Compound Poisson Distribution Research Articles

Compound Poisson distributions for random dynamical systems using probabilistic approximations

We obtain quenched hitting distributions to be compound Poissonian for a certain class of random dynamical systems. The theory is general and designed to accommodate non-uniformly expanding behavior and targets that do not overlap much with the region where uniformity breaks. Based on annealed and quenched polynomial decay of correlations, our quenched result adopts annealed Kac-type time-normalization and finds limits to be noise-independent. The technique involves a probabilistic block-approximation where the quenched hit-counting function up to annealed Kac-normalized time is split into equally sized blocks which are mimicked by an independency of random variables distributed just like each of them. The theory is made operational due to a result that allows certain hitting quantities to be recovered from return quantities. Our application is to a class of random piecewise expanding one-dimensional systems, casting new light on the well-known deterministic dichotomy between periodic and aperiodic points, their usual extremal index formula EI=1−1/JTp(x0), and recovering the Polya-Aeppli case for general Bernoulli-driven systems, but distinct behavior otherwise.

When is the discrete Weibull distribution infinitely divisible?

For the discrete Weibull probability distribution we prove that it is only infinitely divisible if the shape parameter lies in the range 0<β≤1 . The proof is based on some results of Steutel and van Harn (2004). For this case we construct the corresponding compound Poisson distribution and thus the related Lévy process.

Optimal premium pricing in a competitive stochastic insurance market with incomplete information: A Bayesian game-theoretic approach

This paper examines a stochastic one-period insurance market with incomplete information. The aggregate amount of claims follows a compound Poisson distribution. Insurers are assumed to be exponential utility maximizers, with their degree of risk aversion forming their private information. A premium strategy is defined as a mapping between risk-aversion types and premium rates. The optimal premium strategies are denoted by the pure-strategy Bayesian Nash equilibrium, whose existence and uniqueness are demonstrated under specific conditions on the insurer-specific demand functions. Boundary and monotonicity properties for equilibrium premium strategies are derived.

Total variation distance and compound poisson approximations for random sums

We derive new upper bounds for the total variation distance between compound Poisson distributions as well as between a random sum and a compound Poisson distribution, and as a result we also obtain upper bounds for the total variation distance between compound Poisson distributions and a sum of independent random variables. These bounds are generalizations and refinements of some well-known bounds in the literature. We also derive upper bounds for the total variation distance between negative binomial distributions of order k and between the negative binomial distributions of order k and compound Poisson distributions. Upper bounds for the total variation distances between the number of success runs of length k in binary Markovian trials and its limiting distributions for several enumeration schemes, are also given.

Forecasting number of corner kicks taken in association football using compound Poisson distribution

This article presents a holistic compound Poisson regression model framework to forecast number of corner kicks taken in association football. Corner kick taken events are often decisive in the match outcome and inherently arrive in batch with serial clustering pattern. Providing parameter estimates with intuitive interpretation, a class of compound Poisson regression including a Bayesian implementation of geometric-Poisson distribution are introduced. With a varying shape parameter, the corner counts serial correlation between matches is handled naturally within the Bayesian model. In this study, information elicited from cross-market betting odds was used to improve the model predictability. Margin application methods to adjust market inefficiency in raw odds are also discussed.

Uncertainties in TDCR measurement revisited: Contribution of optical effects

In the Triple to Double Coincidence Ratio method in Liquid Scintillation Counting, the detection efficiency is calculated from the value of a free parameter describing the intrinsic light yield of the counting system. This model is generally based on a Poisson distribution of the number of photoelectrons detected and the detection efficiency is obtained from the complement of the non-detection efficiency. In the classical free parameter model, the mean of the Poisson distribution, m, is a constant but some variability of this mean could be expected from optical effects due to internal reflections inside the LS source or from non-homogeneity of the detection efficiency of the photomultiplier tubes. Then, m becomes a random variable and the distribution of the photoelectrons becomes a compound Poisson distribution, with a random variable as mean value.This paper explores the implications of the variance of m, which were, to our knowledge, never considered previously in the uncertainty budget of TDCR measurements.

Convergence to Infinite-Dimensional Compound Poisson Distributions on Convex Polyhedra

Convergence to Infinite-Dimensional Compound Poisson Distributions on Convex Polyhedra

Mass-Dependent Critical Value Expressions for Particle Finding in Single-Particle ICP-TOFMS.

In time-of-flight mass spectrometry (TOFMS), ion detection is often achieved via electron multiplication followed by fast analog-to-digital conversion (ADC). This detection approach is chosen over time-to-digital conversion because it extends the dynamic range of TOFMS measurements, especially for transient analyses. However, fast ADC detection also introduces measurement noise fundamental to the electron multiplication process. In previous research, we demonstrated that TOFMS signals acquired with fast ADC follow a compound Poisson distribution in which the Poisson-distributed arrival of ions at the detector is compounded with the response profile of the electron multiplier. Here, we consider the influence of mass-to-charge (m/z)-dependent detector responses and their impact on particle-finding accuracy in single-particle inductively coupled plasma TOFMS (spICP-TOFMS). In spICP-TOFMS, highly time-resolved ion signals are recorded and particle signals are distinguished from background signals based on thresholding the data at m/z-specific critical values. Through Monte Carlo modeling with measured m/z-dependent detector responses, we generate compound Poisson model distributions and critical values that accurately account for the dispersion of measured signals. We test the accuracy of critical values through the analysis of dissolved element solutions and comparison of measured versus predicted event rates above critical value thresholds. The use of m/z-dependent compound Poisson critical values reduces false-positive particle identifications by one to two orders of magnitude compared to thresholding criteria based on normal or Poisson statistics. The improved accuracy and robustness of compound Poisson critical values enables automated multi-element particle finding in spICP-TOFMS.

Prediction Interval for Compound Conway–Maxwell–Poisson Regression Model with Application to Vehicle Insurance Claim Data

Regression models in which the response variable has a compound distribution have applications in actuarial science. For example, the aggregate claim amount in a vehicle insurance portfolio can be modeled using a compound Poisson distribution. In this paper, we propose a regression model, wherein the response variable is assumed to have a compound Conway–Maxwell–Poisson (CMP) distribution. This distribution is a parsimonious two-parameter Poisson distribution that accounts for both over- and under-dispersed count data, making it more suitable for application in various fields. A two-part methodology in the framework of a generalized linear model is proposed to estimate the parameters. Additionally, a method to obtain the prediction interval of the response variable is developed. The workings of the proposed methodology are illustrated through simulated data. An application of the compound CMP regression model to real-life vehicle insurance claims data is presented.

Subexponential densities of compound Poisson sums and the supremum of a random walk

We characterize the subexponential densities on $(0,\infty)$ for compound Poisson distributions on $[0,\infty)$ with absolutely continuous L\'evy measures. As a corollary, we show that the class of all subexponential probability density functions on $\mathbb R_+$ is closed under generalized convolution roots of compound Poisson sums. Moreover, we give an application to the subexponential density on $(0,\infty)$ for the distribution of the supremum of a random walk.

Delicate Comparison of the Central and Non-Central Lyapunov Ratios with Applications to the Berry–Esseen Inequality for Compound Poisson Distributions

For each t∈(−1,1), the exact value of the least upper bound H(t)=sup{E|X|3/E|X−t|3} over all the non-degenerate distributions of the random variable X with a fixed normalized first-order moment EX1/EX12=t, and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum M:=supL1(X)/L1(X−EX)=17+77/4, where L1(X)=E|X|3/(EX2)3/2 is the non-central Lyapunov ratio, and hence proving S. Shorgin’s (2001) conjecture on the exact value of M. As a corollary, an analog of the Berry–Esseen inequality for the Poisson random sums of independent identically distributed random variables X1,X2,… is proven in terms of the central Lyapunov ratio L1(X1−EX1) with the constant 0.3031·Ht(1−t2)3/2∈[0.3031,0.4517), t∈[0,1), which depends on the normalized first-moment t:=EX1/EX12 of random summands and being arbitrarily close to 0.3031 for small values of t, an almost 1.5 size improvement from the previously known one.

Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case

In the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous analog of deficiency. In the case under consideration, by continuous deficiency, we will mean the difference between the parameter of the Poisson distribution of the number of summands in a Poisson random sum and that of the compound Poisson distribution providing the desired accuracy of the normal approximation. This approach is used for the solution of the problem of determination of the distribution of a separate term in the Poisson sum that provides the least possible value of the parameter of the Poisson distribution of the number of summands guaranteeing the prescribed value of the (1−α)-quantile of the normalized Poisson sum for a given α∈(0,1). This problem is solved under the condition that possible distributions of random summands possess coinciding first three moments. The approach under consideration is applied to the collective risk model in order to determine the distribution of insurance payments providing the least possible time that provides the prescribed Value-at-Risk. This approach is also used for the problem of comparison of the accuracy of approximation of the asymptotic (1−α)-quantile of the sum of independent, identically distributed random variables with that of the accompanying infinitely divisible distribution.

On Discrete Poisson–Mirra Distribution: Regression, INAR(1) Process and Applications

Several pieces of research have spotlighted the importance of count data modelling and its applications in real-world phenomena. In light of this, a novel two-parameter compound-Poisson distribution is developed in this paper. Its mathematical functionalities are investigated. The two unknown parameters are estimated using both maximum likelihood and Bayesian approaches. We also offer a parametric regression model for the count datasets based on the proposed distribution. Furthermore, the first-order integer-valued autoregressive process, or INAR(1) process, is also used to demonstrate the utility of the suggested distribution in time series analysis. The unknown parameters of the proposed INAR(1) model are estimated using the conditional maximum likelihood, conditional least squares, and Yule–Walker techniques. Simulation studies for the suggested distribution and the INAR(1) model based on this innovative distribution are also undertaken as an assessment of the long-term performance of the estimators. Finally, we utilized three real datasets to depict the new model’s real-world applicability.

Refinements of bounds for tails of compound distributions and ruin probabilities

In this paper we derive lower bounds for right-tails of compound geometric distributions and ruin probabilities in the classical compound Poisson risk model in the heavy-tailed cases using the truncated Lundberg condition, which improve all the corresponding known lower bounds. Some upper bounds are also derived. Examples are given and numerical comparison for ruin probabilities when the adjustment coefficient does not exist are also considered, illustrating the effectiveness of the proposed new bounds. In addition, several bounds for tails of negative binomial distributions are obtained in terms of the tail of compound geometric distributions as well as bounds in the heavy-tailed cases. Also, some bounds for the stop-loss premium associated with compound negative binomial distributions are given. Using Chernoff's upper bounds we derive two-sided bounds for tails of compound Poisson distributions. Finally, two-sided bounds are given for tails of compound logarithmic distributions in the heavy-tailed cases.

Multi-echelon inventory optimization in closed-loop supply chain

In this paper, we are interested in the stock control in the closed-loop supply chain and its application to the cylinder packaged gas supply chains of Air Liquide company. Air Liquide is a multinational company producing and distributing industrial and medical gases. Firstly, we introduce a new classification of demand profile and perform a comprehensive study of the goodness of fit of demands sizes and time between successive demands based on real data. As well known in the literature (Boylan and Syntetos (2021)), we find that negative binomial distribution fits better demand sizes and intervals. Secondly, we explain how to use a single echelon model that is easy to implement on the field, to control inventory and compute the stock target in the observable part of the closed-loop supply chain. It is modeled as a multi-echelon serial inventory system. Finally, we perform numerical analysis and comparisons of methods to compute the stock target of the observable part of the closed-loop supply chain with real data. Those methods are derivated from the up-to-level periodic inventory review policy where we vary the statistical distribution of the demand and the measure of the service level. Using the Item Fill Rate (IFR) as a service level measure in the stock target calculation, reduce the stock target by more than 10% compared to the stock target obtained using the Cycle Service Level (CSL). We also investigate methods using compound Poisson distribution for demand models.

Geometric sums, size biasing and zero biasing

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.

Solving inventory routing with transshipment and substitution under dynamic and stochastic demands using genetic algorithm and deep reinforcement learning

In this paper, we investigate a two-level supply chain consisting of a company which manufactures a set of products and distributes them via its central warehouse to a set of customers. The problem is modelled as a dynamic and stochastic inventory routing problem (DSIRP) that considers two flexible instruments of transshipment and substitution to mitigate shortages at the customer level. A new resolution approach, based on the hybridisation of mathematical modelling, Genetic Algorithm and Deep Reinforcement Learning is proposed to handle the combinatorial complexity of the problem at hand. Tested on the 150 most commonly used benchmark instances for single-vehicle-product DSIRP, results show that the proposed algorithm outperforms the current best results in the literature for medium and large instances. Moreover, 450 additional instances for multi-products DSIRP are generated. Different demand distributions are examined in these experiments, namely, Normal distribution, Poisson distribution for demand occurrence, combined with demands of constant size; Stuttering Poisson distribution and Negative Binomial distribution. In terms of managerial insights, results show the advantages of promoting inventory sharing and substitutions on the overall supply chain performance.

Spare parts inventory routing problem with transshipment and substitutions under stochastic demands

We study a two-level spare parts supply chain in which a manufacturer supplies a central warehouse (CW) with original equipment manufacturer (OEM) and replacement or pattern parts (PP). The CW, distantly located from the manufacturer, distributes both OEM parts and PP to a given number of depots facing stochastic demands. The demand for spare parts is intermittent, exhibiting an infrequent rate and extreme dispersal over time periods. Along with lateral transshipment, PP can be used as substitutes for the OEM parts to sidestep shortage at depots. Assuming that emergency shipments are significantly longer and more expensive, we aim at underlining the relative effectiveness of such a new spare parts inventory management policy. A mixed-integer linear programming model is proposed to solve the inventory routing problem with transshipment and substitution under stochastic demands. The objective is to minimise costs of holding inventory, transportation which includes regular shipment and transshipment, substitution and lost sales. To solve the problem, Sample Average Approximation method is used. Based on empirical goodness-of-fit tests, three demand patterns are studied: Poisson distribution, stuttering Poisson distribution and negative binomial distribution. The model is tested on well-known benchmark instances generated for multi-product multi-vehicle IRP. Computational experiments highlight the benefits of promoting transshipment and substitution on the overall supply chain performance. Results also suggest insights, which are of interest to professionals who are willing to develop new decision support models for the most efficient management of such items.

On a limit theorem for a non-linear scaling

In this note we proved that weak limits, of sums of positive, independent and identically distributed random variables which are re-normalized by a non-linear shrinking transform max(0,x−r), are either degenerate or (some) compound Poisson distributions.

Rare event process and entry times distribution for arbitrary null sets on compact manifolds

We establish the general equivalence between rare event process for arbitrary continuous functions whose maximal values are achieved on non-trivial sets, and the entry times distribution for arbitrary measure zero sets. We then use it to show that for differentiable maps on a compact Riemannian manifold that can be modeled by Young’s towers, the rare event process and the limiting entry times distribution both converge to compound Poisson distributions. A similar result is also obtained on Gibbs–Markov systems, for both cylinders and open sets. We also give explicit expressions for the parameters of the limiting distribution, and a simple criterion for the limiting distribution to be Poisson. This can be applied to a large family of continuous observables that achieve their maximum on a non-trivial set with zero measure.