Pliska Studia Mathematica

Volume 24, 2015

Proceedings of the XVI International Summer Conference on Probability and Statistics
Seminar on Statistical Data Analysis
Workshop on Branching Processes and Applications
21−29 June, 2014, Pomorie, Bulgaria

GUEST EDITOR: N. M. Yanev

C O N T E N T S

• Yanev, N. M. Preface. (pp. 3−4)
• Vatutin, V. Scientific and Personal Life of B. A. Sevastyanov (pp. 5−12)
• Yanev, N. M. In Memory of N. A. Dmitriev (pp. 13−20)
  
  Branching Processes and Applications
  
  
• González, M., C. Gutiérrez, R. Martínez, I. del Puerto. Asymptotic Behaviour of Y-linked Genes through Bisexual Branching Processes for Genetic Balanced Sex Determination (pp. 21−34)
• Hyrien, O., K. V. Mitov, N. M. Yanev. Subcritical Markov Branching Processes with Non-Homogeneous Poisson Immigration (pp. 35−54)
• Jagers, P. On the Complete Life Career of Populations in Environments with a Finite Carrying Capacity (pp. 55−60)
• Molina, M., M. Mota, A. Ramos. Contributions to the Class of Branching Processes in Varying Environments (pp. 61−72)
• Staneva, A., V. Stoimenova. Statistical Estimation in Branching Processes with Bivariate Poisson Offspring Distribution (pp. 73−88)
• Topchii, V. A., V. A. Vatutin, A. M. Iksanov.Evolution of a Two-Type Bellman-Harris Process Generated by a Large Number of Particles (pp. 89−98)
• Traynov, P., M. Slavtchova-Bojkova. Estimating the Effect of Economic Crisis with Crump-Mode-Jagers Branching Process (pp. 99−110)
• Yanev, G. P. Critical Controlled Branching Processes and Their Relatives (pp. 111−130)
  
  Stochastic Models and Statistical Inference
  
  
• Atanasov, D. V., D. M. Dimitrov. Equating Test Scores Based on Their IRT Calibration (pp. 131−138)
• Benseghir, R., A. Benchettah. Existence and Uniqueness of the Solution for the Stochastic Equation of Motion of a Viscous Gas in a Discretized One-Dimensional Domain (pp. 139−150)
• Filipova, M., I. Zheleva, A. Lecheva, P. Rusev. Analysis of Surface Water Key Pollutants of the Tributaries of the Danube River in Bulgarian Section (pp. 151−162)
• Pancheva, E. I., A. Gacovska-Barandovska. On Limit Laws for Central Order Statistics under Power Normalization (pp. 163−180)
• Sečkárová, V. Dynamic Parameter Estimation Based on Minimum Cross-Entropy Method for Combining Information Sources (pp. 181−189)
• Zhelyazkova, M., S. Nencheva-Svechtarova, V. Svechtarov. Clinico-Statistical Analysis of the Effect of Gaaias Laser Treatment on Temporomandibular and Myofascial Pain Disorders (pp. 189−194)

A B S T R A C T S

SCIENTIFIC AND PERSONAL LIFE OF B. A. SEVASTYANOV
V. Vatutin vatutin@mi.ras.ru

IN MEMORY OF N. A. DMITRIEV
Nikolay M. Yanev yanev@math.bas

It is an unusual story for Nikolay A. Dmitriev (1924−2000) born in Moscow, whose father was from an eminent Bulgarian family. His grandfather has taken active part in the Bulgarian struggles for the liberation against the Ottoman Empire together with the national poet and hero Hristo Botev. Only 14 years old Nikolay became a winner in the Russian Mathematical Olympiad and the youngest student at Moscow State University. After that he was a PhD student of A. N. Kolmogorov in Steklov Mathematical Institute and the creator together with him of the modern theory of branching stochastic processes which could be interpreted as mathematical models of nuclear reactions and a lot of other real phenomena. N. A. Dmitriev has been working all his life as a researcher in a secret scientific organization and was one of the creators of the atomic and hydrogen bombs. He was a coauthor and collaborator with well-known mathematicians and physicists as Keldish, Gelfand, Dynkin, Zeldovich, Hariton, Kurchatov, Frank-Kameneckii, Saharov, Bogolubov, Tamm and others.

ASYMPTOTIC BEHAVIOUR OF Y-LINKED GENES THROUGH BISEXUAL BRANCHING PROCESSES FOR GENETIC BALANCED SEX DETERMINATION
M. González mvelasco@unex.es,
C. Gutiérrez cgutierrez@unex.es,
R. Martínez rmartinez@unex.es,
I. del Puerto idelpuerto@unex.es

2010 Mathematics Subject Classification: 60J80, 60J85.
Key words: Sex-linked inheritance. Genetic balanced sex determination. Bidimensional two-sex stochastic model. Perfect fidelity mating. Rates of growth.

The limiting genotype behaviour of Y-linked genes is studied in a two-sex monogamous population, where the sex designation is balanced. To this end, a multitype bisexual branching process is considered to model the evolution of the numbers of females and males of each genotype. It is assumed perfect fidelity mating with preference of females for males carrying certain allele of the gene. From this model, conditions for having positive probability of coexistence are investigated. Moreover, genotype growth rates on the coexistence event are established. Hence, the dominant genotype is found. Finally, the main results are illustrated by means of a simulated example.

SUPERCRITICAL MARKOV BRANCHING PROCESSES WITH NON-HOMOGENEOUS POISSON IMMIGRATION
Ollivier Hyrien Ollivier_Hyrien@urmc.rochester.edu,
Kosto V. Mitov kmitov@yahoo.com,
Nikolay M. Yanev yanev@math.bas.bg

2010 Mathematics Subject Classification: 60J80.
Key words: Branching processes, Immigration, Poisson process, Limit theorems.

The paper proposes an extension of Sevastyanov (1957) model based on a Markov branching process allowing an immigration component in the moments of a homogeneous Poisson process. Now Markov branching processes are also considered but assuming a time-nonhomogeneous Poisson immigration. These processes could be interpreted as models in cell proliferation kinetics with stem cell immigration. Limit theorems are proved in the subcritical case and new effects are obtained due to the non-homogeneity.

ON THE COMPLETE LIFE CAREER OF POPULATIONS IN~ENVIRONMENTS WITH A FINITE CARRYING CAPACITY
P. Jagers jagers@chalmers.se

2010 Mathematics Subject Classification: Primary 60J80; Secondary 92D25.
Key words: branching process, population life span, carrying capacity, extinction.

If a general branching process evolves in a habitat with a finite carrying capacity, i. e. a number such that reproduction turns subcritical as soon as population size exceeds that number, then the population may either die out quickly, or else grow up to around the carrying capacity, where it will linger for a long time, until it starts decaying exponentially to extinction.

STATISTICAL ESTIMATION IN BRANCHING PROCESSES WITH BIVARIATE POISSON OFFSPRING DISTRIBUTION
Ana Staneva anastaneva@gmail.com,
Vessela Stoimenova stoimenova@fmi.uni-sofia.bg

2010 Mathematics Subject Classification: Primary 60J80; Secondary 62F15, 62F35.
Key words: Bivariate power series distributions, Bayesian estimation, Trimmed likelihood, Multitype branching processes.

We consider two-type branching stochastic processes with offspring distributions from the bivariate poisson distribution family. We study the parametric estimation under different sampling schemes − when the entire family tree is observed and when observations only on the generation sizes are made. We use a randomized algorithm to switch from the generation sizes scheme to family tree observations and apply it in the context of the Bayesian approach. The considered estimation methods are illustrated via a simulational and computational example.

EVOLUTION OF A TWO-TYPE BELLMAN−HARRIS PROCESS GENERATED BY A LARGE NUMBER OF PARTICLES
V. A. Topchii topchij@ofim.oscsbras.ru,
V. A. Vatutin vatutin@mi.ras.ru,
A. M. Iksanov iksan@univ.kiev.ua

2010 Mathematics Subject Classification: 60J80, 60E10, 60J85, 60K05.
Key words: two-type critical Bellman--Harris branching process; process initiated by a large number of particles; survival probability; limit theorems.

We investigate the evolution of a two-type critical Bellman--Harris branching process with the following properties: the tail of the life-length distribution of the first type particles is of order o(t^-2); the tail of the life-length distribution of the second type particles is regularly varying at infinity with index −β, β ∈ (0,1]; the process is generated at time t=0 by a large number N of the second type particles and no particles of the first type. Letting t = N^γL(N), where γ ∈ [0,∞) and L(N) is a function slowly varying at infinity, we show that the set of triples (β, γ, L(N)) may be divided into several regions within each of which the process at time t exhibits asymptotics (as N, t → ∞) which is different from those in the other regions.

ESTIMATING THE EFFECT OF ECONOMIC CRISIS WITH~CRUMP-MODE-JAGERS BRANCHING PROCESS
Plamen Trayanov plament@fmi.uni-sofia.bg,
Maroussia Slavtchova-Bojkova bojkova@fmi.uni-sofia.bg

2010 Mathematics Subject Classification: 60J85, 92D25.
Key words: branching process, human population, Malthusian parameter.

The Crump-Mode-Jagers branching process is used to model the populations of Bulgaria, Greece, Ireland, Italy, Portugal and Spain, which were influenced greatly by the economic crisis, started in 2008. The social effects were felt shortly after 2008 with different delay for each country. The unemployment rate in some of these countries was very high and constituted mainly of young people around 25 years old. This paper reviews the demographic changes that were experienced during the crisis and its effect on population growth. It also reviews the current retirement policy in these countries and extrapolates the needed changes in order to keep the current percentage of working people constant.

CRITICAL CONTROLLED BRANCHING PROCESSES AND THEIR RELATIVES
George P. Yanev yanevgp@utpa.edu

2010 Mathematics Subject Classification: 60J80.
Key words: critical branching processes, immigration, emigration, control, extinction.

This survey aims at collecting and presenting results for one-type, discrete time branching processes with random control functions. In particular, the subclass of critical migration processes with different regimes of immigration and emigration is reviewed in detail. Critical controlled branching processes with continuous state space are also discussed.

EQUATING TEST SCORES BASED ON THEIR IRT CALIBRATION
D. V. Atanasov datanasov@nbu.bg,
D. M. Dimitrov ddimitro@gmu.edu

2010 Mathematics Subject Classification: 91E10, 91E45.
Key words: key words IRT, score equating.

Common problem arising in the everyday practice of ability evaluation using tests is how one can compare or equate the scores obtained on different tests or different forms of the same test. Under the main assumption that these tests are based on the same unidimensional latent trait, their scores can be compared using the IRT calibration of the items in both tests. In this paper a procedure for test score equating, based on the sequence of tests with common items in each test are considered.

EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR THE STOCHASTIC EQUATION OF MOTION OF A VISCOUS GAS IN A DISCRETIZED ONE-DIMENSIONAL DOMAIN
R. Benseghir benseghirrym@ymail.com,
A. Benchettah abenchettah@hotmail.com

2010 Mathematics Subject Classification: 39A50, 34C25, 37N10, 34A45.
Key words: Stochastic equation, viscous barotropic gas, periodic measure, discretized domain.

A stochastic equation system corresponding to the description of the motion of a barotropic viscous gas in a discretized one-dimensional domain with a weight regularizing the density is considered. In [R. Benseghir, H. Fujita Yashima. Mesure invariante pour léquation stochastique d'un gaz visqueux en une dimension avec la discrétisation du domaine. Romanian Jornal. Pure Appl. Math., 58 (2013), 149−162], [R. Benseghir. Processus stochastique réciproque, Equation stochastique d'un gaz visqueux. Doctoral thesis, Badji Mokhtar university, Annaba, Algeria, 2014], the existence and uniqueness of the solution of this discretized problem in the stationary case was established. In this paper, by applying the technics used in [R. Benseghir, H. Fujita Yashima. Mesure invariante pour léquation stochastique d'un gaz visqueux en une dimension avec la discrétisation du domaine. Romanian Jornal. Pure Appl. Math., 58 (2013), 149−162], we generalize this result in the periodic case.

TIME TO EXTINCTION IN BRANCHING PROCESSES AND ITS APPLICATION IN EPIDEMIOLOGY
M. Filipova mfilipova@uni-ruse.bg,
I. Zheleva izheleva@uni-ruse.bg,
A. Lecheva alecheva@uni-ruse.bg,
P. Rusev prussev@uni-ruse.bg

2010 Mathematics Subject Classification: 62-07, 62P12, 92F99.
Key words: surface water, key pollutants, statistical data, environment.

Based on official data, a comparative analysis of the surface water along the rivers flowing into the Danube River in the transborder area Bulgaria−Romania is presented. The content of dissolved oxygen, nitrate nitrogen and Biological and Chemical Oxygen Demand (BOD₅ and COD) for a five year period 2009−2013 is analyzed. The aim is the dynamics of these indicators and the reasons for the current exceedances to be traced and analyzed. Measures for improving the condition of the surface runoff are also proposed.

ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION
E. I. Pancheva pancheva@math.bas.bg,
A. Gacovska-Barandovska aneta@pmf.ukim.mk

2010 Mathematics Subject Classification: 62G30, 62E20, 39B22.
Key words: k-th upper order statistic, Central rank, Power normalization, Regular norming sequence.

Smirnov (1949) derived four limit classes of distributions for linearly normalized central order statistics. In this paper we investigate the possible limit distributions of the $k$-th upper order statistics with central rank using regular power norming sequences and obtain twelve limit classes.

DYNAMIC PARAMETER ESTIMATION BASED ON~MINIMUM CROSS-ENTROPY METHOD FOR COMBINING INFORMATION SOURCES
Vladimíra Sečkárová seckarov@utia.cas.cz

2010 Mathematics Subject Classification: 94A17, 62L12.
Key words: minimum cross-entropy principle, Kullback-Leibler divergence, dynamic diffusion estimation.

When combining information sources, e.g. measuring devices or experts, we deal with two problems: which combining method to choose (linear combination, geometric mean) and how to measure the reliability of the sources, i.e. how to assign the weights to them. Inspired by [V. Sečkárová. On Supra-Bayesian Weighted Combination Of Available Data Determined By Kerridge Inaccuracy And Entropy. Pliska Stud. Math. Bulgar., 22 (2013), 159−168] we introduce a method which overcomes such shortcomings. Proposed method, based on minimization of the Kullback-Leibler divergence with specific constraints, directly combines data, i.e. probability vectors, thus no additional step to obtain the weights is needed. The detailed description of the proposed method and a comparison with recently introduced dynamic diffusion estimation [K. Dedecius, V. Sečkárová. Dynamic Diffusion Estimation in Exponential Family Models. IEEE Signal Processing Letters, 20, No 11 (2013), 1114−1117], which heavily depends on the determination of the weights, form the core of this contribution.

CLINICO-STATISTICAL ANALYSIS OF THE EFFECT OF GAAIAS LASER TREATMENT ON TEMPORO- MANDIBULAR AND MYOFASCIAL PAIN DISORDERS
Maya Zhelyazkova zhelyazkova@fmi.uni-sofia.bg,
Savina Nencheva-Svechtarova savinanenceva@gmail.com,
Vassil Svechtarov vassilsvechtarov@yahoo.com

2010 Mathematics Subject Classification: Primary 6207.
Key words: temporomandibular joint, laser photobiomodulation, pain, paired t-test.

The tested GaAIAs (870 nm) and superluminescent red light (660 nm)/MedX 1100 phototherapy device demonstrated positive results regarding the relief of painful symptoms in patients with chronic temporomandibular and myofascial complaints. A significant reduction (p < 0.05) in the pain level was observed in the temporomandibular joint and in the masseter muscles using paired samples t-test and Wilcoxon signed rank test.