Erlang Distribution Research Articles

Collective Risk Models with Hierarchical Archimedean Copulas

In actuarial science, collective risk models, in which the aggregate claim amount of a portfolio is defined in terms of random sums, play a crucial role. In these models, it is common to assume that the number of claims and their amounts are independent, even if this might not always be the case. We consider collective risk models with different dependence structures. Due to the importance of such distributions in an actuarial setting, we first investigate a collective risk model with dependence involving the family of multivariate mixed Erlang distributions. Other models based on mixtures involving bivariate and multivariate copulas in a more general setting are then presented. These different structures allow to link the number of claims to each claim amount, and to quantify the aggregate claim loss. Then, we use Archimedean and hierarchical Archimedean copulas in collective risk models, to model the dependence between the claim number random variable and the claim amount random variables involved in the random sum. Such dependence structures allow us to derive a computational methodology for the assessment of the aggregate claim amount. While being very flexible, this methodology is easy to implement, and can easily fit more complicated hierarchical structures.

Erlang circular model motivated by inverse stereographic projection

The Erlang distribution is a special case of the Gamma distribution with the shape parameter is an integer. This paper proposed a new circular model used inverse stereographic projection. The inverse stereographic projection which is a mapping that projects a random variable from a real line onto a circle can be used in circular statistics to construct a distribution on the circle from real domain. From the circular model, then can be derived the characteristics of the Erlang circular model such as the mean resultant length, mean direction, circular variance and trigonometric moments of the distribution.

Bayes Estimators for the Parameter of the Inverted Exponential Distribution Under different Double informative priors

In this paper, we present a comparison of double informative priors which are assumed for the parameter of inverted exponential distribution.To estimate the parameter of inverted exponential distribution by using Bayes estimation ,will be used two different kind of information in the Bayes estimation; two different priors have been selected for the parameter of inverted exponential distribution. Also assumed Chi-squared - Gamma distribution, Chi-squared - Erlang distribution, and- Gamma- Erlang distribution as double priors. The results are the derivations of these estimators under the squared error loss function with three different double priors. 
 Additionally Maximum likelihood estimation method (MLE) was used to estimate the parameter of inverted exponential distribution .We used simulation technique, to compare the performance for each estimator, several cases from inverted exponential distribution for data generating, for different samples sizes (small, medium, and large).Simulation results shown that the best method is the bayes estimation according to the smallest values of mean square errors( MSE) for all samples sizes (n) comparative to the estimated values by using MLE . According to obtained results, we see that when the double prior distribution for is Gamma- Erlang distribution for some values for the parameters a, b & given the best results according to the smallest values of mean square errors (MSE) comparative to the same values which obtained by using Maximum likelihood estimation (MLE) for the assuming true values for and for all samples sizes. 

Statistical Distribution of the Enhanced Backscatter Coefficient in Reverberation Chamber

In an ideal RC , the analytical expressions for the statistical distributions of the enhanced backscatter coefficient ( ${e}_{b}$ ) are derived. It has been found that the superposition of random variables with exponential distributions has a probability density function of Erlang distribution, and the distribution of the ratio of two random variables with Erlang distribution can be derived analytically, which is the distribution of the enhanced backscatter coefficient. The unbiased estimator of ${e}_{b}$ is also given with and without approximations. Measurements have been performed to verify the results.

On the Discounted Kth Moment of the Deficit at Ruin in the Delayed Renewal Risk Model

In this paper, we derive a very general expression for the discounted kth moment of the deficit at ruin in the delayed renewal risk model. The formula would yield some of the earlier relevant results as special cases, and reduce to a mathematically tractable form if the distributions for claim sizes and interclaim times are of exponential variants including a mixture of Erlang distributions. We provide numerical examples with an emphasis on the impact of the time until the first claim on the kth moment of the deficit at ruin.

Optimal Bayesian control policy for gear shaft fault detection using hidden semi-Markov model

Abstract A new optimal Bayesian control approach is presented to predict early fault of a partially observable gear shaft system subject to deterioration and random failure. The gear shaft system deterioration process is modeled as a three-state continuous time hidden semi-Markov process. States 0 and 1 are unobservable and represent the good and warning system states, respectively. Only state 2 is assumed to be observable and represents the failure state. The general Erlang distribution is considered for modeling the sojourn time in each of the hidden states, which is closer to the actual deterioration process modeling of the gear shaft system than the exponential sojourn time distribution assumed in a hidden Markov model (HMM). The optimal maintenance policy represented by a multivariate Bayesian control scheme based on a hidden semi-Markov model (HSMM) is developed. The objective is to maximize the long-run expected average availability per unit time. An effective computational algorithm is designed in the semi-Markov decision process (SMDP) framework to obtain the optimal control limit and the optimal average availability. Using multidimensional data obtained from condition monitoring, the proposed approach can not only predict early fault occurrence of the gear shaft, but also update the remaining useful life (RUL) at each sampling epoch. A comparison with other maintenance policies is given, which illustrates the effectiveness of the proposed approach.

Event patterns extracted from top quark-related spectra in proton-proton collisions at 8 TeV* * Supported by National Natural Science Foundation of China (11575103, 11747319), the Shanxi Provincial Natural Science Foundation (201701D121005), the Fund for Shanxi “1331 Project” Key Subjects Construction and the US DOE (DE-FG02-87ER40331.A008)

We analyze the transverse momentum (p T) and rapidity (y) spectra of top quark pairs, hadronic top quarks, and top quarks produced in proton-proton (pp) collisions at center-of-mass energy TeV. For spectra, we use the superposition of the inverse power-law suggested by the QCD (quantum chromodynamics) calculus and the Erlang distribution resulting from a multisource thermal model. For y spectra, we use the two-component Gaussian function resulting from the revised Landau hydrodynamic model. The modelling results are in agreement with the experimental data measured at the detector level, in the fiducial phase-space, and in the full phase-space by the ATLAS Collaboration at the Large Hadron Collider (LHC). Based on the parameter values extracted from p T and y spectra, the event patterns in three-dimensional velocity (βx -βy -βz ), momentum (px -py -pz ), and rapidity (y 1-y 2-y) spaces are obtained, and the probability distributions of these components are also obtained.

Approximation of ruin probabilities via Erlangized scale mixtures

In this paper, we extend an existing scheme for numerically calculating the probability of ruin of a classical Cramér–Lundbergreserve process having absolutely continuous but otherwise general claim size distributions. We employ a dense class of distributions that we denominate Erlangized scale mixtures (ESM) that correspond to nonnegative and absolutely continuous distributions which can be written as a Mellin–Stieltjes convolution Π⋆G of a nonnegative distribution Π with an Erlang distribution G. A distinctive feature of such a class is that it contains heavy-tailed distributions.We suggest a simple methodology for constructing a sequence of distributions having the form Π⋆G with the purpose of approximating the integrated tail distribution of the claim sizes. Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modeled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.

Asymptotic Approximations to the Non-Isothermal Distributed Activation Energy Model for Bio-Mass Pyrolysis

Abstract This paper describes the influence of some parameters significant to biomass pyrolysis on the numerical solutions of the non-isothermal nth order distributed activation energy model (DAEM) using the Gamma distribution and discusses the special case for the positive integer value of the scale parameter (λ), i.e. the Erlang distribution. Investigated parameters are the integral upper limit, the frequency factor, the heating rate, the reaction order, and the shape and rate parameters of the Gamma distribution. Influence of these parameters has been considered for the determination of the kinetic parameters of the non-isothermal nth order Gamma distribution from the experimentally derived thermoanalytical data of biomass pyrolysis. Mathematically, the effect of parameters on numerical solution is also used for predicting the behaviour of the unpyrolysized fraction of biomass with respect to temperature. Analysis of the mathematical model is based upon asymptotic expansions, which leads to the systematic methods for efficient way to determine the accurate approximations. The proposed method, therefore, provides a rapid and highly effective way for estimating the kinetic parameters and the distribution of activation energies.

Discrete stochastic analogs of Erlang epidemic models

ABSTRACTErlang differential equation models of epidemic processes provide more realistic disease-class transition dynamics from susceptible (S) to exposed (E) to infectious (I) and removed (R) categories than the ubiquitous SEIR model. The latter is itself is at one end of the spectrum of Erlang SEIR models with concatenated E compartments and concatenated I compartments. Discrete-time models, however, are computationally much simpler to simulate and fit to epidemic outbreak data than continuous-time differential equations, and are also much more readily extended to include demographic and other types of stochasticity. Here we formulate discrete-time deterministic analogs of the Erlang models, and their stochastic extension, based on a time-to-go distributional principle. Depending on which distributions are used (e.g. discretized Erlang, Gamma, Beta, or Uniform distributions), we demonstrate that our formulation represents both a discretization of Erlang epidemic models and generalizations thereof. We consider the challenges of fitting SEIR models and our discrete-time analog to data (the recent outbreak of Ebola in Liberia). We demonstrate that the latter performs much better than the former; although confining fits to strict SEIR formulations reduces the numerical challenges, but sacrifices best-fit likelihood scores by at least 7%.

Explicit expressions for gamma spacings and their moments

ABSTRACTThis work is concerned with the distributions of spacings from a two-parameter gamma distribution, when the shape parameter is a positive integer (or Erlang Distribution). We express the probability density functions of spacings and their moments in closed forms that are easy to implement using computer algebra systems like Mathematica.

Ruin probabilities in a Sparre Andersen model with dependency structure based on a threshold window

AbstractWe analyse ruin probabilities for an insurance risk process with a more generalised dependence structure compared to the one introduced in Constantinescu et al. (2016). In this paper, we assume that a random threshold window is generated every time after a claim occurs. By comparing the previous inter-claim time with the threshold window, the distributions of the current threshold window and the inter-arrival time are determined. Furthermore, the statuses for the previous and current inter-arrival times give rise to the current claim size distribution as well. Like Constantinescu et al. (2016), we first identify the embedded Markov additive process where all the randomness takes a general form. Inspired by the Erlangisation technique, the key message of this paper is to analyse such risk process using a Markov fluid flow model where the underlying random variables follow phase-type distributions. This would further allow us to approximate the fixed observation windows by Erlang random variables. Then ruin probabilities under the process with Erlang(n) observation windows are proved to be Erlangian approximations for those related to the process with fixed threshold windows at the limit. An exact form of the limit can be obtained whose application will be illustrated further by a numerical example.

Event patterns extracted from anisotropic spectra of charged particles produced in Pb-Pb collisions at 2.76 TeV

Event patterns extracted from anisotropic spectra of charged particles produced in lead-lead collisions at 2.76 TeV are investigated. We use an inverse power-law resulted from the QCD calculus to describe the transverse momentum spectrum in the hard scattering process, and a revised Erlang distribution resulted from a multisource thermal model to describe the transverse momentum spectrum and anisotropic flow in the soft excitation process. The pseudorapidity distribution is described by a three-Gaussian function which is a revision of the Landau hydrodynamic model. Thus, the event patterns at the kinetic freeze-out are displayed by the scatter plots of the considered particles in the three-dimensional velocity, momentum, and rapidity spaces.

랜들리 및 어랑 수명분포에 의존한 소프트웨어 신뢰성 모형에 대한 신뢰도 속성 비교 연구

소프트웨어 개발시행 과정에서 소프트웨어 신뢰성은 매우 기본적이고 필수적인 문제 중에 하나이다. 소프트웨어 고장현상을 파악하기 위하여 비동질적인 포아송 과정에서 순간 고장률인 강도함수가 고장시간에 독립적으로 일정하거나, 증가형 혹은, 감소형 추세를 가질 수 있다. 본 논문에서는 소프트웨어 설계 과정에서 강도형태가 감소패턴을 따르는 랜들리 수명분포와 증가하다가 감소하는 어랑수명 분포를 활용한 소프트웨어 신뢰속성 모형에 대하여 신뢰도 장단점에 관한 연구를 하였다. 소프트웨어 고장현상을 파악하기 위하여 모수추정은 최우추정법을 적용하였다. 따라서, 본 논문에서는 소프트웨어 고장시간 자료를 적용하여 소프트웨어 신뢰도를 비교하고, 평가하였다. 그 결과, 랜들리 모형이 어랑분포 모형보다 신뢰도가 상승하는 것으로 나타났으나, 어랑분포 모형에서는 형상모수가 높을수록 높은 신뢰도를 나타내는 추세를 보였다. 본 논문를 통하여 소프트웨어 기획 부서에서는 특정한 수명분포와 형상모수를 활용함으로서 소프트웨어 고장분석을 활용한 소프트웨어 신뢰성 모형에 대한 신뢰성 속성을 적용한 데이터 및 기본 지식을 제공함으로서 소프트웨어 설계에 실질적인 도움을 줄 수 있다. Software reliability is one of the most basic and essential problems in software development. In order to detect the software failure phenomenon, the intensity function, which is the instantaneous failure rate in the non-homogeneous Poisson process, can have the property that it is constant, non-increasing or non-decreasing independently at the failure time. In this study, was compared the reliability performance of the software reliability model using the Landely lifetime distribution with the intensity function decreasing pattern and Erlang lifetime distribution from increasing to decreasing pattern in the software product testing process. In order to identify the software failure phenomenon, the parametric estimation was applied to the maximum likelihood estimation method. Therefore, in this paper, was compared and evaluated software reliability using software failure interval time data. As a result, the reliability of the Landely model is higher than that of the Erlang distribution model. But, in the Erlang distribution model, the higher the shape parameter, the higher the reliability. Through this study, the software design department will be able to help the software design by applying various life distribution and shape parameters, and providing software reliability attributes data and basic knowledge to software reliability model using software failure analysis.

Erlang renewal models for genetic recombination

Erlang renewal models, also called chi-squared models, provide a tractable model for genetic recombination that exhibits positive interference. Closed form expressions for multilocus probabilities are derived for the crossover process when it is a renewal process with the distance between crossovers modeled by a Erlang distribution. These expressions yield explicit formulas for the map functions, coincidence functions and distributions of the identity-by-descent process, giving exact results for a class of models that better model observed biological data.

Weighted Erlang and Lomax Distributions

Two new families of weighted Erlang and Lomax probability distributions are developed. Various properties including the moments and the bias in the estimated mean of a weighted distribution are derived. As applications of these distributions, time truncated acceptance sampling plans are designed to obtain the minimum sample size and the acceptance number for un-weighted and weighted distributions, where various pre-assigned producer’s and consumer’s risks, various ratios of true mean to the specified mean life, and various test termination time multipliers are considered. A real data set is analyzed to illustrate the proposed time-truncated sampling plan.

The number of key carcinogenic events can be predicted from cancer incidence

The widely accepted multiple-hit hypothesis of carcinogenesis states that cancers arise after several successive events. However, no consensus has been reached on the quantity and nature of these events, although “driver” mutations or epimutations are considered the most probable candidates. By using the largest publicly available cancer incidence statistics (20 million cases), I show that incidence of 20 most prevalent cancer types in relation to patients’ age closely follows the Erlang probability distribution (R2 = 0.9734–0.9999). The Erlang distribution describes the probability y of k independent random events occurring by the time x, but not earlier or later, with events happening on average every b time intervals. This fits well with the multiple-hit hypothesis and potentially allows to predict the number k of key carcinogenic events and the average time interval b between them, for each cancer type. Moreover, the amplitude parameter A likely predicts the maximal populational susceptibility to a given type of cancer. These parameters are estimated for 20 most common cancer types and provide numerical reference points for experimental research on cancer development.

An availability-based system with general repair via Bayesian aspect

We study the Bayesian inferences of an availability system with reboot delay and standby switching failures, in which the system consists of one active component and one warm standby. The system is studied under the assumption that the time-to-failure and the time-to-repair are assumed to follow an exponential and a general distribution. The reboot time is assumed to be exponentially distributed with parameter β. There is always the failure possibility c during the switching process from standby-component state to the active-component state. To implement the simulation inference for the system availability, two repair-time distributions, namely, the lognormal and Erlang distributions characterized by their shape parameters are considered. Finally, all simulation results are displayed by appropriate tables and curves to analyze the performance of the statistical inference procedures.

Determining Steady-State Characteristics of Some Queuing Systems with Erlangian Distributions

This article proposes a method to study M / E s / 1 / m, E r E s /1 / m, and E r / M / n / m queuing systems including the case when m = ∞. Recurrence relations are obtained to compute the stationary distribution of the number of customers in a system and its steady-state characteristics. The developed algorithms are tested on examples using simulation models constructed with the help of the GPSS World tools.

Quantized/Saturated Control for Sampled-Data Systems Under Noisy Sampling Intervals: A Confluent Vandermonde Matrix Approach

In this paper, a unified framework is established to investigate both the quantized and the saturated control problems for a class of sampled-data systems under noisy sampling intervals. A random variable obeying the Erlang distribution is used to describe the noisy sampling intervals. In virtue of the matrix exponential, the sampled-data control system is transformed into an equivalent discrete-time stochastic system, and the aim of this paper is to design a quantized/saturated sampled-data controller such that the resulting discrete-time stochastic system is stochastically stable when the sampling error follows the Erlang distribution. In order to deal with the case of multiple control inputs, a confluent Vandermonde matrix approach is proposed in the design process. By using the Kronecker product operation and the matrix inequality techniques, the desired quantized/saturated controller gains are designed in terms of the solution to certain matrix inequalities. Finally, a simulation example is exploited to verify the effectiveness of the proposed design approach.