rv, then there exists, under certain conditions, a rvwhere {N (t)} is a standard (homogeneous) Poisson process, and obtain the Laplace-Stieltjes transform of Y . This leads to a new characterization for the gamma distribution. It is also shown that a G(α, k) distribution may arise as the distribution of S k , where the components are not necessarily exponential. Several typical examples are discussed.

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Section: Discussionmentioning

confidence: 99%

Some Intrinsic Properties of the Gamma Distribution

Vellaisamy

Sreehari²

2010

JJSS

Self Cite

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“…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 24,25 . Thus, the approximation of D(v) in equations (3) and (4) with sums of negative binomial distributions that have success probability of 1 − 1 S , suggests the empirical observation implemented in the Backtrack algorithm 20 .…”

Section: Methodsmentioning

confidence: 99%

On statistical modeling of sequencing noise in high depth data to assess tumor evolution

Rabadán

Bhanot

Marsilio

et al. 2017

Preprint

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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 emerges 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 pretreatment clinical setting to determine appropriate therapy and prepare for potential recurrence pretreatment.

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“…In the literature, there are many different representations for PDF or CDF of such distribution f.e. in terms of zonal polynomials and confluent hypergeometric functions [48], single gamma-series [54], Lauricella multivariate hypergeometric functions [1], extended Foxs functions [3] and others [76]. Here we present the formulas for PDF and CDF according to [54].…”

Section: Probability Distribution Of Dma Statisticmentioning

confidence: 99%

Statistical test for fractional Brownian motion based on detrending moving average algorithm

Sikora

2018

Chaos, Solitons & Fractals

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Motivated by contemporary and rich applications of anomalous diffusion processes we propose a new statistical test for fractional Brownian motion, which is one of the most popular models for anomalous diffusion systems. The test is based on detrending moving average statistic and its probability distribution. Using the theory of Gaussian quadratic forms we determined it as a generalized chi-squared distribution. The proposed test could be generalized for statistical testing of any centered non-degenerate Gaussian process. Finally, we examine the test via Monte Carlo simulations for two exemplary scenarios of subdiffusive and superdiffusive dynamics.

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On the Sums of Compound Negative Binomial and Gamma Random Variables

Cited by 12 publications

References 9 publications

Some Intrinsic Properties of the Gamma Distribution

Some Intrinsic Properties of the Gamma Distribution

On statistical modeling of sequencing noise in high depth data to assess tumor evolution

Statistical test for fractional Brownian motion based on detrending moving average algorithm

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