On discrete Gibbs measure approximation to runs

Kumar, Amit N.; Upadhye, N. S.

doi:10.1080/03610926.2020.1765256

Cited by 5 publications

(3 citation statements)

References 31 publications

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“…For a crisp overview of Stein's method related to classical distributions, we refer the reader to [26,27] and references therein. The method is also extended to some families of distributions, such as the Pearson [32], variance-gamma [16] and discrete Gibbs measure families [15,25]. We refer the reader to the web page of Yvik Swan [sites.google.com/site/steinsmethod/home] for the exhaustive historical development of Stein's method.…”

Section: Introductionmentioning

confidence: 99%

A unified approach to Stein’s method for stable distributions

Upadhye

Barman²

2022

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In this article, we first review the connection between Lévy processes and infinitely divisible random variables, and discuss the classification of infinitely divisible distributions. Next, we establish a Stein identity for an infinitely divisible random variable via the Lévy-Khintchine representation of the characteristic function. In particular, we establish and unify the Stein identities for an α-stable random variable available in the existing literature. Next, we derive the solutions of the Stein equations and its regularity estimates. Further, we derive error bounds for α-stable approximations, and also obtain rates of convergence results in Wasserstein-δ, δ < α for α ∈ (0, 1) and Wasserstein-type distances for α ∈ (1, 2). Finally, we compare these results with existing literature.

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Section: Introductionmentioning

confidence: 99%

A unified approach to Stein’s method for stable distributions

Upadhye

Barman²

2022

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“…Stein's method for various families of distributions is also keen interest for researchers such as Pearson [36], variance-gamma [19,20,18], discrete Gibbs measure [27,13] family and so on. Recently, Arras and Houdré [3,4], Chen et.…”

Section: Introductionmentioning

confidence: 99%

Stein's Method for Tempered Stable Distributions

Barman¹,

Upadhye²

2020

Preprint

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In this article, we develop Stein characterization for two-sided tempered stable distributions using its characteristic function. It enables us to give the Stein characterizations for normal, gamma, Laplace, product of two normal, difference of two gamma, and variance-gamma distributions from the existing literature. Further, it also enables us to give the Stein characterizations for truncated Lévy flight, CGMY, KoBol, and bilateral-gamma distributions. We prove the existence of additive size bias for the one-sided case of tempered stable distributions, in particular, the gamma distribution. We also show that the Stein characterization for the convolution of independent tempered stable distributions can be derived from its characteristic function. Finally, we derive an error bound for tempered stable approximation.

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“…To address this issue, the negative binomial distribution can be used as an alternative to the Poisson distribution. To study more in this field, you can refer to [4][5][6]. Additionally, it is important to note that the mean and variance of count data must be equal for accurate fitting of the Poisson distribution; especially, the mean should be less than or equal to the variance.…”

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confidence: 99%

The Poisson–Lindley Distribution: Some Characteristics, with Its Application to SPC

2023

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Statistical process control (SPC) is a significant method to monitor processes and ensure quality. Control charts are the most important tools in SPC. As production processes and production parts become more complex, there is a need to design control charts using more complex distributions. One of the most important control charts to monitor the number of nonconformities in production processes is the C-chart, which uses the Poisson distribution as a quality characteristic distribution. However, to fit the Poisson distribution to the count data, equality of mean and variance should be satisfied. In some cases, such as biological and medical sciences, count data exhibit overdispersion, which means that the variance of data is greater than the mean. In such cases, we can use the Poisson–Lindley distribution instead of the Poisson distribution to model the count data. In this paper, we first discuss some important characteristics of the Poisson–Lindley distribution. Then, we present parametric and bootstrap control charts when the observations follow the Poisson–Lindley distribution and analyze their performance. Finally, we provide a simulated example and a real-world dataset to demonstrate the implementation of control charts. The results show the good performance of the proposed control charts.

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On discrete Gibbs measure approximation to runs

Cited by 5 publications

References 31 publications

A unified approach to Stein’s method for stable distributions

A unified approach to Stein’s method for stable distributions

Stein's Method for Tempered Stable Distributions

The Poisson–Lindley Distribution: Some Characteristics, with Its Application to SPC

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