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Image Quality
Published in Harry E. Martz, Clint M. Logan, Daniel J. Schneberk, Peter J. Shull, X-Ray Imaging, 2016
Harry E. Martz, Clint M. Logan, Daniel J. Schneberk, Peter J. Shull
where Φ is the measured signal, G is an energy weighting factor, and Ω(Φ/G)⊗N is the N-fold convolution of the energy spectrum and λ is the mean. Notice that this takes the form of a compound Poisson distribution.
A Tweedie Compound Poisson Model in Reproducing Kernel Hilbert Space
Published in Technometrics, 2023
Yi Lian, Archer Yi Yang, Boxiang Wang, Peng Shi, Robert William Platt
The distribution of Y is referred to as the compound Poisson distribution. For example, in insurance applications, N is the number of claims for a risk, Zd is the amount of losses for the dth claim, and v is the exposure (e.g., duration of the policy in years), thus, Y represents the aggregate loss amount per unit at risk (Jørgensen and de Souza 1994; Smyth and Jorgensen 2002; Shi 2016). Note that Y = 0 if and only if N = 0, thus, Y has a probability mass at 0, that is, . Additionally, Y conditional on N = n follows a gamma distribution with shape and scale . It has been shown that the compound Poisson distribution is related to a special class of the ED family known as the Tweedie distribution (Tweedie 1984; Jørgensen 1987; Smyth 1996). The density function of the ED family follows with the natural parameter and the dispersion parameter . The normalizing function and the cumulant function are both known. By the property of the ED family, we have that where and are the first and second order derivative of , respectively. For the Tweedie distribution, θ, and have the specific forms: for some index parameter (also called power parameter) and the dispersion parameter . Using (2), the density of the Tweedie distribution can be written as where the exact form of can be found in Section 3 of Yang, Qian, and Zou (2018). The mean and variance relationship of the Tweedie distribution becomes .
Some Structural Properties Related to the Borel-Taner Distribution and its’ Application
Published in American Journal of Mathematical and Management Sciences, 2023
There are several useful references on the efficacy of the Borel-Tanner distribution and a non-exhaustive list is presented as follows. For example, for a detailed discussion on the applications to count data with a high frequency of zeros as well as uses of the BT distribution relating to queuing theory, see Tanner (1961), Haight et al. (1960) and the references cited therein. Miller (1961) found that the BT distribution model performed well when fitted to observed queue length frequencies for their data. However, after stating this claim, Miller argues that there is little explanation as to why the distribution fits well. For this reason, it is important to discuss some structural properties of the model. Gómez-Déniz et al. (2017) studied the application of a modified BT distribution (reparameterization) in the context of a regression. Noticeably, the Borel-Tanner distribution is valid for any positive integral starting point. There have also been some developments on the bivariate extension of this distribution, for example, in the works of Shenton and Consul (1973). However, in this paper, we shall focus our attention in the univariate version of this distribution. In the next, we aim to provide several points that serves as a motivation of our current work.The Borel-Tanner distribution subsumes (under appropriate limiting conditions) several well-known univariate discrete distributions, such as negative binomial, Poisson Consul distribution, compound Poisson distribution with Borel summands. For example, if and then unconditionally, Therefore, by exploring structural results for a Borel-Tanner distribution will help us achieve useful properties for the limiting (alias derived) distributions. Application wise this would be very important. For example, compound Poisson distribution plays an important role as a model for the total size of insurance claims in the context of actuarial science.As indicated in Daly and Shneer (2019), in stochastic modeling context, the Borel-Tanner distribution can be considered as an approximation for the number of customers served in a busy period of an M/G/1 queue.The Borel-Tanner distribution appears as a limiting distribution in some settings, such as Erdos-Renyi random graph with n vertices. Therefore, understanding of the structural properties will be quite important and useful to study the nature of such random graphs in the limiting case, i.e., under the distributional assumption as Borel-Tanner.