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We propose a novel mortality improvement model with the difference of death counts follows the Skellam distribution. We extend Mitchell et al. (2013) by considering the difference in Poisson death counts instead of the ratio of subsequent mortality rate, which does not have a known distribution. We derive the iterative estimators of the model from the Skellam distribution. Our model can employ maximum likelihood estimation for estimation issues such as missing data and provides a better fit than Mitchell et al. (2013) . Using English and Wales mortality rate age 0-89 data during 1950-2016, the model estimate suggests that the age-dependent mortality improvement is slower than the benchmark, which coincides with a recent observation by Office for National Statistics (2018) . The forecasting performance outperforms the Poisson and M10 model. We make inferences on the price of longevity swaps and analyze how the volatility shock of mortality improvement affects the premium of longevity swaps.

Estimation of latent network flows is a common problem in statistical network analysis. The typical setting is that we know the margins of the network, i.e. in- and outdegrees, but the flows are unobserved. In this paper, we develop a mixed regression model to estimate network flows in a bike-sharing network if only the hourly differences of in- and outdegrees at bike stations are known. We also include exogenous covariates such as weather conditions. Two different parameterizations of the model are considered to estimate 1) the whole network flow and 2) the network margins only. The estimation of the model parameters is proposed via an iterative penalized maximum likelihood approach. This is exemplified by modeling network flows in the Vienna Bike-Sharing Network. Furthermore, a simulation study is conducted to show the performance of the model. For practical purposes it is crucial to predict when and at which station there is a lack or an excess of bikes. For this application, our model shows to be well suited by providing quite accurate predictions., Comment: 27 pages, 20 figures

A discrete version of the Gumbel distribution (Type-I Extreme Value distribution) has been derived by using the general approach of discretization of a continuous distribution. Important distributional and reliability properties have been explored. It has been shown that depending on the choice of parameters the proposed distribution can be positively or negatively skewed; possess long-tail(s). Log-concavity of the distribution and consequent results have been established. Estimation of parameters by method of maximum likelihood, method of moments, and method of proportions has been discussed. A method of checking model adequacy and regression type estimation based on empirical survival function has also been examined. A simulation study has been carried out to compare and check the efficacy of the three methods of estimations. The distribution has been applied to model three real count data sets from diverse application area namely, survival times in number of days, maximum annual floods data from Brazil and goal differences in English premier league, and the results show the relevance of the proposed distribution.

The trust of Mobile Nodes inside the network is considered on the grounds that the potential factor that needs to be explored for encouraging prevalent network execution. This network execution impacting trust is corrupted by the presence of deliberately carrying on selfish Nodes that doesn't advance information parcels to their local agreeable Mobile Nodes. These narrow minded Nodes decrease the parcel forward action depending on its own vitality state for moderating its vitality in order to gauge dynamic inside the network. Here, a Modified Skellam Distribution - Based Selfish Node Detection Technique (MSD-SNDT) is proposed for authorizing powerful and productive location and segregation of selfishly carrying on portable Nodes from the network. The reproduction tests is directed for evaluating the greatness of the proposed MSD-SNDT approach regarding parcel conveyance, throughput, complete overhead and vitality utilizations under differing quantities of portable Nodes and narrow minded Nodes inside the network.

The trust of nodes in Mobile Ad hoc NETwork (MANET) is the potential factor that needs to be investigated for facilitating predominant network performance. The network performance influencing trust is degraded due to the presence of intentionally behaving selfish nodes that do not forward data packets to their adjoining cooperative mobile nodes. The selfish nodes lessen packet forwarding based on the energy level for conserving their energy and remain active in the network. In this paper, Skellam Distribution Inspired Trust Factor-based Selfish Node Detection Technique (SDITF-SNDT) is propounded for enforcing effective and efficient detection, and isolation of selfishly behaving mobile nodes from the network. The proposed SDITF-SNDT initiates the process of selfish node detection by determining the mean packet deviation through which the Standard Deviation (SD) and Variance are estimated for computing the Skellam Distribution Inspired Trust Factor (SDITF). This computation aids in reliability estimation of nodes in order to categorize them into selfish and cooperative nodes. From the investigations conducted for the proposed SDITF-SNDT approach, excellent improvement in Packet Delivery Ratio (PDR) and phenomenal minimization in energy consumption are confirmed for increased number of mobile nodes in the network.

Using European soccer data sets, which contain data related to common European soccer leagues, players basic information, and teams’ goals, etc., this paper analyzes the characteristics of European soccer and players, explores data visualization regarding European soccer, and makes predictions of results of matches. Based on Python 3 and some of the packages inside, such as numpy, the author improves the data set to make it clear and user-friendly. Visualizations of data and basic statistics, including Poisson Distribution, are then utilized to determine the results. Finally, this paper analyzes the attacking and defending abilities of different leagues and teams in Europe, ascertains distributions of players’ attributes, and predicts match results by using Poisson distribution and Skellam Distribution. Generally, this paper analyzes data from leagues to matches to players. All these analyses are meaningful for the public to understand the characteristics of European soccer and the world behind the numbers.

The degree that the environment determines what genes contribute towards adaptation is a fundamental question in microbial evolution. Microbial populations are often experimentally passaged in different environments and sequenced in order to identify candidates for adaptation in a particular environment. However, there remains the need to develop an appropriate statistical framework to identify genes that acquired more mutations in one environment over the other (i.e., divergent evolution). Here we demonstrate how the evolutionary outcomes among replicate populations in the same environment, known as parallel evolution, can be leveraged to construct an intuitive statistical test for identifying the genes that contribute towards divergent evolution. To accomplish this task, we examined publicly available evolve-and-resequence experiment datasets and found that the distribution of mutation counts among genes can be predicted using an ensemble of independent Poisson random variables. Building on this result, we propose that the degree of divergent evolution at a given gene between populations from two different environments can be modeled as the difference between two Poisson random variables, known as the Skellam distribution. We then propose and apply a statistical test to identify specific genes that contribute towards divergent evolution. IMPORTANCE: There is currently no existing framework that can be leveraged to identify genes that contribute towards divergent evolution in microbial evolution experiments. To correct for this absence, we investigated the distribution of mutation counts among genes in order to identify an appropriate null model. Our observations suggest that divergent evolution within a given gene can be modeled as the difference in the total number of mutations observed between two environments. This quantity is described by a probability distribution known as the Skellam distribution, providing an appropriate statistical test for researchers seeking to identify the set of genes that contribute towards divergent evolution in evolution experiments.

Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

Photon-limited Poisson image denoising is urgent demand in many application fields, but is particularly challenging because the image structures are often damaged seriously. The effective using of the image prior is very important for improving the quality of the denoised image. In this paper, we exploit the prior of inner statistical relationships among pixels in local spatial neighborhood of the Poisson noisy image and introduce a new Skellam distribution based inner interaction potential function as a distance measurement between the pixels. In the framework of conditional random field (CRF) modeling, we propose a novel Poisson-Skellam distribution based regularization CRF model for photon-limited Poisson noise removal. Using the alternating direction method of multipliers technique, we extend our method to a flexible plug-and-play scheme in which we can combine powerful Gaussian denoising method to improve the denoising performance better.

We investigate the network selection problem over heterogeneous networks. We formulate the decision making problem as a Markov decision process (MDP). Two Poisson processes are introduced to model user arrival and departure, and the change of the number of user in a specific network is described by Skellam Distribution. We apply the Total order preference by similarity to the ideal solution (TOPSIS) method to address the issue of state space explosion in MDP model. To acquire the optimal network selection scheme, the value iteration algorithm is employed to solve the MDP model. We take simulations in Matlab to prove the effectiveness of the proposed scheme. From the experiment results, it is found that our proposed scheme can reduce handover probability by 7.5% and 40.4% at the early stage of connection, compared with the Infinite MDP-based (IMDP) strategy and TOPSISonly strategy. It can also reduce blocking probability by 6.1% compared with IMDP, while maintaining desirable throughput for the target user.

The Inter-Symbol Interference (ISI) in Diffusive Molecular Communication (DMC) is unavoidable due to the stochastic behavior of molecular Brownian motion in a biological environment. In this paper, we analyze the channel capacity of DMC model by considering the ISI from all the previous time slots and the channel transmission probability in each time slot. We first use the Poisson distribution to approximate the Binomial distribution in order to obtain the total number of molecules received in the current time slot. Then we derive the mathematical expressions of optimal decision threshold by using the Skellam distribution. On this basis, we formulate the mutual information and bit error probability. Our numerical results are presented to show the channel capacity can be maximized with optimal lower bit error probability by controlling the channel transmission probability in each time slot and using the Skellam distribution. More importantly, under the same physical parameters, a higher capacity is achieved with using fewer molecules in our work.

We determine the net-baryon multiplicity distribution which reproduces all cumulants measured so far by lattice QCD. We present the dependence on the volume and temperature of this distribution. We find that for temperatures and volumes encountered in heavy ion reactions, the multiplicity distribution is very close to the Skellam distribution, making the experimental determination of it rather challenging. We further provide estimates for the statistics required to measure cumulants of the net-baryon and net-proton distributions., Comment: 13 pages, 6 figures; Extended version. Now include statistics estimate for RHIC and LHC based on delta method

Two bivariate extensions of the Skellam distribution were introduced in 2014 by a subset of the present authors, who also proposed moment estimators for the dependence parameters in these models. The limiting distribution of these estimators is established here, and their asymptotic efficiency is compared with that of the corresponding maximum likelihood estimators. © 2017 The Authors. Stat Published by John Wiley & Sons Ltd.

We derive a set of recurrence relations for the calculation of the probabilities of a large class of integer-valued random variables. We show that the probability mass function can be recursively computed for random variables with a probability generating function satisfying certain functional form.

The tick structure of the financial markets entails that price changes observed at very high frequency are discrete. Departing from this empirical evidence we develop a new model to describe the dynamic properties of multivariate time-series of high frequency price changes, including the high probability of observing no variations (price staleness). We assume the existence of two independent latent/hidden Markov processes determining the dynamic properties of the price changes and the excess probability of the occurrence of zeros. We study the probabilistic properties of the model that generates a zero-inflated mixture of Skellam distributions and we develop an EM estimation procedure with closed-form M step. In the empirical application, we study the joint distribution of the price changes of four assets traded on NYSE. Particular focus is dedicated to the precision of the univariate and multivariate density forecasts, to the quality of the predictions of quantities like the volatility and correlations across assets, and to the possibility of disentangling the different sources of zero price variation as generated by absence of news, microstructural frictions or by the offsetting positions taken by the traders.

In this paper, an integer-valued autoregressive model of order one (INAR(1)) with time-varying parameters and driven by a periodic sequence of innovations is introduced. The proposed INAR(1) model is based on the signed thinning operator defined by Kachour and Truquet (2011) and conveniently adapted to the periodic case. Basic notations and definitions concerning the periodic signed thinning operator are provided. Based on this thinning operator, Chesneau and Kachour (2012) established a signed INAR(1) model. Motivated by the work of Chesneau and Kachour (2012), we introduce a periodic model, denoted by S-PINAR(1), with period s. In contrast to conventional INAR(1) models, these models are defined in \(\mathbb {Z}\) allowing for negative values both for the series and its autocorrelation function. For a proper \(\mathbb {Z}\)-valued time series, a distribution for the innovation term defined on \(\mathbb {Z}\) is required. The S-PINAR(1) model assumes a specific innovation distribution, the Skellam distribution. Regarding parameter estimation, two methods are considered: conditional least squares and conditional maximum likelihood. The performance of the S-PINAR(1) model is assessed through a simulation study.

The probability of informed trading (PIN) is a commonly used market microstructure measure for detecting the level of information asymmetry. Estimating PIN can be problematic due to corner solutions, local maxima and floating point exceptions (FPE). Yan and Zhang [J. Bank. Finance, 2012, 36, 454–467] show that whilst factorization can solve FPE, boundary solutions appear frequently in maximum likelihood estimation for PIN. A grid search initial value algorithm is suggested to overcome this problem. We present a faster method for reducing the likelihood of boundary solutions and local maxima based on hierarchical agglomerative clustering (HAC). We show that HAC can be used to determine an accurate and fast starting value approximation for PIN. This assists the maximum likelihood estimation process in both speed and accuracy.

The pixel values of images taken by an image sensor are said to be corrupted by Poisson noise. To date, multiscale Poisson image denoising techniques have processed Haar frame and wavelet coefficients—the modeling of coefficients is enabled by the Skellam distribution analysis. We extend these results by solving for shrinkage operators for Skellam that minimizes the risk functional in the multiscale Poisson image denoising setting. The minimum risk shrinkage operator of this kind effectively produces denoised wavelet coefficients with minimum attainable $L^{2}$ error.

In this work we investigate the problem of private statistical analysis of time-series data in the distributed and semi-honest setting. In particular, we study some properties of Private Stream Aggregation (PSA), first introduced by Shi et al. 2011. This is a computationally secure protocol for the collection and aggregation of data in a distributed network and has a very small communication cost. In the non-adaptive query model, a secure PSA scheme can be built upon any key-homomorphic weak pseudo-random function as shown by Valovich 2017, yielding security guarantees in the standard model which is in contrast to Shi et al. We show that every mechanism which preserves \((\epsilon ,\delta )\)-differential privacy in effect preserves computational \((\epsilon ,\delta )\)-differential privacy when it is executed through a secure PSA scheme. Furthermore, we introduce a novel perturbation mechanism based on the symmetric Skellam distribution that is suited for preserving differential privacy in the distributed setting, and find that its performances in terms of privacy and accuracy are comparable to those of previous solutions. On the other hand, we leverage its specific properties to construct a computationally efficient prospective post-quantum protocol for differentially private time-series data analysis in the distributed model. The security of this protocol is based on the hardness of a new variant of the Decisional Learning with Errors (DLWE) problem. In this variant the errors are taken from the symmetric Skellam distribution. We show that this new variant is hard based on the hardness of the standard Learning with Errors (LWE) problem where the errors are taken from the discrete Gaussian distribution. Thus, we provide a variant of the LWE problem that is hard based on conjecturally hard lattice problems and uses a discrete error distribution that is similar to the continuous Gaussian distribution in that it is closed under convolution. A consequent feature of the constructed prospective post-quantum protocol is the use of the same noise for security and for differential privacy.

Two-stage non-local methods represented by the Poisson non-local means (PNLM) method [1] and the non-local principal component analysis (NLPCA) [2] method perform well for photon-limited Poisson image reconstruction, where patch-similarity computation in the non-local reconstruction stage is guided and affected by a pre-reconstructed image obtained at the first stage. In this paper, we propose a new method to provide a better pre-reconstructed image with low computational cost. Firstly, we propose an adaptive method for fitting the linear relationship between image intensity and Poisson parameter; Secondly, we obtain an initial estimated image according to this relationship. Lastly, we obtain new pre-reconstructed image by adjusting the initial estimation according to Skellam distribution. Numerical experiments show that our method can provide a better pre-reconstructed image, and therefore can improve the performance of the PNLM method and reduce the computational cost of the NLPCA method efficiently.

What are the most surprising, most impressive and most exciting matches in the history of the highest division of English football? Fans, clubs, critics and broadcasters often attempt to interpret and compare the outcomes of football matches. This can be useful as a means of building a reputation for and interest in a particular club, match or competition. However, such analysis is often subjective with fan bias, personal preference and the prize on offer influencing expectation and interpretation of the results. In this paper statistical measures are developed to identify the most surprising, most impressive and most exciting outcomes from an objective ex post point of view. To achieve this it is assumed that goal scoring in soccer can approximately be considered as a Poisson process. First, the most surprising outcomes are estimated as those with the lowest Poisson probability to occur. Second, the most impressive outcomes are identified by their so-called Skellam percentile position. This measure varie...

While models for integer valued time series are now abundant, there is a shortage of similar models when the time series refer to data defined on Z, i.e., in both the positive and negative integers. Such data occur in certain disciplines and the need for such models also appear when taking differences of a positive integer count time series. In addition one would often like to include covariates to explain variations in the variable of interest. In this article we construct a model doing all these assuming a specific innovation distribution and provide fully parametric inference, including prediction. Real data applications on accidents and financial returns are given. Finally we also discuss alternative models and extensions.

The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.

Skellam's name is traditionally attached to the distribution of the difference of two independent Poisson random variables. Many bivariate extensions of this distribution are possible, e.g., through copulas. In this paper, the authors focus on a probabilistic construction in which two Skellam random variables are affected by a common shock. Two different bivariate extensions of the Skellam distribution stem from this construction, depending on whether the shock follows a Poisson or a Skellam distribution. The models are nested, easy to interpret, and yield positive quadrant-dependent distributions, which share the convolution closure property of the univariate Skellam distribution. The models can also be adapted readily to account for negative dependence. Closed form expressions for Pearson's correlation between the components make it simple to estimate the para-meters via the method of moments. More complex formulas for Kendall's tau and Spearman's rho are also provided.

In recent years, there has been a growing interest in modelling integred-valued time series. In this article, we propose a modified and generalized version of the first order rounded integer-valued autoregressive RINAR(1) model, originally introduced by Kachour and Yao (2009). Indeed, this class can be considered as an alternative of classical models based on the thinning operators. Using a Markov chain method, conditions for stationarity and the existence of moments are investigated. Least squares estimator of the model parameters is considered and its consistence is established. Finally, we describe the price change data using a model of the new class.

Financial prices are usually modelled as continuous, often involving geometric Brownian motion with drift, leverage and possibly jump components. An alternative modelling approach allows financial observations to take integer values that are multiples of a fixed quantity, the ticksize - the monetary value associated with a single change during the price evolution. In the case of high-frequency data, the sample exhibits diverse trading operations in a few seconds. In this context, the observables are assumed to be conditionally independent and identically distributed from either of two flexible likelihoods: the Skellam distribution - defined as the difference between two independent Poisson distributions - or a mixture of Geometric distributions. Posterior inference is obtained via adaptive Gibbs sampling algorithms. Comparisons of the models applied to high-frequency financial data is provided.

Rankings based on counts are often presented to identify popular items, such as baby names, English words, or Web sites. This article shows that, in some examples, the number of correctly identified items can be very small. We introduce a standard error versus rank plot to diagnose possible misrankings. Then to explain the slowly growing number of correct ranks, we model the entire set of count data via a Zipf–Poisson ensemble with independent Xi ∼ Poi(Ni − α) for α > 1 and N > 0 and integers i ⩾ 1. We show that as N → ∞, the first n′(N) random variables have their proper order relative to each other, with probability tending to 1 for n′ up to (AN/log (N))1/(α + 2) for A = α2(α + 2)/4. We also show that the rate N 1/(α + 2) cannot be achieved. The ordering of the first n′(N) entities does not preclude for some interloping m > n′. However, we show that the first n″ random variables are correctly ordered exclusive of any interlopers, with probability tending to 1 if n″ ⩽ (BN/log (N))1/(α + 2) for any B < A....

By the laws of quantum physics, pixel intensity does not have a true value, but should be a random variable. Contrary to the conventional assumptions, the distribution of intensity may not be an additive Gaussian. We propose to directly model the intensity difference and show its validity by an experimental comparison to the conventional additive model. As a model of the intensity difference, we present a Skellam distribution derived from the Poisson photon noise model. This modeling induces a linear relationship between intensity and Skellam parameters, while conventional variance computation methods do not yield any significant relationship between these parameters under natural illumination. The intensity-Skellam line is invariant to scene, illumination, and even most of camera parameters. We also propose practical methods to obtain the line using a color pattern and an arbitrary image under natural illumination. Because the Skellam parameters that can be obtained from this linearity determine a noise distribution for each intensity value, we can statistically determine whether any intensity difference is caused by an underlying signal difference or by noise. We demonstrate the effectiveness of this new noise model by applying it to practical applications of background subtraction and edge detection.

Motivated by features of low latency data in financial econometrics we study in detail integer-valued Levy processes as the basis of price processes for high-frequency econometrics. We propose using models built out of the difference of two subordinators. We apply these models in practice to low latency data for a variety of different types of futures contracts.

The ubiquity of integrating detectors in imaging and other applications implies that a variety of real-world data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vector-valued signal of interest. In this article, we first show how the so-called Skellam distribution arises from the fact that Haar wavelet and filterbank transform coefficients corresponding to measurements of this type are distributed as sums and differences of Poisson counts. We then provide two main theorems on Skellam shrinkage, one showing the near-optimality of shrinkage in the Bayesian setting and the other providing for unbiased risk estimation in a frequentist context. These results serve to yield new estimators in the Haar transform domain, including an unbiased risk estimate for shrinkage of Haar-Fisz variance-stabilized data, along with accompanying low-complexity algorithms for inference. We conclude with a simulation study demonstrating the efficacy of our Skellam shrinkage estimators both for the standard univariate wavelet test functions as well as a variety of test images taken from the image processing literature, confirming that they offer substantial performance improvements over existing alternatives., 27 pages, 8 figures, slight formatting changes; submitted for publication

We propose a method that compares Poisson distributed outcomes. Our method uses the exact distribution of the difference between two Poisson variables to calculate the sample size required to detect a given difference with prespecified power. When the true difference between the two Poisson rates is more than 1.2 units, the number of subjects and events needed at the desired power and Type I error rate is 5–10% less than that computed by simulation based on the normal approximation method. The normal approximation method is more comparable to the exact sample size method when the difference between the rates is less than 1.2 units. The proposed method is more intuitive, efficient, and less subjective than the normal approximation method. A simple code is developed in R to estimate the sample size and critical values.

For its unprecedented level of spatial resolution, chromatin immunoprecipitation combined with λ exonuclease digestion followed by high-throughput sequencing (ChIP-exo) has the potential to replace ChIP-seq as the standard approach for genome-wide mapping of protein-DNA interactions. In this assay, the midpoint between the strand-specific paired peaks, formed in the forward and reverse strands, is typically delimited by the exonuclease stop-sites, within which the protein-binding events are located. Although numerous algorithms have been developed for peak-calling in ChIP-seq data, none of them is fully adjusted for the analysis of ChIP-exo. This is because those statistical models do not make use of ChIP-exo’s strand-specificity for the identification of protein-DNA binding sites. Here, we present the CexoR algorithm, which aims to ease the analysis of replicated ChIP-exo data in BAM alignment format. The detection algorithm relies on the Skellam distribution (cross-correlation of two Poisson distributions) to calculate probabilities of consecutive punctate-sources of read-enrichment located nearby at Watson-and-Crick strands. ChIP-exo peak-pairs are identified and ranked by their irreproducible discovery rate estimated across biological replicates, and finally reported in BED format files. CexoR can potentially be applied to other ChIP-exo-based protocols, such as ChIP-nexus. Availability and implementation: CexoR has been implemented in R, and is freely available at http://bioconductor.org.

Paired count data usually arise in medicine when before and after treatment measurements are considered. In the present paper we assume that the correlated paired count data follow a bivariate Poisson distribution in order to derive the distribution of their difference. The derived distribution is shown to be the same as the one derived for the difference of the independent Poisson variables, thus recasting interest on the distribution introduced by Skellam. Using this distribution we remove correlation, which naturally exists in paired data, and we improve the quality of our inference by using exact distributions instead of normal approximations. The zero-inflated version is considered to account for an excess of zero counts. Bayesian estimation and hypothesis testing for the models considered are discussed. An example from dental epidemiology is used to illustrate the proposed methodology.

We investigate the intraday dependence pattern between tick data of stock price changes using a new time-varying model for discrete copulas. We let parameters of both the marginal models and the copula vary over time using an observation driven autoregressive updating scheme based on the score of the conditional probability mass function with respect to the time-varying parameters. We apply the model to high-frequency stock price changes expressed as discrete tick-size multiples for four liquid U.S. financial stocks. Our modeling framework is based on Skellam densities for the marginals and a range of different copula functions. We find evidence of intraday time-variation in the dependence structure. After the opening and before the close of the stock market, dependence levels are lower. We attribute this finding to more idiosyncratic trading at these times. The introduction of score driven dynamics in the dependence structure significantly increases the likelihood values of the time-varying copula model. By contrast, a fixed daily seasonal dependence pattern clearly fits the data less well.

We investigate Yang-Lee zeros of grand partition functions as truncated fugacity polynomials of which coefficients are given by the canonical partition functions $Z(T,V,N)$ up to $N \leq N_{\text{max}}$. Such a partition function can be inevitably obtained from the net-baryon number multiplicity distribution in relativistic heavy ion collisions, where the number of the event beyond $N_{\text{max}}$ has insufficient statistics, as well as canonical approaches in lattice QCD. We use a chiral random matrix model as a solvable model for chiral phase transition in QCD and show that the closest edge of the distribution to real chemical potential axis is stable against cutting the tail of the multiplicity distribution. The similar behavior is also found in lattice QCD at finite temperature for Roberge-Weiss transition. In contrast, such a stability is found to be absent in the Skellam distribution which does not have phase transition. We compare the number of $N_{\text{max}}$ to obtain the stable Yang-Lee zeros with those of critical higher order cumulants., Comment: 15 pages, 9 figures. Version to appear in Phys. Rev. D. Comments on lattice results are added. Typos are corrected

We explore the influence of deuteron formation in the late stage of nucleus-nucleus reactions on the fluctuations observed in the final net-proton yields around midrapidity. At each investigated energy, the produced (anti-)proton yield at chemical freeze-out is assumed to fluctuate according to a Poisson distribution and in each event the probability for deuteron formation by coalescence is proportional to $(dN_{\mathrm{proton}}/dy)^2$. The protons that are then clustered in deuterons are usually not included in the experimental measurement of the net-proton fluctuations, therefore, we subtract these clustered protons from the final state proton number for the calculation of the net-proton fluctuations (the same is done in the anti-proton sector). Due to the non-linear deuteron formation probability the resulting distribution is not a Skellam distribution, but shows the interesting feature of a decrease in the kurtosis $\kappa \sigma^2$ and a local maximum in the skewness $S \sigma$ observables as the collision energy decreases., Comment: 5 pages, 5 figures

We theoretically investigate the coefficient of performance (COP) of a mesoscopic thermoelectric refrigerator realized by using a tunnel junction. We analyze the influence of particle and heat current fluctuations on the COP out of the equilibrium regime. We calculate the average COP by using full counting statistics and find that it depends on the measurement time $\tau$. The deviation from the macroscopic COP value can be expressed with the Skellam distribution at all times. This result enables us to improve the Gaussian approximation valid within the linear response regime, which cannot predict the average COP in the limit of $\tau \to 0$. We illustrate the time dependence of the average COP and find that in the short-time regime, the average COP possesses a minimum. In order to confirm the physical consistency far from equilibrium, we propose checking the correlation coefficient between the particle and the heat currents in addition to the positivity of the entropy production rate., Comment: 12 pages, 5 figures. The paper was expanded

Background RNA-seq, based on deep-sequencing techniques, has been widely employed to precisely measure levels of transcripts and their isoforms expressed under different conditions. However, robust statistical tools used to analyze these complex datasets are lacking. By grouping genes with similar expression profiles across treatments, cluster analysis provides insight into gene functions and networks that have become increasingly important. Results We proposed and verified a cluster algorithm based on a skellam model for grouping genes into distinct groups based on the pattern of gene expression in response to changing conditions or in different tissues. This algorithm capitalizes on the skellam distribution to capture the count property of RNA-seq data and clusters genes in different environments. A two-stage hierarchical expectation-maximization (EM) algorithm was implemented to estimate the optimal number of groups and mean expression levels of each group across two environments. A procedure was formulated to test whether and how a given group shows a plastic response to environmental changes. The model was used to analyze an RNA-seq dataset measured from reciprocal crosses of early Arabidopsis thaliana embryos that respond differently based on the extent of maternal and paternal genome contributions, from which genes associated with maternal and paternal contributions were identified. Simulation studies were also performed to validate the statistical behavior of the model. Conclusions This model is a useful tool for clustering gene expression data by RNA-seq, thus facilitating our understanding of gene functions and networks.

We introduce a dynamic statistical model for Skellam distributed random variables. The Skellam distribution can be obtained by taking differences between two Poisson distributed random variables. We treat cases where observations are measured over time and where possible serial correlation is modeled via stochastically time-varying intensities of the underlying Poisson counts. The likelihood function for our model is analytically intractable and we evaluate it via a multivariate extension of numerically accelerated importance sampling techniques. We illustrate the new model by two empirical studies and verify whether our framework can adequately handle large data sets. First, we analyze long univariate high-frequency time series of U.S. stock price changes, which evolve as discrete multiples of a fixed tick size of one dollar cent. In a second illustration, we analyze the score differences between rival soccer teams using a large, unbalanced panel of seven seasons of weekly matches in the German Bundesliga.In both empirical studies, the new model provides interesting and non-trivial dynamics with a clear interpretation.

We study a minimal model of active transport in crowded single-file environments which generalizes the emblematic model of single-file diffusion to the case when the tracer particle (TP) performs either an autonomous directed motion or is biased by an external force, while all other particles of the environment (bath) perform unbiased diffusions. We derive explicit expressions, valid in the limit of high density of bath particles, of the full distribution P((n))(X) of the TP position and of all its cumulants, for arbitrary values of the bias f and for any time n. Our analysis reveals striking features, such as the anomalous scaling [proportionality] √[n] of all cumulants, the equality of cumulants of the same parity characteristic of a Skellam distribution and a convergence to a Gaussian distribution in spite of asymmetric density profiles of bath particles. Altogether, our results provide the full statistics of the TP position and set the basis for a refined analysis of real trajectories of active particles in crowded single-file environments.

In fluoroscopy quantum noise is dominant with respect to other common noise sources, whose effects can be often neglected. Quantum noise is originated by the limited number of photons (low-dose X-ray) involved in fluoroscopic image formation; this noise is commonly modeled as Poisson distributed. Estimation of noise-parameters is required for evaluation of X-ray digital imaging sensors and in several image processing applications (e.g. denoising). The first aim of this work is to validate the analytically derived noise-parameter models by real fluoroscopic image sequences, also considering non-linear gray level mappings currently employed by fluoroscopic devices. A plain procedure for estimation of noise pixel-intensity variance as a function of mean pixel-intensity, which does not require specific test-objects but only some images screening a still scene, has been provided. Besides, a procedure for noise-parameter estimation by differencing fluoroscopic static images has been proposed. It computes image-pair differences, estimates concise parameters of the resulting Skellam distribution and, then, quotes Poisson noise-parameters. Image sequences of a simple step-phantom, acquired with a conventional fluoroscopic device, were utilized for performing the noise measurements. Experimental results confirmed a great agreement of the measured noise-parameters with those analytically derived and the possibility to use static images to estimate noise.

Motivated by features of low latency data in finance we study in detail discrete-valued Levy processes as the basis of price processes for high frequency econometrics. An important case of this is a Skellam process, which is the difference of two independent Poisson processes. We propose a natural generalisation which is the difference of two negative binomial processes. We apply these models in practice to low latency data for a variety of different types of futures contracts.

In this article we address the issue of denoising photon-limited image data by deriving new and efficient multivariate Bayesian estimators that approximate the conditional expectation of Haar wavelet and filterbank transform coefficients of Poisson data—coefficients that take the so-called Skellam distribution. We show that in this setting, the posterior mean under a Bayesian model forms the solution to a linear differential equation, owing in part to the recursive property of the Skellam distribution. We then propose a practical approach to solve—approximately—this differential equation, and arrive at a near mean-square-optimal Skellam mean estimator that is both computationally efficient and amenable to an Empirical Bayes approach. We then derive three approaches to shrinkage based on smoothing the marginal likelihood of the data, and demonstrate their superior performance relative to state-of-the-art approaches for both natural test images and examples from computed tomography scans.

Owing to the stochastic nature of discrete processes such as photon counts in imaging, a variety of real-world data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vector-valued signal of interest. Certain wavelet and filterbank transform coefficients corresponding to measurements of this type are distributed as sums and differences of Poisson counts, taking in the simplest case the so-called Skellam distribution. We show that a Skellam mean estimator provides a Poisson intensity estimation method based on shrinkage of filterbank coefficients, and a means of estimating the risk of any Skellam mean estimator is derived in closed form under a frequentist model.

Owing to the stochastic nature of discrete processes such as photon counts in imaging, real-world data measurements often exhibit heteroscedastic behavior. In particular, time series components and other measurements may frequently be assumed to be non-iid Poisson random variables, whose rate parameter is proportional to the underlying signal of interest-witness literature in digital communications, signal processing, astronomy, and magnetic resonance imaging applications. In this work, we show that certain wavelet and filterbank transform coefficients corresponding to vector-valued measurements of this type are distributed as sums and differences of independent Poisson counts, taking the so-called Skellam distribution. While exact estimates rarely admit analytical forms, we present Skellam mean estimators under both frequentist and Bayes models, as well as computationally efficient approximations and shrinkage rules, that may be interpreted as Poisson rate estimation method performed in certain wavelet/filterbank transform domains. This indicates a promising potential approach for denoising of Poisson counts in the above-mentioned applications.

Conventional edge detectors suffer from inherent image noise and threshold determination. In this paper, we propose a noble edge detector based on the noise distribution for CCD or CMOS cameras. By assuming the dominant photon noise, we model the distribution of intensity differences between two neighborhood pixels. Since it is well known that photon noise follows a Poisson distribution, we introduce a Skellam distribution, which is the difference of two Poisson random variables. We show experimentally that the Skellam distribution can be used to model the noise distribution of pixels that are captured from the same scene radiance. For estimating the noise distribution given a single pixel, we find the important property that the Skellam parameters are linearly related to the intensity value of pixels. This linearity enables us to determine noise parameters according to the intensity value. In addition, parameters of the line are preserved under illumination, scene and camera setting changes except for only a gain change. Based on the noise distributions, we calculate intensity allowances of three channels for each pixel given a confidence interval. We propose a noble edge detector by skipping a pre-processing step of conventional Gaussian smoothing which is the main obstacle for robust and accurate edge detection. If the difference of intensity exceeds the intensity allowance at least in a single channel, the in- between pixel is marked as an edge pixel. We demonstrate that without conventional Gaussian smoothing the noise-model based approach can automatically extract the fine details of image structures, such as edge and corners, independent of camera setting.

In this paper, we introduce the Skellam distribution as a sensor noise model for CCD or CMOS cameras. This is derived from the Poisson distribution of photons that determine the sensor response. We show that the Skellam distribution can be used to measure the intensity difference of pixels in the spatial domain, as well as in the temporal domain. In addition, we show that Skellam parameters are linearly related to the intensity of the pixels. This property means that the brighter pixels tolerate greater variation of intensity than the darker pixels. This enables us to decide automatically whether two pixels have different colors. We apply this modeling to detect the edges in color images. The resulting algorithm requires only a confidence interval for a hypothesis test, because it uses the distribution of image noise directly. More importantly, we demonstrate that without conventional Gaussian smoothing the noise model-based approach can automatically extract the fine details of image structures, such as edges and corners, independent of camera setting.