We consider subcritical Bellman-Harris branching processes with two types of immigration - one appears whenever the process hits zero state and another one is in accordance of an independent renewal process. The law of large numbers (LLN) for the total progeny of theese processes and Anscombe's type central limit theorem (CLT) for the total number of particles in the cycles completely finished by the moment t are obtained. 

References

1. Slavtchova-Bojkova, M. (2002). On the subcritical age-dependent branching processes with two types of immigration. Proc. 31st Spring Conf. of Bulgarian Mathematisians, Borovets , 187-191.

2. Weiner, H.(1991). Age-dependent branching processes with two types of immigration. J. Inform. Optim. Sci., 12, 207-218.

3. Nadarajah, S. (2002). Reliability for beta models. Serdica Mathematical Journal, 28, 1001-1016.

4. Nadarajah, S. (2003). Reliability for lifetime distributions. Mathematical and Computer Modelling, to appear.

KEY WORDS: central limit theorem, total progeny, Bellman-Harris branching processes, law of large numbers, renewal processes.

2000 AMS Subject Classification: 60J80, 60F05

We introduce confidence residuals and standartised confidence residuals. These residuals may be especially useful for asymmetric and multimodal distributions. 

A new definition for a d -fullness of a set of functions is proposed and its equivalence to the original one given by Vandev is proved. The breakdown point of the WTL(k) estimator of Vandev and Neykov for a grouped binary linear regression model with generalized logistic link is studied.

KEY WORDS: Breakdown Point, Subcompact Function, d-fullness, Robustness, Trimmed Likelihood Estimator, generalized logistic model. 

2000 AMS Subject Classification: 62J12, 62F35

This paper describes a statistics-based methodology for document unsupervised clustering and cluster topics extraction.

For this purpose, multiword lexical units (MWUs) of any length are automatically extracted from corpora using the LiPXtractor - a language independent statistics-based tool.

The MWUs are taken as base features to characterize documents. These features are transformed and a document similarity matrix is constructed. From this matrix, a reduced set of features is selected using an approach based on Principal Component Analisys. Then, using the Model Based Clustering Analisys software, it is possible to obtain the best number of clusters. Precision and Recall for document-cluster assignment range above 90 %.

Most important MWUs are extracted from each cluster and taken as document cluster topics.

Results on new document classification will just be mentioned.

KEY WORDS: cluster analysis, applied statistics, document clustering, text mining, topics extraction.

2000 AMS Subject Classification: 62H30

The present paper considers the problem how to construct the unknown density having N realizations of a random variable using B-splines approximation, least squares method and Monte Carlo method. It is shown that B-splines are appropriate for density modeling. The results from approximation of an unknown density distribution for the considered algorithm are compared with some non-parametric statistical methods such as histogram and kernel density estimation. A large number of experiments are made using Matlab 6.

References

[1]

I. Dimov, A. Karaivanova, Overconvergent Monte Carlo Methods for Density-function Modelling Using B-Splines, Advances in Numerical Methods and Appl., World Scientific, pp. 85-93, 1994.

[2]

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods, Jonh Wiley & Sons, inc., New York, London, Methuen, 1964.

[3]

A. Karaivanova, I. Dimov, Error Analysis of an Adaptive Monte Carlo Method for Numerical Integration, Mathematics and Computers in Simulation 47, Elsevier, pp.201-213, 1998.

[4]

I. Sobol, Monte Carlo Numerical Methods, Nauka, Moscow, 1973.

KEY WORDS: Monte Carlo algorithms

2000 AMS Subject Classification: 65C05

We consider a general single-type size-dependent branching process {N_n}_n such that the offspring mean converges to a limit m ³ 1 with a rate of convergence of order N_n^a, as the population size N_n grows to ¥, and the variance may increase at the rate N_n^b, where -1 £ b < 1. We assume that m(N)=m₁(N)+m₂(N), where m₁(N) depends on an unknown asymptotically identifiable parameter q₀ that belongs either to the limit model or to the transient model, and m₂(N) is a nuisance term that is assumed asymptotically negligible relatively to m₁ (N).

We estimate q₀ on the non-extinction set from the observations {N_h,¼,N_n}, by using the conditional least squares method weighted by {N_n-1^-g}_n. We study the strong consistency of the estimator according to g, with either h or n-h remaining constant as n®¥, by using either the minimum contrast method or a Taylor's approximation of the first derivative of the contrast. The main sufficient and probably also necessary condition for the strong consistency of the transient parameter is that b+2a £ 1. We also give the asymptotic distribution of the estimator by using Rahimov's central limit theorem for random sums and we show that the best rate of convergence is reached for g = 1+b. All the results are independent of the value taken by the nuisance term m₂ (N).

KEY WORDS: size-dependent branching process, controlled branching process. 

2000 AMS Subject Classification: 60J80, 62F12, 62P10

We review results obtained in [13] and [14] on a one-dimensional Burgers-type stochastic differential equation involving fractional power of the Laplacian in its linear part, perturbed by a whole noise term, with Dirichlet boundary conditions. We discuss existence of weak solutions and regularity of solutions.

KEY WORDS: Burgers equation, white noise, weak and strong solutions, Hilbert space regularity.

2000 AMS Subject Classification: 60H15, 60H40

Gait analysis is one of the methods used for estimation of the degree of restoration of motor recovery after stroke. The purpose of the present study was to examine the disgnostic value of the footprint parameters and their relationship with the functional ambulation profile (FAP) scores provided automatically by the pressure sensor walkway system for gait examination. The patterns of walking were studied in a group of 23 patients with chronic unilateral stroke and 72 healthy subjects. Their predictive value was compared with some other gait indicators for motor recovery after stroke.

KEY WORDS:  gait analysis, recovery after stroke.

2000 AMS Subject Classification: 62P10, 92C20

We consider the risk model in which the claim counting process {N(t)} is a modified renewal process. {N(t)} is governed by a sequence of independent and identically distributed inter-occurrence times with a common distribution function with mass at zero equal to r > 0. In the case of r = 0 this model is called the Sparre - Andersen model. The particular case of the Pólya - Aeppli risk model is studied. The Cramér - Lundberg approximation and the martingale approach of the model are given.

KEY WORDS: Pólya - Aeppli risk model, ruin probability, Cramér - Lundberg approximation.

2000 AMS Subject Classification: 60K10, 62P05

The spent life time and the residual life time are well investigated characteristics of an ordinary renewal process. In the present paper a generalization of these lifetime processes associated with an alternating renewal process is considered. Limiting distributions are presented in the case of infinite mean renewal periods.

References

1. W. Feller, An Introduction to Probability Theory and its Applications, vol.2, Wiley, New York, 1966.

2. K.B. Erickson, Strong renewal theorems with infinite mean, Trans. Amer. Math. Soc., 151, 1970, 263-291.

3. K.V. Mitov and N.M. Yanev, Limit theorems for alternating renewal processes in the infinite mean case, Adv. Appl. Prob. 33, 2001, 896-911.

4. K.V. Mitov and N.M. Yanev, Regenerative processes in the infinite mean cycle case, J. Appl. Prob. 38, 2001, 165-179.

KEY WORDS: alternating renewal processes, spent working time, spent waiting time, residual working time, limiting disributions, infinite mean renewal periods.

2000 AMS Subject Classification: 60K05

 This paper concerns with the bisexual branching process in varying enviroments introduced in [2]. For such a model a survey of results is provided. Previously, brief descriptions about the bisexual branching process and some bisexual models derived from it are given.

KEY WORDS:  discrete time branching processes, bisexual processes, branching processes in varying environments .

2000 AMS Subject Classification: 60J80

(1) D. J. Daley. Extinction conditions for certain bisexual Galton-Watson branching processes. Z. Wahrscheinlichkeitst., (1968), 9, 315-322.
(2) M. Molina, M. Mota and A. Ramos. Bisexual Galton-Watson branching process in varying environments. Stoch. Anal. Appl., (2003), 21, 1353-1367.

In the area of stress-strength models there has been a large amount of work as regards estimation of the probability R = Pr(X < Y) when X and Y are independent random variables belonging to the same univariate family of distributions. In this paper we propose an estimate of this quantity based on a simple property of the uniform distribution. We illustrate the use of the estimate with bootstrap confidence intervals for four commonly known distributions (normal, exponential, gamma and beta).

References

1. Downton, F. (1973). On the estimation of Pr(Y < X) in the normal case. Technometrics, 15 , 551-558.

2. Ihaka, R. and Gentleman, R. (1996). R: A language for data analysis and graphics. Journal of Computational and Graphical Statistics, 5, 299-314.

3. Nadarajah, S. (2002). Reliability for beta models. Serdica Mathematical Journal, 28, 1001-1016.

4. Nadarajah, S. (2003). Reliability for lifetime distributions. Mathematical and Computer Modelling, to appear.

KEY WORDS: beta distribution, bootstrap confidence intervals, exponential distribution, gamma distribution, normal distribution, stress-strength, uniform disribution.

2000 AMS Subject Classification: 33C90, 62E99

There is no universally accepted agreement in the literature defining the Markov property for random fields and various definitions are in use. Specifically, if F[( def) || =] {f(x) : x Î Rⁿ } is a random field and G Ì Rⁿ, define the sharp s-field of F as F(F, G)[( def) || =] s{f(x): x Î G} and the germ s-field of F as [`(F)](F,G) [( def) || =]Ç<sub>e > 0</sub>F(F,G<sub>e</sub>) , where G<sub>e</sub> is the uniform neighborhood { x:   dist(x, G) < e} of G. When G is a closed set separating R<sup>n</sup> into two complementary open sets D<sub>+</sub> and D<sub>-</sub>, the following definitions were introduced by Pitt in [1]: F is said to satisfy the germ field Markov property at G if the s-fields [`(F)](F, D₊) and [`(F)]( F,D<sub>-</sub>) are conditionally independent given [`(F )](F,G). If the random field F satisfies the more restrictive condition that [`(F)]( F,D<sub>+</sub>) and [`(F)]( F,D_-) conditionally independent given F (F,G), F is said to satisfy the sharp Markov property at G. In [2] on the other hand, Dalang and Walsh use slightly different definitions: F is said to satisfy the germ field Markov property at G if the s-fields F(F, D₊) and F( F,D_-) are conditionally independent given [`(F )]( F,G) and to satisfy the sharp Markov property at G when F( F,D₊) and F( F,D_-) are conditionally independent given F(F,G ).

Thus, to determine when the definitions in [1] and [2] coincide and to find conditions that imply the equivalence of the germ field and the sharp Markov properties, the following question needs consideration: If S Ì Rⁿ, what conditions on the set S imply that F(F, S) = [`(F)](F ,S)?

For Gaussian random fields F, we present a general answer to this question as a necessary and sufficient condition for spectral synthesis in the reproducing kernel Hilbert space associated with F . More detailed conditions are derived for Gaussian fields that arise as solutions of certain pseudo-differential equations.

References

[1] Pitt, L. D. (1971). A Markov Property for Gaussian Processes with Multidimensional Parameter. Arch. Rat. Mech. Anal. 43 367-391.

[2] Dalang, R.C., Walsh, J.B. (1992). The Sharp Markov Property of The Brownian Sheet and Related Processes. Acta Math. 168 153-218.

KEY WORDS: 60G15, 60G60

2000 AMS Subject Classification: Gaussian processes, Gaussian fields, germ fields, sharp Markov property, spectral syntesis.

A method for estimating a transition parameter in two-phase regression is described. The two phases are fitted and simultaneously the transition point is estimated. Practical application of the method is demonstrated on the data for determining soil hydraulic properties.

KEY WORDS: two-phase regression, transition point,air entry value.

2000 AMS Subject Classification: 62F10, 62J05, 62P30

This study is focused on the comparison and modification of different estimates arising in the branching processes. Simulations of models with or without migration are put through. Due to the complexity of the computations the algorithms are designed with the language of technical computing MATLAB. Using the simulations, estimates of the offspring mean of the generated processes are calculated. It is well known in the literature that under certain conditions the asymptotic distribution of the estimates is proved to be normal. Using the asymptotic normality a modified method of maximum likelihood is proposed. The aim is to obtain trimmed maximum likelihood estimates based on several sample paths with the same number of generations. Thus in a natural way the observations, inconsistent with the aprior information about the asymptotic normality are excluded from the model. The computation of the standard error allows the comparison of different types of estimates.

Electrical activity of the human brain can be recorded on the scalp. One of the advantages of the electrical recordings is the high temporal resolution by which e.g. cognitive processes can be followed from millisecond to millisecond. It is not uncommon to record simultaneously 128 electrode sites with high sampling rate and thus advanced mathematical and statistical methods are needed to sufficiently process the obtained data. Here an example of the analysis of data recorded during a complex visual processing task is presented. Using advanced methods large amounts of data can be reduced and new information of the function of the human brain can be invstigated.

KEY WORDS:  electroencephalography, source modelling, visual recignition.

2000 AMS Subject Classification: 62P10, 92C20

The estimation of statistical distance between populations arises in many multivariate analysis techniques. Whereas distance measures for continuous data are well developed, those for mixed discrete and continuous data are less so because of the lack of a standard model for such data. Such mixture of variables arise frequently in the field of medicine, biometry, psychology, econometrics and only comparatively few models have been developed for evaluating distance between populations. The aim of the presented paper is to apply methods for analysis of dissimilarity between 44 populations of 13 species of Ghenopodium genus, presented by 15 variables – 10 continuous and 5 categorical. The previously developed by another authors distance measures between populations presented by mixed attributes turned out not appropriate for the available data of Ghenopodium genus, . The matrices with distances between populations and species were used as input for Hierarchical Cluster Analysis to explore the taxonomic structure of the Ghenopodium genus.

KEY WORDS: distance between populations based on continuous and discrete variables, genus Ghenopodium, cluster analysis.

2000 AMS Subject Classification: 62P10, 62H30

Th program ldagui.m is an interactive tool for linear and quadratic discriminant analysis. The reason for developing such a tool consists in failing of conformity with conventional statistical programs in following aspects: treating of missing data; interaction with the user; testing the quality of obtained models.

KEY WORDS: stepwise discriminant analysis linear quadratic MATLAB

2000 AMS Subject Classification:

Component amounts in a mixture formulation are often set with errors in mass production. That causes variations in the product performance characteristic and the product quality of such products is low. A quality improvement problem arises, which is defined so that to minimize this variation while keeping the performance characteristic on a preliminary given target.

The component proportions in mixture models are ratios of component amounts. Therefore they are nonlinear functions of the errors in the component amounts. This makes the problem of error transmission in mixture experiments quite specific and the known models of error transmission cannot be directly applied.

In this paper we propose explicit models of mean and variance of a mixture performance characteristic for the case when the mixture amounts are set with errors in the production process. They are based on Taylor expansion up to third order terms of a mixture model expressed through the component amounts. Properties of these models are studied.

An example from rubber industry is given.