Pliska Studia Mathematica Bulgarica
Volume 20, 2011
Proceedings of the XIV International Summer Conference on Probability and Statistics and Seminar on Statistical Data Analysis, SDA, 2010, Workshop on Branching Processes
GUEST EDITOR: N. Yanev
Sofia, 2011
C O N T E N T S
- Atanasov, D. Many Facet, Common Latent Trait Polytomous IRT Model and EM Algorithm. (pp. 5-12)
- Balkema, G. Densities with Spherical Level Sets in the Gauss-exponential Domain. (pp. 13-38)
- Daskalova, N., Bonev, B. About the Oral Health of Bulgarian Population over 20 Years Old. (pp. 39-44)
- Daskalova, N. EM Estimation of the Offspring Distribution in Mutitype Branching Processes - a Model in Cell Kinetics. (pp. 45-52)
- Dimitrov, M. Single-Server Queueing System with Markov-Modulated Arrivals and Service Times. (pp. 53-62)
- Gonzales, M., Molina, M., Mota, M., del Puerto, I. Estimation of the Offspring and Immigration Mean Vectors for Bisexual Branching Processes with Immigration of Females and Males. (pp. 63-80)
- Hyrien, O., Yanev, N. Two-Type Age-Dependent Branching Pro-cesses with Inhomogeneous Immigration as Models of Renewing Cell Population. (pp. 81-108)
- Jordanova, P., Mitov, I. Maximal Number of Successors in a NGINAR(l) Process. (pp. 109-120)
- Kolkovska, E. Risk Measures for Classical and Perturbed Risk Processes - A Survey. (pp. 121-134)
- Ma, S., Molina, M., Xing, Y. Some Contributions to the Class of Two-Sex Branching Processes Depending on the Number of Couples in the Population. (pp. 135-148)
- Mitova-Bobcheva, M., Slavtchova-Bojkova, M., Kojouharova, M., Kurchatova, A. Analysing and Monitoring Surveillance Data of Mumps in Bulgaria. (pp. 149-154)
- Mitov, K., Mitov, G. Subcritical Randomly Indexed Branching Processes. (pp. 155-168)
- Pancheva, E., Mitov, K., Nadarajah, S. Nonlinear Normalization in Limit Theorems for Extremes. (pp. 169-186)
- Prodanova, K. Optimization Problem in a Class of Linear System. Application to Multiple Drug Administration. (pp. 187-202)
- Stoimenova, V., Atanasov, D. Robust Estimation in Multitype Branching Processes Based on Their Asymptotic Properties. (pp. 203-220)
- Veleva, E. Properties of the Bellman Gamma Distribution. (pp. 221-232)
- Yanev, G. Characterizations of Exponential Distribution Via Conditional Expectations of Record Values. (p. 233-242)
- Zeifman, A., Korotysheva, A. Weak Ergodicity of Mt/Mt/N/N + R Queue. (pp. 243-254)
A B S T R A C T S
Many Facet, Common Latent Trait Polytomous IRT Model and Em Algorithm
Dimitar Atanasov datanasov@fmi.uni-sofia.bg
2000 Mathematics Subject Classification: 91E45
Key words: item response theory, many facets, EM algorithm.
There are many areas of assessment where the level of performance can not
be determined using ordinal, "objective" test. For example, there are aspects
of writing language skills which should be subjectively qualified by judges
based on some criteria. Usually such data is studied using the Many Facet
Rasch Model. To address the problem with Item Response Theory (IRT)
approach a Many Facet IRT model for polytomous item response in the case
of common latent trait is presented. For estimation of the parameters of the
model the EM-algorithm is derived.
Densities With Spherical Level Sets in the Gauss-Exponential Domain
Guus Balkema
2000 Mathematics Subject Classification: 60F05, 60B10
Key words: Gauus-exponential, high risk scenario, hyperbolic metric, level set, spherical,
unimodal.
For a sequence of independent observations Z_{n} = (X_{n}, Y_{n}) from the bivariate
standard normal density there are at least three asymptotic descriptions of
the sample clouds. This paper looks at sample clouds from light-tailed unimodal densities with
spherical level sets. What conditions will give the asymptotic behaviour
above? We do not assume that the level sets are concentric.
About the Oral Health of Bulgarian Population over 20 Years Old
Nina Daskalova
Boyko Bonev
2000 Mathematics Subject Classification: 62P10, 62J12.
Key words: epidemiological research, caries, statistical data analysis, generalized linear
models, logistic regression.
Caries is the most widespread dental disease around the world. Thus, caries
prevalence and intensity are repeatedly studied. Epidemiological research
has been conducted covering 1636 persons from Bulgaria. A statistical model
for the number of teeth damaged from caries, depending on the age and sex
of the patients has been considered in this work.
Em Estimation of the Offspring Distribution in Mutitype Branching Processes - a Model in
Cell Kinetics
Nina Daskalova
2000 Mathematics Subject Classification: 60J80, 60J85, 62P10, 92D25.
Key words: multitype branching processes, offspring distribution, maximum likelihood estimation, expectation maximization, stochastic context-free grammars, inside-outside algorithm,
cell kinetics modelling.
Multitype branching processes (MTBP) have been proven to be very useful
models in cell kinetics. A typical example is the process of oligodendrocyte
generation in cell culture, which is regarded as two-type branching process.
Usually, such a process is not observable in the sense of the whole tree,
but only as the "generation" at given moment in time, which consist of the
number of cells of every type. An EM-type algorithm is used to obtain
a maximum likelihood (ML) estimation of the offspring distribution. The
performance of the presented algorithm is assessed using simulated data.
Single-Server Queueing System with Markov-Modulated Arrivals and Service Times
Mitko Dimitrov
2000 Mathematics Subject Classification: 60K25.
Key words: Markov-modulated queues, waiting time, heavy traffic.
Markov-modulated queueing systems are those in which the input process
or service mechanism is influenced by an underlying Markov chain. Several
models for such systems have been investigated .
In this paper we present heavy traffic analysis of single queueing system
with Poisson arrival process whose arrival rate is a function of the state of
Markov chain and service times depend on the state of the same Markov
chain at the epoch of arrivals.
Estimation of the Offspring and Immigration Mean Vectors for Bisexual Branching
Processes with Immigration of Females And Males
M. Gonzalez
M. Molina
M. Mota
I. del Puerto
2000 Mathematics Subject Classification: 60J80, 62M05.
Key words: Bisexual branching processes, immigration, moment method, least squares
method, asymptotic properties.
This paper concerns with the bisexual branching model with immigration
of females and males in each generation introduced in [4]. For this model
the problem of estimating the offspring and immigration mean vectors is
dealt. The estimation is considered in two sample situations, depending on
the ability to observe the number of female and male immigrants or the
impossibility to do it. The asymptotic properties of the proposed estimators
are investigated in the supercritical case. The behaviour of the estimators
is illustrated through a simulated example.
Two-Type Age-Dependent Branching Processes with Inhomogeneous
Immigration as Models of Renewing Cell Population
Ollivier Hyrien
Nikolay M. Yanev
2000 Mathematics Subject Classification: 60J80, 60J85, 92C37.
Key words: Age-dependent branching processes, immigration, cell proliferation models,
limiting moments and correlation.
Two-type reducible age-dependent branching processes with inhomogeneous
immigration are considered to describe the kinetics of renewing cell populations. This class of processes can be used to model the generation of
oligodendrocytes in the central nervous system in vivo or the kinetics of
leukemia cells. The asymptotic behavior of the first and second moments,
including the correlation, of the process is investigated.
Maximal Number of Successors in a Nginar(1)Process
Pavlina K. Jordanova
Ivan K. Mitov
2000 Mathematics Subject Classification: 60J80, 60J20, 60J10, 60G10, 60G70, 60F99.
Key words: Markov chains; NGINAR process; Branching processes; Processes of maxima;
Stationary sequences.
In this paper we obtain upper and lower bounds for normalized sequences of
maxima, associated with a stationary integer-valued autoregressive process
of first order with geometricmarginals (NGINAR(1) process). These processes
are a special case of AR(1) processes and strictly stationary ergodic Markov
chains.
Risk Measures for Classical and Perturbed Risk Processes - a Survey
Ekaterina T. Kolkovska
2000 Mathematics Subject Classification: 60B10, 60G17, 60G51, 62P05.
Key words: Classical risk process, α-processes, ruin probability, severity of ruin,
expected discounted penalty Gerber-Shiu function.
In this review paper we consider several risk measures in actuarial mathematics, such as the ruin probability,
the ruin time, the severity of ruin, the
surplus immediately before ruin, and the Gerber-Shiu penalty function as a
generalization of these measures. We discuss results on these measures for
classical and perturbed classical risk processes.
Some Contributions to the Class of Two-Sex Branching Processes Depending on the Number
of Couples in the Population
Shixia Ma
Manuel Molina
Yongsheng Xing
2000 Mathematics Subject Classification: 60J80.
Key words: Two-sex branching processes, limiting behavior.
We consider the class of two-sex branching processes with offspring and
mating depending on the number of couples in the population introduced
in Molina et al. (2008). In addition to its theoretical interest, this class
also has clear practical implications, especially in population dynamics. We
investigate its extinction probability and limiting behavior. By considering
different probabilistic approaches, necessary and sufficient conditions for its
almost sure extinction are determined. Assuming the nonextinction, some
limiting results are derived.
Analysing and Monitoring Surveillance Data of Mumps in Bulgaria
M. Mitova-Bobcheva
M. Slavtchova-Bojkova
M. Kojouharova
A. Kurchatova
2000 Mathematics Subject Classification: 62P10.
Key words: Bayesian approach, Outbreak, Surveillance of Infectious Diseases.
This paper aims to present the applicability of the Bayesian methodology
for monitoring surveillance data collected for vaccine-preventable diseases,
like mumps, measles and possibly others.
We investigate the ability of implemetation of the R-Package "surveillance"
especially written for monitoring surveillance data to the mumps data
kindly provided by the National Center of Infectious and Parasitic Diseases
in Bulgaria.
Subcritical Randomly Indexed Branching Processes
Kosto V. Mitov
Georgi K. Mitov
2000 Mathematics Subject Classification: 60J80, 62P05.
Key words: Galton-Watson branching process; Random time change; Moments; Extinction;
Limit theorems
The paper continues the study of the randomly indexed branching processes
in the subcritical case. The asymptotic behavior of the moments and the
probability for non-extinction is investigated. Conditional limiting distributions are obtained.
Nonlinear Normalization in Limit Theorems for Extremes
E. I. Pancheva
K. V. Mitov
S. Nadarajah
2000 Mathematics Subject Classification: 60G70, 60F05.
Key words: Extreme values, Nonlinear normalization, Limit theorems, Domain of attraction.
It is well known that under linear normalization the maxima of iid random
variables converges in distribution to one of the three types of max-stable
laws: Frechet, Gumbel and Weibull. During the last two decades the first
author and her collaborators worked out a limit theory for extremes and extremal processes under non-linear but monotone normalizing mappings. In
this model there is only one type of max-stable distributions and all continuous and strictly increasing df's belong to it. In a recent paper on General
max-stable laws, Sreehari points out two "confusing" results in Pancheva
(1984). They concern the explicit form of a max-stable df with respect to
a continuous one-parameter group of max-automorphisms, and domain of
attraction conditions. In the present paper the first claim is answered by
a detailed explanation of the explicit form, while for the second we give a
revised proof. The rate of convergence is also discussed.
Optimization Problem in a Class of Linear System. Application to Multiple Drug
Administration
Krasimira Prodanova
2000 Mathematics Subject Classification: 62H15, 62P10.
Key words: optimal control, application to medical science.
The optimal control settings are useful tools in studying the behavior of
the plasma concentration of a drug (medicine) after its application. Linear
compartmental systems are widely used as mathematical models for studying
this behavior. In this paper we develop a strategy for minimizing the
total amount of applied drug under the restriction: the mean plasma concentrations
belongs to prescribed therapeutic interval. The aim of the article
is to give to the therapeutist an simple and applicable formula for the optimal
input sequence of multiple doses as a function of the time and rate
constant of drug elimination. The desired formula is given as a result of the
theorems for optimal input function which we proved. The optimal input is
important specially for patients, which needs of treatment with antibiotics
but they have kidney shortage function. Because of this the relationship between
the optimal dose and the time intervals of administration to maintain
a effective drug concentration in plasma is considered too. The results are
applied to the two compartment stochastic model using experimental data
of plasma concentration after single administration of antibiotic Tobramicin
to a patient with kidney shortage function.
Robust Estimation in Multitype Branching Processes Based on their Asymptotic
Properties
Vessela Stoimenova
Dimitar Atanasov
2000 Mathematics Subject Classification: 60J80.
Key words: Trimmed likelihood, multitype branching process, asymptotic properties, para-
meter estimation.
In this work we propose two procedures for robust estimation of the individual
distributions of multitype discrete-time Galton-Watson branching
processes with an increasing number of ancestors, using the relative frequencies
of the process and their asymptotic distributions. The study is based
on simulations and numerical results.
Properties of the Bellman Gamma Distribution
Evelina Veleva
2000 Mathematics Subject Classification: 62H10.
Key words: exact distribution, sample covariance matrix, monotone missing data, Wishart
distribution.
The Bellman gamma distribution is a matrix variate distribution, which is a
generalization of the Wishart distribution. In practice it arises as a distribution
of the empirical normal covariance matrix for samples with monotone
missing data. The exact distributions of determinants and quotient of determinants
of some submatrices of Bellman gamma distributed random matrices
are obtained. The method, considered in this paper, gives the possibility
to derive the distribution of products and quotient of products of principal
minors of a Bellman gamma matrix, and in particular, of a Wishart matrix.
Characterizations of Exponential Distribution via Conditional Expectations
of Record Values
George P. Yanev
2000 Mathematics Subject Classification: 62G30, 62E10.
Key words: characterizations, exponential distribution, record values.
We prove that the exponential distribution is the only one which satisfies
a regression identity. This identity involves conditional expectation of the
sample mean of record values given two record values outside of the sample.
Weak Ergodicity of M_{t} /M_{t} /N /N + R Queue
Alexander Zeifman
Anna Korotysheva
2000 Mathematics Subject Classification: 60J27, 60K25.
Key words: nonstationary birth and death processes, rate of convergence, M_{t}/M_{t}/N/N + R
queue.
We consider nonstationary birth and death processes on finite state space
and study the bounds of the rate of convergence to the limit regime. We also
obtain some bounds on the rate of convergence for the queue-length process
of M_{t} /M_{t} /N /N + R queue.