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The Limit Theorems for Transportation Networks
L.G. Afanasieva
A. Sergeev
2000 Mathematics Subject Classification: 60K25, 60F05, 37A50
Key words: polling model, ergodicity conditions, stationary distribution, large transportation
network, regenerative process, stochastic boundedness.
The questions of ergodicity and of existence of explicit formulas for the
stationary distribution are examined for various types of transportation net-
works which can be viewed as polling models. Also several limit theorems
are proved both for large symmetric and asymmetric networks.
Estimators in Branching Processes with Immigration
Dimitar Atanasov
Vessela Stoimenova
Nikolay Yanev
2000 Mathematics Subject Classification: 60J80
Key words: immigration mean, asymptotic normality, robust estimator
In the present paper we consider the branching process with immigration
and its relationship to the Bienayme - Galton - Watson process with a ran-
dom number of ancestors. Several estimators of the immigration component
are considered - the conditional least squares estimator of Heyde - Seneta,
the conditional weighted least squares estimator of Wei - Winnicki and the
estimator of Dion and Yanev. Their comparison is based on simulations of
the entire immigration family trees and computational results. The asymp-
totic normality of the estimator of Dion and Yanev is combined with the
general idea of the trimmed and weighted maximum likelihood. As a result,
robust modifications of the immigration component estimator is proposed.
They are based on one and several realizations of the entire family tree and
are studied via simulations and numerical results.
A Stochastic Control Approach to a Parabolic Equation, Reciprocal Processes
A. Benchettah abenchettah@hotmail.com
AMS 2000 Subject Classification: 49L60, 60J60, 93E20
Key words: Fokker-Planck equation, reciprocal process, entropy distance, stochastic optimal
control, Markov process, transition function.
A controllability problem for a Fokker-Planck equation is considered. A
solution (v*, fifi) to that problem is constructed by a theorem of Jamison,
under proper assumptions. We give a suficiency condition concerning the
initial and terminal data for that solution to exist. We show that v* is an
optimal feedback control for a stochastic optimal control problem. Further,
we prove that the corresponding optimally controled stochastic process is a
reciprocal process which is Markov.
Sensitivity Analysis of Some Applied Probability Models
Ekaterina V. Bulinskaya
2000 Mathematics Subject Classification: 90C31, 62C12, 62P05, 93C41
Key words: Asymptotically optimal policy; Incomplete information; Input-output model;
Insurance; Inventory; Risk measures; Sensitivity analysis; Stability
The aim of the paper is two-fold, namely, to give a brief survey of sensitivity
analysis methods and to use them for investigation of two input-output
models arising in applied probability.
Estimation of Fraction of Distinguished Elements in Population Based on Partially
Realized Random Sample
Wieslawa Dabala
Bronislaw Lednicki
2000 Mathematics Subject Classification: 62D05.
Key words: applications of mathematical statistics, applications of representative method
in public opinion polls.
This article presents an application of representative method in public opinion polls.
Controlled Multitype Branching Models: Geometric Growth
Miguel Gonzalez mvelasco@unex.es
Rodrigo Martinez rmartinez@unex.es
Manuel Mota mota@unex.es
2000 Mathematics Subject Classification: 60J80, 60F25.
Key words: Controlled branching processes. Multitype branching processes. Geometric
growth. L α-convergence.
In this work we deal with a multitype branching process that puts together
control in the number of reproductive units of each type and populationsize-
dependent reproduction. Moreover, unlike other branching models, it is
possible interaction between individuals at reproduction time. We investigate
suficient conditions for such a model to have asymptotically a geometric
growth, considering almost sure and Lα, 1 ≤ α ≤ 2, convergences. We
pay special attention to L2 convergence, taking advantage of the Hilbertian
properties of this space.
Nonparametric Versus Parametric Statistical Approaches for Genetic Anticipation: The
Pancreatic Cancer Case
Gleb R. Haynatzki
Vera R. Haynatzka
Randall E. Brand
Henry T. Lynch
Simon A. Sherman
2000 Mathematics Subject Classification: 62N01, 62N05, 62P10, 92D10, 92D30
Key words: genetics, genetic epidemiology, anticipation, pancreatic cancer, PCCR
Genetic anticipation for a particular disease can involve an earlier age of
onset, greater severity, and/or a higher number of afiected individuals in
successive generations within a family. Comparison between nonparametric
and semiparametric tests is studied for matched data, and is one of the main
focuses of this study. This comparison is investigated for the variable age
of diagnosis among difierent birth cohorts, before and after adjustment for
time under observation. The comparison is illustrated on an example of
familial pancreatic cancer, which example is the second main focus of this
study. The nonparametric test performed on our example better than the
two semiparametric tests, and was less sensitive to right censoring. After
adjusting for follow up time, all methods detected genetic anticipation.
A New Class of Processes for Formalizing and Generalizing Individual-Based Models: The
Semi-Semi-Markov Processes
C. Jacob christine.jacob@jouy.inra.fr
A. F. Viet af.viet@vet-nantes.fr
2000 Mathematics Subject Classification: 60K15, 60K20, 60G20,60J75, 60J80, 60J85, 60-08,
90B15.
Key words: Individual-Based Model; Multi-Agent Model; Random Graph; Complex System;
Branching Process; Semi-Markov Process; Markov Renewal Process.
Individual-based models are a \bottom-up" approach for calculating em-
pirical distributions at the level of the population from simulated individual
trajectories. We build a new class of stochastic processes for mathemati-
cally formalizing and generalizing these simulation models according to a
\top-down" approach, when the individual state changes occur at countable
random times. We allow individual population-dependent semi-Markovian
transitions in a non closed population such as a branching population. These
new processes are called Semi-Semi-Markov Processes (SSMP) and are gen-
eralizations of Semi-Markov processes. We calculate their kernel and their
probability law, and we build a simulation algorithm from the kernel.
(G, λ)-Extremal Processes and Their Relationship with Max-Stable Processes
Pavlina Kalcheva Jordanova
2000 Mathematics Subject Classification: 60G70, 60G18
Key words: G-extremal processes, max-stable processes, self-similar processes
The study of G-extremal processes was initiated by S. Resnick and M.
Rubinovich (1973). Here we transform these processes by a non-decreasing
and right-continuous function λ : [0, ∞) → [0, ∞) and investigate relationship
between (G; λ)-extremal processes and max-stable processes. We prove that
for the processes with independent max-increments if one of the following
three statements is given, the other two are equivalent:
a) Y is a max-stable process;
b) Y is a (G; λ)-extremal process;
c) Y is a self-similar extremal process.
The Number of Parts of Given Multiplicity in a Random Integer Partition
Emil Kamenov
2000 Mathematics Subject Classification: 05A16, 05A17
Key words: Random Integer Partition
Let Xm,n denote the number of parts of multiplicity m in a random partition
of the positive integer n. We study the asymptotic behaviour of the variance
of Xm,n as n → ∞ and fixed m.
Critical Exponents of Semilinear Equations Via the Feynman-Kac Formula
Ekaterina T. Kolkovska
Jose Alfredo L'opez-Mimbela jalfredo@cimat.mx
2000 Mathematics Subject Classification: 60H30, 35K55, 35K57, 35B35.
Key words: Semilinear partial difierential equations, Feynman-Kac representation, critical
exponent, finite time blow-up, non-global solution
On the Moving Boundary Hitting Probability for the Brownian Motion
Dobromir P. Kralchev
2000 Mathematics Subject Classification: 60J65
Key words: Brownian motion, hitting time, Laplace transformation
Consider the probability that the Brownian motion hits a moving two-sided
boundary by a certain moment. In some special cases we find formulae for
this probability.
Entropy Based Approach to Finding Interacting Genes Responsible for Complex Human Disease
Valentin Milanov vmilanov@uncfsu.edu
Radoslav Nickolov
2000 Mathematics Subject Classification: 62P10, 92D10, 92D30, 94A17, 62L10
Key words: entropy, SNP, genotype, genomewide, association, adaptive search
A challenging problem in human genetics is the identification and charac-
terization of susceptibility genes for complex human diseases such as car-
diovascular disease, cancer, hypertension and obesity. These conditions are
likely due to the efiects of high-order interactions among multiple genes and
environmental factors. Genome-wide association studies, where hundreds
of thousands of single-nucleotide polymorphisms (SNPs) are genotyped in
samples of cases and controls, ofier a powerful approach for mapping of com-
plex disease genes. The classical statistical methods, parametric and non-
parametric, are usually limited to small number of SNPs. Here we propose
a new method based on a classical search algorithm - "sequential forward
oating search", utilizing entropy based criterion function. Using simulated
case-control data we demonstrate that the method has a high discovery rate
under difierent models of gene-gene interaction, including pure interaction
without main efiects of the genes. The performance of the proposed method
is also compared to a method recently advocated in the literature: multifac-
tor dimensionality reduction (MDR).
Option Pricing by Branching Process
Georgi Mitov
Kosto Mitov
2000 Mathematics Subject Classification: 60J80, 62P05
Key words: Branching process; Galton-Watson process; Geometric distribution; Option
pricing; Stock-price process
The randomly indexed Galton-Watson branching process has been used for
the model of daily stock prices. Using this stock price process we derive a
new formula for the price of European call options.
Some Probabilistic Results in A Bisexual Branching Process with Immigration
M. Molina mmolina@unex.es
I. del Puerto idelpuerto@unex.es
A. Ramos aramos@unex.es
2000 Mathematics Subject Classification: 60J80
Key words: Branching processes, bisexual processes, immigration processes.
A bisexual branching process with immigration of females and males is
introduced. It is allowed, in each generation, that the mating function and
the probability distributions associated to the ofispring and the immigration
may change depending on the number of progenitor couples. Relationships
among the probability generating functions involved in the model and some
transition and stochastic monotony properties are established.
Combination of Global and Local Attributional Similarities for Synonym
Detection
Rumen Moraliyski
Gael Dias
2000 Mathematics Subject Classification: 68T50
Key words: Synonym discovery, similarity measure, discourse
In this paper, we present a new methodology for synonym detection based
on the combination of global and local distributional similarities of pairs of
words. The methodology is evaluated on the noun space of the 50 multiple-
choice synonym questions taken from the ESL and reaches 91.30% accuracy
using a conditional probabilistic model associated with the cosine similarity
measure.
Stability of the Inventory-Backorder Process in the (R; S) Inventory/Production Model
Zahir Mouhoubi z_mouhoubi@yahoo.fr
Djamil Aissani
2000 Mathematics Subject Classification: 60G52, 90B30
Key words: Uniform ergodicity, Strong Stability, Perturbation, backorder process, (R; S)
inventory/production model.
The aim of this paper is to obtain the suficient conditions for the uniform
ergodicity and the strong stability of the inventory-backorder process in a
single-item, single location, (R; S) inventory/production model with limited
capacity of production per period and uncertain demands. In this order
some intermediate results are established and an overview about the main
stability methods for stochastic processes and the performance measure in
the inventory models are also considered.
Extraction of Fraud Schemes from Trade Series
Charalambos Moussas charalambos.moussas@jrc.it
Veska Noncheva wesnon@pu.acad.bg
2000 Mathematics Subject Classification: 62H30, 62M10, 62M20, 62P20, 94A13
Key words: Fraud Detection, Time Series Analysis, Forecasting, Cluster Analysis
It is very often the case that the patterns of a fraudulent activity in trade
are hidden within existing trade data time series. Furthermore, with the ad-
vent of powerful and afiordable computing hardware, relatively big amounts
of available trade data can be quickly analyzed with a view to assisting anti-
fraud investigations in this field. In this paper, based on the availability of
such import/export data series, we present a statistical method for the iden-
tification of potential fraud schemes, by extracting and highlighting those
cases which lend themselves to further investigation by anti-fraud domain
experts. The proposed method consists in applying time series analysis for
prediction purposes, calculating the resulting significant deviations, and fi-
nally clustering time series with similar patterns together, thus identifying
suspect or abnormal cases.
A Test of Association Between Qualitative Trait and a Set of SNPs
Radoslav Nickolov
Valentin Milanov
2000 Mathematics Subject Classification: 62P10, 92D10, 92D30, 62F03
Key words: Case-control study; Genotypes; Gibbs distribution; Likelihood ratio test.
In this article, we propose a novel candidate-gene association test that
utilizes a set of tightly linked single nucleotide polymorphisms (SNPs). This
is a powerful likelihood ratio test based on Gibbs random field model. We
use simulation studies to evaluate the type I error rate of our proposed test,
and compare its power with that of other candidate-gene association tests.
The simulation results show that our proposed test has correct type I error
rate, and is more powerful than the other tests in most cases considered in
our simulation studies.
Upper and Lower Bounds for Ruin Probability
E. Pancheva
Z. Volkovich
L. Morozensky
Key words: compound extremal processes; α-stable approximation; ruin probability.
In this note we discuss upper and lower bound for the ruin probability in
an insurance model with very heavy-tailed claims and interarrival times.
Application of Regularized Discriminant Analysis
Ute Roemisch
Henry Jager
Dimitar Vandev
2000 Mathematics Subject Classification: 62H30, 62P99
Key words: Discrimination of wines; Regularization; Classfication.
The method of regularized discriminant analysis (RDA) was used for iden-
tifying the geographical origin of wines on the base of chemical-analytical
parameters in the scope of a European project \WINE DB"1. A data
base with 63 measured parameters of 250 authentic wine samples from
five countries of the vintage 2003 was taken as a basis for classifying and
discriminating wines. Uni- and multivariate methods of data analysis were
applied. By using a Matlab-program, which allows an interactive stepwise
discriminant model building, some difierent models for authentic wines with
corresponding classification and prediction error rates (resubstitution, clas-
sical and modified \Leave-one-out", simulation and test) will be presented.
The goodness of our preferred model was analysed by classifying a test
sample that was created by splitting the data set based on Duplex-algorithm
of Snee.
The method of regularized discriminant analysis (RDA) was used for
identifying the geographical origin of wines on the base of chemical-analytical
parameters in the scope of a European project \WINE DB". A data base
with 63 measured parameters of 250 authentic wine samples from five coun-
tries of the vintage 2003 was taken as a basis for classifying and discrimina-
ting wines. Uni- and multivariate methods of data analysis were applied. By
using a Matlab-program, which allows an interactive stepwise discriminant
model building, some difierent models for authentic wines with corresponding
classification and prediction error rates (resubstitution, classical and modified
\Leave-one-out", simulation and test) will be presented. The goodness of our
preferred model was analysed by classifying a test sample that was created
by splitting the data set based on Duplex-algorithm of Snee.
Using Covariance as a Similarity Measure for Document Language Identification in Hard
Contexts
Joaquim Ferreira da Silva fjfs@di.fct.unl.pt
Gabriel Pereira Lopes gplg@di.fct.unl.pt
2000 Mathematics Subject Classification: C2P99
Key words: Statistical Applications
Existing Language Identification (LID) approaches achieve 100% precision
in most common situations, dealing with suficiently large documents, writ-
ten in just one language. However, there are many situations where text
language is hard to identify and where current LID approaches do not pro-
vide a reliable solution. One such situation occurs when it is necessary to
discriminate the correct variant of the language used in a text. In this pa-
per, we present a fully statistics-based LID approach which is shown to be
correct for common texts and maintains its robustness when classifying hard
LID documents. For that, character sequences were used as base features.
The Discriminant Ability of each sequence, in each training situation, is
measured and used to filter out less important character sequences. Docu-
ment similarity measure, based on the covariance concept, was defined. In
the training phase, document clusters are built in a reduced k uncorrelated
dimensions space. In the classification phase the Quadratic Discriminant
Score decides which cluster (language) must be assigned to the documents
one needs to classify.
Limit Theorems for Maxima of Heavy{Tailed Terms with Random Dependent Weights
Stilian Stoev sstoev@umich.edu
Murad S. Taqqu
2000 Mathematics Subject Classification: 60F17, 60G52, 60G70, 60E07,
62E20.
Key words: weighted maxima, random weights, limit theorems, extremal Frfiechet process
Joint Densities of Correlation Coefficients for Samples from Multivariate Standard
Normal Distribution
Evelina Veleva eveleva@abv.bg
2000 Mathematics Subject Classification: 62H10
Key words: Multivariate normal distribution, sample correlation coeficients, independence,
conditional independence.
We consider the joint distribution of the correlation coeficients for samples
from multivariate standard normal distribution. Some marginal densities
are obtained. Independence and conditional independence between sets of
sample correlation coeficients are established.
Branching Populations of Cells Bearing aContinuous Label
A. Y. Yakovlev Andrei Yakovlev@urmc.rochester.edu
N. M. Yanev yanev@math.bas.bg
2000 Mathematics Subject Classification: 60J80
Key words: branching processes, continuous label, cell proliferation, label distribution
This paper is concerned with an age-dependent branching process with
particles (cells) bearing a label, the latter being treated as a continuous
parameter. The proposed stochastic model is motivated by applications in
cell biology. It is assumed that the mitotic division results in a random
distribution of the label among daughter cells in accordance with some
bivariate probability distribution. In the event of cell death the label borne
by that cell disappears. The main focus is on the label distribution as
a function of the time elapsed from the moment of label administration.
Explicit expressions for this distribution are derived in some particular cases
which are of practical interest in the analysis of cell cycle. The Markov
branching process with the same evolution of a continuously distributed
label is considered as well.
Revisiting Offspring Maxima in Branching
Processes
George P. Yanev gyanev@cas.usf.edu
2000 Mathematics Subject Classification: 60J80; 60G70
Key words: branching processes, varying environments, bisexual processes, geometric arrays,
maximum family sizes
We present a progress report for studies on maxima related to ofispring in
branching processes. We summarize and discuss the findings on the subject
that appeared in the last ten years. Some of the results are refined and
illustrated with new examples.
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