Pliska Studia Mathematica Bulgarica

Volume 18, 2007



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

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
Rodrigo Martinez
Manuel Mota

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
A. F. Viet

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

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
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
I. del Puerto
A. Ramos

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
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
Veska Noncheva

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
Gabriel Pereira Lopes

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
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

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
N. M. Yanev

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

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.