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Semicontinuous data frequently occur in longitudinal studies. The popular two‐part modelling approach deals with longitudinal semicontinuous data by analyzing the occurrence of positive values and the intensity of positive values separately; however, this separation may break down the natural sequence of semicontinuous data within a subject and destroy its serial dependence structure. In this article, we introduce a Tweedie compound Poisson mixed model to study the occurrence of positive values and the quantity of the semicontinuous response simultaneously. In our approach, covariate effects on the semicontinuous response are assessed directly. The correlation within a subject and the unobserved heterogeneity are incorporated with serially correlated nonparametric random effects. Our model unifies subject‐specific and population‐averaged interpretations. We illustrate the approach with applications to a Brief Symptom Inventory study and an infants' fluoride intake study. [ABSTRACT FROM AUTHOR]

We propose a nonparametric compound Poisson model for underreported count data that introduces a latent clustering structure for the reporting probabilities. The latter are estimated with the model's parameters based on experts' opinion and exploiting a proxy for the reporting process. The proposed model is used to estimate the prevalence of chronic kidney disease in Apulia, Italy, based on a unique statistical database covering information on m = 258 municipalities obtained by integrating multisource register information. Accurate prevalence estimates are needed for monitoring, surveillance, and management purposes; yet, counts are deemed to be considerably underreported, especially in some areas of Apulia, one of the most deprived and heterogeneous regions in Italy. Our results agree with previous findings and highlight interesting geographical patterns of the disease. We compare our model to existing approaches in the literature using simulated as well as real data on early neonatal mortality risk in Brazil, described in previous research: the proposed approach proves to be accurate and particularly suitable when partial information about data quality is available. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let S = ω1S1 + ω2S2 + ⋯ + ωNSN. Here Sj is the sum of identically distributed random variables and ωj > 0 denotes weight. We consider the case, when Sj is the sum of independent random variables satisfying Cramer’s condition. Upper-bounds for the accuracy of compound Poisson first and second order approximations in uniformmetric are established.

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 E X 1 / E X 1 2 = t , and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum M : = sup L 1 (X) / L 1 (X − E X) = 17 + 7 7 / 4 , where L 1 (X) = E | X | 3 / (E X 2) 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 X 1 , X 2 , ... is proven in terms of the central Lyapunov ratio L 1 (X 1 − E X 1) with the constant 0.3031 · H t (1 − t 2) 3 / 2 ∈ [ 0.3031 , 0.4517) , t ∈ [ 0 , 1) , which depends on the normalized first-moment t : = E X 1 / E X 1 2 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Poisson-stopped sum of the Hurwitz–Lerch zeta distribution is proposed as a model for interarrival times and rainfall depths. Theoretical properties and characterizations are investigated in comparison with other two models implemented to perform the same task: the Hurwitz–Lerch zeta distribution and the one inflated Hurwitz–Lerch zeta distribution. Within this framework, the capability of these three distributions to fit the main statistical features of rainfall time series was tested on a dataset never previously considered in the literature and chosen in order to represent very different climates from the rainfall characteristics point of view. The results address the Hurwitz–Lerch zeta distribution as a natural framework in rainfall modelling using the additional random convolution induced by the Poisson-stopped model as a further refinement. Indeed the Poisson contribution allows more flexibility and depiction in reproducing statistical features, even in the presence of very different climates. [ABSTRACT FROM AUTHOR]

Bivariate time-series counts with excessive zeros frequently occur in medical, biological, social, and environmental sciences. An appropriate analysis of such data must account for serial correlation over time, cross correlation between bivariate responses, and excessive zeros. We first introduce a serially correlated random effect series shared by both responses to account for a common temporal trend. The shared temporal random effects can also characterize the correlation between two responses. Given the common temporal random effects, we incorporate a compound Poisson random effect series into a Poisson model for each response to capture excessive zeros and additional variation. Our approach models zero and positive parts of the responses in an integral way. Our approach also unifies marginal modeling and conditional modeling analyses. We develop a quasi-likelihood approach for the estimation of the model parameters. Our method is illustrated with the analysis of musculoskeletal and non-musculoskeletal workplace injuries recorded every four weeks on a group of hospital cleaners over a seven-year period. Performance of the proposed model is evaluated through simulation studies. [ABSTRACT FROM AUTHOR]

Faced with the need to adjust public pension systems to meet changing demographic, economic and social conditions, most developed countries have created government reserve funds to ensure macroeconomic sustainability. This paper aims to study the importance that this reserve fund plays in the sustainability of the Spanish public pension system. Using data for the 2000 to 2019 period (20 observations) on the main variables impacting on the system, we calculate probabilities and other indicators of its unsustainability in relation to the reserve fund. Our model accurately reflects certain aspects of the data, and suggests that the probability of unsustainability is inversely associated with the size of the reserve fund, but that this relation is moderated by the heterogeneity of the members of the pension system. Moreover, the probability of unsustainability increases in line with the pension system deficit, the time elapsed until unsustainability is reached is shorter when the Reserve Fund balance falls, and the size of this fund at which the system becomes unsustainable diminishes with the probability of unsustainability. [ABSTRACT FROM AUTHOR]

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.

The distribution of the sum of 1-dependent lattice vectors with supports on coordinate axes is approximated by a multivariate compound Poisson distribution and by signed compound Poisson measure. The local and ℓ α -norms are used to obtain the error bounds. The Heinrich method is used for the proofs. [ABSTRACT FROM AUTHOR]

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.

Local limit theorems for compound discrete distributions (a normal theorem and a large deviation theorem) are put forward. Some examples of such distributions, including compound Poisson, compound binomial, and negative binomial distributions, are considered. Our analysis is based on elementary asymptotic methods in combination with the saddle-point method. [ABSTRACT FROM AUTHOR]

The Poisson-stopped sum of the Hurwitz–Lerch zeta distribution is proposed as a model for interarrival times and rainfall depths. Theoretical properties and characterizations are investigated in comparison with other two models implemented to perform the same task: the Hurwitz–Lerch zeta distribution and the one inflated Hurwitz–Lerch zeta distribution. Within this framework, the capability of these three distributions to fit the main statistical features of rainfall time series was tested on a dataset never previously considered in the literature and chosen in order to represent very different climates from the rainfall characteristics point of view. The results address the Hurwitz–Lerch zeta distribution as a natural framework in rainfall modelling using the additional random convolution induced by the Poisson-stopped model as a further refinement. Indeed the Poisson contribution allows more flexibility and depiction in reproducing statistical features, even in the presence of very different climates.

Faced with the need to adjust public pension systems to meet changing demographic, economic and social conditions, most developed countries have created government reserve funds to ensure macroeconomic sustainability. This paper aims to study the importance that this reserve fund plays in the sustainability of the Spanish public pension system. Using data for the 2000 to 2019 period (20 observations) on the main variables impacting on the system, we calculate probabilities and other indicators of its unsustainability in relation to the reserve fund. Our model accurately reflects certain aspects of the data, and suggests that the probability of unsustainability is inversely associated with the size of the reserve fund, but that this relation is moderated by the heterogeneity of the members of the pension system. Moreover, the probability of unsustainability increases in line with the pension system deficit, the time elapsed until unsustainability is reached is shorter when the Reserve Fund balance falls, and the size of this fund at which the system becomes unsustainable diminishes with the probability of unsustainability.

We develop a modelling framework for estimating and predicting weighted network data. The edge weights in weighted networks often arise from aggregating some individual relationships between the nodes. Motivated by this, we introduce a modelling framework for weighted networks based on the compound Poisson distribution. To allow for heterogeneity between the nodes, we use a regression approach for the model parameters. We test the new modelling framework on two types of financial networks: a network of financial institutions in which the edge weights represent exposures from trading Credit Default Swaps and a network of countries in which the edge weights represent cross-border lending. The compound Poisson Gamma distributions with regression fit the data well in both situations. We illustrate how this modelling framework can be used for predicting unobserved edges and their weights in an only partially observed network. This is for example relevant for assessing systemic risk in financial networks. [ABSTRACT FROM AUTHOR]

The sum of symmetric three-point 1-dependent nonidentically distributed random variables is approximated by a compound Poisson distribution. The accuracy of approximation is estimated in the local and total variation norms. For distributions uniformly bounded from zero, the accuracy of approximation is of the order O(n–1). In the general case of triangular arrays of identically distributed summands, the accuracy is at least of the order O(n–1/2). Nonuniform estimates are obtained for distribution functions and probabilities. The characteristic function method is used.

This paper considers an extension of the classical discrete time risk model for which the claim numbers are assumed to be temporal dependence and overdispersion. The risk model proposed is based on the first-order integer-valued autoregressive (INAR(1)) process with discrete compound Poisson distributed innovations. The explicit expression for the moment generating function of the discounted aggregate claim amount is derived. Some numerical examples are provided to illustrate the impacts of dependence and overdispersion on related quantities such as the stop-loss premium, the value at risk and the tail value at risk. [ABSTRACT FROM AUTHOR]

Communication networks have been recognized as substantial energy consumer. However, the ubiquity of Ethernet links provides opportunity for energy savings with Energy-Efficient Ethernet standard and packet coalescing. So far, theoretical analysis of coalescing algorithms for Energy-Efficient Ethernet has assumed that the coalescing limit is expressed in packets; however, as Ethernet links are byte-congestible resources in nature, we argue that byte-based coalescing algorithms should be employed. To that goal, we propose an M/M/1 model for byte-based coalescing on Ethernet links compliant to Energy-Efficient Ethernet standard. The model is based on compound Poisson distribution and provides single formula for calculation of expected low-power state duration and, in turn, of energy savings. The model is applicable to 10GBASE-T Ethernet and to emerging 2.5GBASE-T and 5GBASE-T standards. Detailed simulation results show a good match of achieved energy savings to those predicted by the model. The paper discusses application of the proposed model for the evaluation of energy efficiency of 10GBASE-T Ethernet links in future IoT data centers within Smart Cities. [ABSTRACT FROM AUTHOR]

The sum of symmetric three-point 1-dependent nonidentically distributed random variables is approximated by a compound Poisson distribution. The accuracy of approximation is estimated in the local and total variation norms. For distributions uniformly bounded from zero, the accuracy of approximation is of the order O(n−1). In the general case of triangular arrays of identically distributed summands, the accuracy is at least of the order O(n−1/2). Nonuniform estimates are obtained for distribution functions and probabilities. The characteristic function method is used. [ABSTRACT FROM AUTHOR]

In the most general terms, tax evasion is a breach of tax law. Such a practice is characterised by the economic effect of tax burden elimination or reduction. This paper is aimed at adapting the fixed effects models that are applied in the non-life insurance ratemaking process to modelling tax abuses. The common characteristics of tax risk assessment and non-life insurance ratemaking include for both cases: the number of a given type of events is counted in the examined time period, the distribution of the amount levied as a tax liability for a single event is examined, with the distribution characterised by strong right-skewed asymmetry, and finally the expected amount levied as a tax liability on a single insured/entity is estimated based on these results. In the case of insurance ratemaking, the obtained expected value serves directly the purpose of estimating the insurance premium. In the area of tax risk, the obtained expected values serve the purpose of estimating the scale of abuses arising from the application of a given behaviour pattern. [ABSTRACT FROM AUTHOR]

Using the estimates of the accuracy of the normal approximation to distributions of Poisson-binomial random sums from [I. G. Shevtsova, Theory Probab. Appl., 62 (2018), pp. 278-294], we obtain moment-type estimates of the rate of convergence in the central limit theorem for Poisson and mixed Poisson random sums in the uniform and mean metrics. As corollaries, we provide estimates of the accuracy of the approximation to distributions of negative binomial random sums by the normal law (with the growth of the shape parameter) and by the variance-gamma mixture of the normal law (as the "success probability" tends to zero); in particular, we present estimates of the accuracy of the Laplace approximation to distributions of geometric random sums. [ABSTRACT FROM AUTHOR]

Assuming conditions on factorial cumulants, we estimate the closeness of distribution of a sum of nonnegative integer-valued m-dependent random variables to the class of all infinitely divisible laws. The accuracy of approximation is measured in total variation and local metrics. Our results are exemplified by an analogue of the first uniform Kolmogorov theorem for the statistic of (k1,k2) events. [ABSTRACT FROM AUTHOR]

Abstract Background Implementation of REDD+ requires measurement and monitoring of carbon emissions from forest degradation in developing countries. Dry forests cover about 40 % of the total tropical forest area, are home to large populations, and hence often display high disturbance levels. They are susceptible to gradual but persistent degradation and monitoring needs to be low cost due to the low potential benefit from carbon accumulation per unit area. Indirect remote sensing approaches may provide estimates of subsistence wood extraction, but sampling of biomass loss produces zero-inflated continuous data that challenges conventional statistical approaches. We introduce the use of Tweedie Compound Poisson distributions from the exponential dispersion family with Generalized Linear Models (CPGLM) to predict biomass loss as a function of distance to nearest settlement in two forest areas in Tanzania. Results We found that distance to nearest settlement is a valid proxy variable for prediction of biomass loss from fuelwood collection (p

In spatial disease surveillance, geographic areas with large numbers of disease cases are to be identified, so that targeted investigations can be pursued. Geographic areas with high disease rates are called disease clusters and statistical cluster detection tests are used to identify geographic areas with higher disease rates than expected by chance alone. In some situations, disease-related events rather than individuals are of interest for geographical surveillance, and methods to detect clusters of disease-related events are called event cluster detection methods. In this paper, we examine three distributional assumptions for the events in cluster detection: compound Poisson, approximate normal and multiple hypergeometric (exact). The methods differ on the choice of distributional assumption for the potentially multiple correlated events per individual. The methods are illustrated on emergency department (ED) presentations by children and youth (age < 18 years) because of substance use in the province of Alberta, Canada, during 1 April 2007, to 31 March 2008. Simulation studies are conducted to investigate Type I error and the power of the clustering methods.

Starting from the compound Poisson INGARCH models, we introduce in this paper a new family of integer-valued models suitable to describe count data without zeros that we name zero-truncated CP-INGARCH processes. For such class of models, a probabilistic study concerning moments existence, stationarity and ergodicity is developed. The conditional quasi-maximum likelihood method is introduced to consistently estimate the parameters of a wide zero-truncated compound Poisson subclass of models. The conditional maximum likelihood method is also used to estimate the parameters of ZTCP-INGARCH processes associated with well-specified conditional laws. A simulation study that compares some of those estimators and illustrates their finite distance behaviour as well as a real-data application conclude the paper. [ABSTRACT FROM AUTHOR]

The paper develops new methods of nonparametric estimation of a compound Poisson process. Our key estimator for the compounding (jump) measure is based on series decomposition of functionals of a measure and relies on the steepest descent technique. Our simulation studies for various examples of such measures demonstrate flexibility of our methods. They are particularly suited for discrete jump distributions, not necessarily concentrated on a grid nor on the positive or negative semi-axis. Our estimators also applicable for continuous jump distributions with an additional smoothing step. [ABSTRACT FROM AUTHOR]

Paired data have been widely collected in the efficiency studies of a new method against an established method in environmental, ecological and medical studies. For example, in comparative fishing studies, ability of catching target species (fish catch) or reducing the catch of non-target species (fish bycatch) is usually investigated through a paired design. These paired fish catches by weight are generally skewed and continuous, but with a significant portion of exact zeros (no catch). Such zero-inflated continuous data are traditionally handled by two-part models where the zero and positive components are handled separately; however, this separation generally destroys paired structure, and thus may result in substantial difficulty in characterizing the relative efficiency between two methods. To overcome this problem, we consider compound Poisson mixed model for paired data with which the zero and non-zero components are characterized in an integral way. In our approach, the clustering effects by pair are captured by incorporating relevant random effects. Our model is estimated using orthodox best linear unbiased predictor approach. Unlike two-part models, our approach unifies inferences of zero and positive components. Our method is illustrated with analyses of winter flounder bycatch data and ultrasound safety data. [ABSTRACT FROM PUBLISHER]

A statistical test to distinguish between a Poisson INARCH model and a Compound Poisson INARCH model is proposed, based on the form of the probability generating function of the compounding distribution of the conditional law of the model. For first-order autoregression, the normality of the test statistics' asymptotic distribution is established, either in the case where the model parameters are specified, or when such parameters are consistently estimated. As the test statistics' law involves the moments of inverse conditional means of the Compound Poisson INARCH process, the analysis of their existence and calculation is performed by two approaches. For higher-order autoregressions, we use a bootstrap implementation of the test. A simulation study illustrating the finite-sample performance of this test methodology in what concerns its size and power concludes the paper. [ABSTRACT FROM AUTHOR]

The multitarget version of the traditional target theory based on the Poisson distribution is still used to describe the dose-survival curves of cells after ionizing radiation in radiobiology and radiotherapy. However, noting that the usual ionizing radiation damage is the result of two sequential stochastic processes, the probability distribution of the damage number per cell should follow a compound Poisson distribution, like e.g. Neyman's distribution of type A (N. A.). In consideration of that the Gaussian distribution can be considered as the approximation of the N. A. in the case of high flux, a multitarget model based on the Gaussian distribution is proposed to describe the cell inactivation effects in low linear energy transfer (LET) radiation with high dose-rate. Theoretical analysis and experimental data fitting indicate that the present theory is superior to the traditional multitarget model and similar to the Linear - Quadratic (LQ) model in describing the biological effects of low-LET radiation with high dose-rate, and the parameter ratio in the present model can be used as an alternative indicator to reflect the radiation damage and radiosensitivity of the cells. [ABSTRACT FROM AUTHOR]

In the reinsurance market, the cedent and the reinsurer make their subjective evaluations of the utility and riskiness of the financial operations covered by a treaty by deriving preference rankings of the different kinds of agreements. If both the cedent and the reinsurer evaluate the operation underlying the treaty by means of their utility functions and adopt the criterion based on the comparison between the expected utilities in order to take into account the subjective attitude toward risk, then the indifference prices play a crucial role because they define the preference thresholds of the exchange, thus restricting the range of variability of the reinsurance price. By using an analytical approach, this paper examines the above problem identifying the margins for achieving Pareto-optimal solutions in trading insurance risks. [ABSTRACT FROM PUBLISHER]

The univariate compound Poisson distribution has many applications in various areas such as biology, seismology, risk theory, forestry, health science, etc. In this paper, a bivariate compound Poisson distribution is proposed and the joint probability function of this model is derived. Expressions for the product moments, cumulants, covariance and correlation coefficient are also obtained. Then, an algorithm is prepared in Maple to obtain the probabilities quickly and an empirical comparison of the proposed probability function is given. Bivariate versions of the Neyman type A, Neyman type B, geometric-Poisson, Thomas distributions are introduced and the usefulness of these distributions is illustrated in the analysis of earthquake data.La distribución compuesta de Poisson univariada tiene muchas aplicaciones en diversas áreas tales como biología, ciencias de la salud, ingeniería forestal, sismología y teoría del riesgo, entre otras. En este artículo, una distribución compuesta de Poisson bivariada es propuesta y la función de probabilidad conjunta de este modelo es derivada. Expresiones para los momentos producto, acumuladas, covarianza y el coeficiente de correlación respectivos son obtenidas. Finalmente, un algoritmo preparado en lenguaje Maple es descrito con el fin de calcular probabilidades asociadas rápidamente y con el fin de hacer una comparación de la función de probabilidad propuesta. Se introducen además versiones bivariadas de las distribuciones tipo A y tipo B de Neyman, geométrica-Poisson y de Thomas y se ilustra la utilidad de estas distribuciones aplicadas al análisis de datos de terremoto.