Pliska Studia Mathematica
Volume 27, 2017
Proceedings of the XVII International Summer Conference on
Probability and Statistics
Seminar on Staticstical Data Analysis
Workshop on Branching Processes and Application
25 June − 1 July, 2016, Pomorie, Bulgaria
GUEST EDITOR: E. Stoimenova, M. Slavtchova-Bojkova
C O N T E N T S
- Stoimenova, E., M. Slavtchova-Bojkova. Preface (p. 3)
- Jordanova, P. Multivariate Compounds with Equal Number of Summands (pp. 5−22)
- Kharin, Y., M. Zhurak. Statistical Forecasting Based on Binomial Conditional Autoregressive Model of Spatio-Temporal Data (pp. 23−36)
- Koleva-Petkova, D., M. Milev. Application of Discrete Dividends to American Option Pricing (pp. 37−46)
- Leri, M. M. Forest Fire Model on Configuration Graphs with Random Node Degree Distribution (pp. 47−54)
- Mitov, K. V., N. M. Yanev. Critical Markov Branching Processes with Non-homogeneous Poisson Immigration
- Mitov, K. V., N. M. Yanev. Limit Theorems for Some Classes of Alternating Regenerative Branching Processes
- Stoimenova, E. Power of Exceedance-type Tests against Location Shift Alternative
- Tchorbadjieff, A. Using Branching Processes to Simulate Cosmic Rays Cascades
- Yanev, G., R. Colson. Monotone Empirical Bayes Estimators for the Reproduction Number in Borel-Tanner Distribution
- Zhelyazkova, M., R. Yordanova, M. Slavtchova-Bojkova. A Bayesian Spatial Analysis of Mumps Data in Bulgaria
A B S T R A C T S
MULTIVARIATE COMPOUNDS WITH EQUAL NUMBER OF SUMMANDS
2010 Mathematics Subject Classification: 62E15.
Key words: Compound distributions, Multinomial distribution, Negative multinomial distribution.
The paper considers multivariate discrete random sums with equal number of summands.
Such distributions describe the total claim amount received by a company in a fixed time point. In Queuing theory they characterize cumulative waiting times of customers up to time t. In Theory of branching processes they model the number of heirs at a fixed point in time.
Here some general properties and formulae for numerical characteristics of these distributions are derived and some particular cases are considered.
STATISTICAL FORECASTING BASED ON BINOMIAL CONDITIONAL AUTOREGRESSIVE MODEL OF SPATIO-TEMPORAL DATA
2010 Mathematics Subject Classification: 62-07, 62M20, 62M30.
Key words: spatio-temporal data, vector Markov chain, maximum likelihood estimator, statistical hypotheses testing, optimal forecasting statistic.
Binomial conditional autoregressive model of spatio-temporal data is presented. Asymptotic properties of the maximum likelihood estimators of parameters for the binomial conditionally autoregressive model of spatio-temporal data are studied. Statistical tests on the values of true unknown
parameters are constructed. Results of computer experiments on simulated and real data are given.
APPLICATION OF DISCRETE DIVIDENDS TO AMERICAN OPTION PRICING
2010 Mathematics Subject Classification: 35K10, 65N06, 91G60, 62P05.
Key words: Finite differences, discrete dividends, American options.
Dividends are a detail of financial instruments pricing which is often being oversimplified. However, companies do declare (and pay out) flows which can be significant. In this paper we briefly review some known approaches to this topic. We analyse a few known drawbacks with application to American option pricing. Due to the fact that these options rely on numerical methods for their pricing, applying discrete dividends to the chosen approach may affect the solution quality. As we will show shortly, for some methods there are flaws affecting positivity and smoothness of the numerical solution while others are too computationally heavy. We find that applying discrete dividends to an exponentially fitted scheme (the Duffy scheme) overcomes these problems and we manage to obtain a smooth and sensible solution.
FOREST FIRE MODEL ON CONFIGURATION GRAPHS WITH RANDOM NODE DEGREE DISTRIBUTION
Marina M. Leri
2010 Mathematics Subject Classification: 05C80, 05C82, 62G35.
Key words: Random graphs, Complex networks, Robustness, Simulation modeling, Forest
We consider two types of configuration graphs with node degrees being
i.i.d. random variables following either the power-law or the Poisson distribution. The distribution parameter is a random variable following the uniform distribution on a predefined interval.
We consider a destructive process which can be interpreted as a fire spreading over the graph links, and could be used for modeling forest fires as well as banking system defaults. This process is often referred to as a "forest fire model". The probability of fire transfer over a graph link either possesses a predefined value and is fixed for all the graph links or is a random variable following the standard uniform distribution. By computer simulation we study the robustness of such graphs from a viewpoint of node survival in the two cases of starting a fire propagation process: the "random ignition" and the "targeted lightning-up". The results on finding the optimal interval of the node degree distribution parameter that would ensure maximum survival of trees in case of a fire are presented. A comparative analysis of various graph models in terms of their robustness to various fire propagation processes was performed.
CRITICAL MARKOV BRANCHING PROCESSES WITH NON-HOMOGENEOUS POISSON IMMIGRATION
Kosto V. Mitov
Nikolay M. Yanev
2010 Mathematics Subject Classification: 60J80.
Key words: Branching processes, Immigration, Poisson process, Limit theorems.
The paper deals with critical Markov branching processes with infinite offspring variance allowing an immigration component at the jump points of a time inhomogeneous Poisson process. The asymptotic formulas for the probability for non extinction are obtained depending on the rate of change of the intensity of the Poisson process. Proper limiting distributions are proved under the appropriate normalization.
LIMIT THEOREMS FOR SOME CLASSES OF ALTERNATING REGENERATIVE BRANCHING PROCESSES
Kosto V. Mitov
Nikolay M. Yanev
2010 Mathematics Subject Classification: 60J80, 60K05.
Key words: Branching Processes, Immigration, Regenerative Processes,
Alternating Renewal Processes.
In this paper we propose and study three new classes of alternating
regenerative (AR) branching processes. Limiting distributions are obtained
for AR Sevastyanov processes, for AR Sevastyanov processes with
non-homogeneous Poisson immigration and for AR randomly indexed branching
processes. All these processes are investigated applying renewal and
regenerative methods developed in Mitov and Yanev (Limit theorems for alternating renewal processes in the infinite mean case. Adv. in Appl. Probab., 33, No 4 (2001), 896−911;
Regenerative processes in the infinite mean
cycle case. J. Appl. Probab., 38, No 1 (2001), 165−179).
POWER OF EXCEEDANCE-TYPE TESTS AGAINST LOCATION SHIFT ALTERNATIVE
2010 Mathematics Subject Classification: 62G10, 62E15, 62G20.
Key words: Two-sample problem, Exceedance statistics, Precedence statistics, Lehmann alternative; Location-shift alternative, Stochastic ordering.
This paper deals with a class of nonparametric two-sample tests for ordered alternatives. The test statistics proposed are based on the number of observations from one sample that precede or exceed a threshold specified by the other sample, and they are extensions of Šidák's test. We study their power properties against the location-shift alternative for distributions from the uniform, normal and exponential families. We give the corresponding power functions, obtained by Monte Carlo simulation, and make some comparative comments.
USING BRANCHING PROCESSES TO SIMULATE COSMIC RAYS CASCADES
2010 Mathematics Subject Classification: 60-04.
Key words: Branching processes, Simulations, Cosmic Rays, Multi-type Branching Process, Immigration.
The cosmic rays (CR) cascades are one of the most famous examples of
branching processes in physics. They consist of many different types of
secondary high energy particles which are offsprings of leading primary
one, usually high energy nucleon or gamma photon after collision with atmosphere. The rate of expansion of cascade depends on multiple different
conditional probabilities as the chance of survival in atmosphere without
interactions, particle lifetime, the number of daughter particles, etc.
The paper presents quantitative simulated results with a specially written in R software for parallel simulation of distributions of multiple types of daughter particles. The processes are based on simplified models of Hadron cascades simulated by age-dependent and imbedded Markov Galton-Watson branching processes. For the sake of simplicity in modelling the probability dependencies on particle's energy and free path in atmosphere are not always constrained strictly to the available experimental results. Moreover, the scattering angles also are not considered in this version of software.
MONOTONE EMPIRICAL BAYES ESTIMATORS FOR~THE~REPRODUCTION NUMBER IN BOREL-TANNER DISTRIBUTION
George P. Yanev
2010 Mathematics Subject Classification: Primary: 62C12; Secondary: 60J80.
Key words: Monotone empirical Bayes estimators, Borel-Tanner distribution, branching process, reproduction number of epidemics.
We construct a monotone version of an empirical Bayes estimator for the parameter of the Borel-Tanner distribution.
Some properties of the estimator's regret risk are illustrated through simulations.
A BAYESIAN SPATIAL ANALYSIS OF MUMPS DATA \\IN BULGARIA
2010 Mathematics Subject Classification: 62-07.
Key words: Conditional autoregressive models, Disease mapping, spatial correlation.
Bayesian spatial methods have been widely applied in different scientific areas such as epidemiological studies, image processing and many others. In this work we use Bayesian hierarchical model with Gaussian conditionally autoregressive prior to a collection of weekly mumps data from 2007 outbreak in Bulgaria. We generate a disease mapping of the crude standardized incidence ratio across all regional centers. Similar mapping is also produced for the smoothed relative risk. The combination of methods for estimates of the relative risk is a powerful tool to identify high risk regions and may be used to guide local authorities and programs.