Inverse Gaussian Autoregressive Models B. Abraham University of Waterloo Ontario, Canada N. Balakrishna Cochin University of Science and Technology Kochin, India ABSTRACT A first-order autoregressive process with inverse gaussian marginals is introduced. The innovation distributions are obtained under certain special cases. The unknown parameters are estimated using different methods and these estimators are shown to be consistent and asymptotically normal. The behavior of the estimators for small samples is studied through simulation experiments. On Sums of Triangular Numbers Chandrashekar Adiga and K. R. Vasuki Department of Mathematics University of Mysore Manasagangothri, Mysore 570 006, India ABSTRACT In this note we show how, W.N.Baily's 2(2 summation formula can be employed to obtain formulas for t2(n), t3(n), t4(n), t2*(n) and t3*(n) where tk(n) is the number of representations of n ( 1 as a sum of k triangular numbers and ts*(n) is the number of representations of n in the form [x(x+1)] / 2 + s[y(y+1)]/2 where x and y are nonnegative integers. Our formulas for t3(n), t2*(n) and t3*(n) seem to be new. A Connection Between Burge's Restricted Partition Pairs And Frobenius Partitions A. K. Agarwal Department of Mathematics Punjab University Chandigarh 160 014 Padmavathamma Department of Studies in Mathematics University of Mysore Manasagangothri, Mysore 570 006 ABSTRACT By giving combinatorial arguments we obtain generating functions for three restricted Frobenius partition functions. A connection between these restricted Frobenius partition functions and Burge's restricted partition pairs (J. Combin. Theory Ser A, 63, (1993), 210-222) is shown. This connection and Burge's Theorem 1 give us three new analytic identities. A comparison of these analytic identities with three known identities from Slater's compendium (Proc. London Math. Soc. (2) 54 (1952), 147-167) leads us to Rogers-Ramanujan type identities for Burge's restricted partition pairs. Progressive Censoring and Warranty Design Rita Aggarwala University of Calgary Calgary, AB, CANADA ABSTRACT The design of effective warranties is of concern to producers in any manufacturing industry, particularly in today's ever-changing high technology markets. In order to determine a cost-effective yet attractive warranty period, one must have an accurate description of a product's failure time distribution (where “failure” is not necessarily complete break down, depending upon the terms of the warranty). If n items are placed on life test, a practitioner will rarely wait for all n items to fail before making inference about the life time distribution of interest, rather he or she will censor the sample at some point in time before all of the items fail. In this talk, we will consider using “Progressive Censoring schemes” as a versatile extension to conventional censoring practices. In light of the number of ways to employ progressive censoring schemes, we will discuss selection of optimal censoring schemes and provide an example related to warranty design based on the extreme value distribution. A Genetic Algorithm for the Traveling Salesman Problem with Some Restrictions Z. H. Ahmed and M. Borah Department of Mathematical Sciences Tezpur University Sonitpur, ASSAM 784 001 S. N. N. Pandit Former Visiting Professor Department of Computer Science ABSTRACT The travelling salesman problem with some restrictions (TSP-R) is defined on complete symmetric digraph G = (V,E). A cost matrix which may not satisfy the triangle inequality is defined on the edge set E. The TSP-R consists of determining a least-cost Hamiltonian tour on G of n vertices starting at any given vertex, say v1 visiting each vertex in his tour exactly once and returning to the vertex v1 such that any particular vertex say vi must be visited before visiting another particular vertex, say vk. In this paper a genetic algorithm is developed for obtaining heuristically optimal solution for the TSP-R. Also, this paper develops a data guided lexi-search approach, which gives us an exact solution, and another heuristic approach called sequential random sampling approach. The efficiency of the genetic algorithm to the TSP-R as against data guided lexi-search approach and sequential random sampling approach has been examined. Casa VIRTUALE ( VIRTUALE Home ) Giorgio Albertini [email protected] Casa di Cura S. Raffaele Via della Pisana, 235 00100 ROMA Paola Baccetti Casa di Cura S. Raffaele - Via della Pisana, 235 00100 ROMA and TECNOByte STUDIO (Soc. Applicazioni Grafiche) Via Aladino Govoni 15 - 00136 Roma (Tel. 039 0635343265 'CASA VIRTUALE' (Virtual Home) is a multimedial prototype produced in collaboration with Prof. Giorgio ALBERTINI director of the 'CHILD, ADULT DEVELOPMENT CENTER' at SAN RAFFAELE HOSPITAL in ROME. The product, designed in a multidisciplinary contest (psychologists, pedagogists, logopedists, etc..), has been thought for children with medium or little mental retardation and/or experiencing learning difficulties at the primary school. Exercises, included in the program as games, are bound to consolidate some cognitive capacities as oculo-manual coordination, memory, attention, time and space concepts. Also icluded are metaphonological and reading and writing exercises. Besides of contents, anywise accurately designed, the emotional impact on the child has been privileged. Proposed activities are inserted in a familiar contest very close to the real life style. In the ambient of 'CASA VIRTUALE' the child has an active part in proceeding of 'virtual exploration' and builds his own navigation route. Very peculiar flexibility has been inserted, allowing the adult operator to select level and characteristics of any single exercise compared to the abilities of any child. He can decide whether to play a game and check the results of any activity proposed to the child. This section of the program is hidden to the child so that he is not aware of been controlled by the adult. Graphoidal Covering Number of Graphs With Ki = 1 and on Graphs With ( = (a S. Arumugam Dept. of Mathematics M. S. University Tirunelveli 627 012, India Indra Rajasingh Department of Mathematics Loyola College, Chennai 600 034 P. Roushini Leely Pushpam Department of Mathematics D.B.Jain College Chennai 600 096 ABSTRACT A graphoidal cover of a graph G is a collection (of (not necessarily open ) paths in G such that every vertex of G is an internal vertex of at most one path in ( and every edge of G is in exactly one path in (. If further no member of ( is a cycle, then ( is called an acyclic graphoidal cever of G. The minimum cardinality of a graphoidal cover is called a graphoidal covering number of G and is denoted by ( and the minimum cardinality of an acyclic graphoidal cover is called an acyclic graphoidal covering number and is denoted by (a. In this paper we characterize the class of graphs G with k1 = 1 for which (=q-p where p and q denote the order and size of G respectively. We also obtain a characterization of graphs with (= (a. We also determine the value of (a - ( for any graph G. Equality of Edge Domination Parameters in Graphs S. Arumugam Department of Mathematics Ms University Tirunelveli 627 012, India S. Velammal Department of Mathematics Mepco Shlenk Engineering Sivakasi, India ABSTRACT Let G = (V, E) be a graph. A subset D of E is called an edge dominating set of G if every edge not in D is adjacent to at least one edge in D. An edge dominating set D is called a connected edge dominating set if the edge induced subgraph < D > is connected. It is called a total edge dominating set if < D > has no isolated edges. It is called an independent edge dominating set if no two edges in D are adjacent. The minimum cardinality of an edge dominating set in G is called the edge domination number of G and is denoted by (. . Similarly we define the connected edge domination number (c., the total edge domination number (t. and the independent edge domination number (i. In this paper we investigate graphs in which some of the edge domination parameters are equal. Pseudo-Complete Color Critical Graphs S. Arumugam and J. Suresh Kumar Department of Math Ms University Tirunelveli 627 012, India ABSTRACT A pseudo-complete coloring of a graph G is an assignment of colors to the vertices of G such that for any two distinct colors, there exist adjacent vertices having those colors. The maximum number of colors used in a pseudo-complete coloring of G is called the pseudo-achromatic number of G and is denoted by (s(G). A graph G is called k –edge critical if (s(G) = k and (s(G – e) < k for every edge of G . A graph G is called k-vertex critical if (s(G) = k and (s(G - v) < k for every vertex v of G . In this paper we study the degrees, diameter and traversability concepts of these critical graphs. Fuzzy Techniques in Pattern Recognition S. Arumugam Department of Mathematics Ms University Tirunelveli 627 012, India R. M. Suresh Department of Computer Science Ms University Tirunelveli 627 012, India ABSTRACT Fuzzy sets were introduced in 1965 by Lotfi Zadeh as a new way to represent vagueness in everyday life. The recently propounded theory of fuzzy sets has attracted the attention of researchers in various disciplines since this theory is apparently a generalization of classical set theory . Since 1965, a great deal of work has been done on the development of this theory and on its applications. Computational Pattern Recognition has played a central role in the development of Fuzzy models because Fuzzy interpretations of date structures are a very natural and intuitively plausible way to formulate and solve various problems. In this paper, we attempt to discuss some of the Fuzzy methodologies that have suggested for Pattern Recognition. Asymptotic Theory for Random Permutations with Applications to Genetics G. Jogesh Babu Department of Statistics The Pennsylvania State University University Park, PA 16802-2111, USA ABSTRACT In the last few decades, mathematical population geneticists have been exploring the mechanisms that maintain diversity in a population. Some geneticists believe that much of genetic diversity occurs mainly due to mutation and random fluctuations that are inherent in the reproductive process. Ewens (1972) established an approximation to the sampling distribution of a sample of genes from a population that was evolved over several generations, by a family of measures on the set of partitions of an integer. The derivation ignores the selective effects and assumes that there is no meaningful way of labelling the alleles. In this case the allelic partition contains all the information available in a sample of genes. Ewens formula can be used to test if the popular assumptions are consistent with data, and to estimate the parameters. The statistics that are useful in this connection will generally be expressed as functions of the sums of transforms of allelic partition. Such statistics can be viewed as functions of a process on the permuation group of integers. A functional limit theorem for such a partial sum process will be presented using ideas and concepts from Probabilistic Number Theory. Ewens sampling formula also arises in Bayesian statistics via mixtures of Dirchlet processes. Semi-Parametric Approach to Bayesian Estimation of Reliability Under Ranked Sampling with Dirichlet Processes: Ravija Badarinathi University of North Carolina, Wilmington Wilmington, North Carolina, USA Ram C. Tiwari University of North Carolina, Charlotte Charlotte, North Carolina, USA ABSTRACT Modern nonparametric schools of thought are moving towards Bayesian Models using Dirichlet Processes. The standard hierarchical models are being used to incorporate the nonparametric ideas. In this work some of the simulation results are used to estimate the approximate priors, posteriors, and predictive distributions. Application of EM Algorithm in a Multistate Proportional Hazards Model S. Bae, K. P. Singh, A. A. Bartolucci Department of Biostatistics, School of Public Health University of Alabama at Birmingham Birmingham, AL 35294-0022, USA R. I. Chowdhury Department of Health Information Administration Kuwait University, KUWAIT M. A. Islam Department of Statistics Dhaka University, BANGLADESH ABSTRACT In longitudinal studies we observe the repeated observation of outcome and prognostic factors over time. The study of a subject over time may show changes from one outcome state to another and the histories of a group of individuals may include the partially censored data. The transition from one state to another can be categorized into three distinct types: Transition, Reverse transition, and Repeated transition. In this paper a method is exploited for estimating parameters of the model for reverse and repeated transition developed by Kay (1982) and further extended by Islam (1997.) For estimating parameters, the algorithm expectation-Maximization (EM) was utilized. We apply the model to longitudinal data on Diabetes mellitus (DM) collected at diabetes hospital in Bangladesh. Optimality of Partial Geometric Designs Bhaskar Bagchi and Sunanda Bagchi Stat-Math Unit Indian Statistical Institute Bangalore 560 059, India ABSTRACT In the present paper, we prove optimality properties of the partial geometric designs defined in Bose, R.C.; Shrikhande, S.S. and Singhi, N.M. (1973) [Edge Regular Multigraphs and Partial Geometric Designs, Proc. Intern. Colloq. Combin. Theory, Rome]. First we prove that whenever the nullity of the concurrence matrix of a partial geometric design d* is small enough, d* is better (with respect to all convex decreasing optimality criteria) than all unequally replicated designs (binary or not) with the same parameters b, k, v as d*. Combining this result with existing optimality results we obtain the following results. A linked block design with parameters b = q2 , v = q2 + q, k = q2 -1 is optimal with respect to all the criteria mentioned above in the unrestricted class of all connected designs with the same b, k, v. Many of the regular partial geometric designs (i.e., partial geometric designs which are also regular graph designs) are type I optimal among all the connected binary designs (with the given parameters). This class included many singular and semi-regular group divisible designs and complements of partial geometries. Many more regular partial geometric designs are A-optimal amongst all connected binary designs. This class includes the complements of many partial geometries and the linked block designs which are the dual of the complement of a BIBD satisfying v ( (k2 + 4) ( 2. Order Statistics from Non-identically Distributed Random Variables and Applications to Robustness N. Balakrishnan [email protected] McMaster University Hamilton, Ontario, CANADA ABSTRACT In this talk, I will introduce the distributions of order statistics arising from n independent and non- identically distributed random variables in terms of permanents of some matrices. Making use of these convenient expressions, I will establish some basic properties of order statistics in this general context. Next, I will turn my attention to some special cases including exponential, logistic and Pareto, and show how the calculation of single and product moments of order statistics can be done efficiently in a recursive process. These results, incidentally, generalize the corresponding well-known results for the i.i.d. case. I will then make use of these results in order to discuss the robustness properties of various linear estimators of the parameters of the distributions (mentioned earlier) when multiple outliers are present in the sample. Some numerical calculations and examples will be presented at the end. Three Isomorphic Vector Spaces and Their Application to Statistics K. Balasubramanian Stat/Math Unit ISI, Delhi Centre, NEW DELHI ABSTRACT LThree vector spaces are defined and they are shown to be isomorphic. The isomorphisms lead to very interesting applications to (i) identities among binomial coefficients with their group of automorphims, (ii) distributions arising from sampling with and without replacement, (iii) order statistics for arbitrary random variables, (iv) generalized binomial distributions. Generalizations of the above results for multinomial case are indicated. It is hoped that the techniques presented will find extensive applications in future. Assessing Therapeutic Equivalence (TE) in Clinical Trials Using the Beta Binomial Distribution Al Bartolucci, Karan Singh, and Delicia Carey Department of Biostatistics, School of Public Health University of Alabama at Birmingham Birmingham, Alabama 35294-0022, USA ABSTRACT Non rejection of the null hypothesis when comparing two response rates in a clinical trial does not necessarily imply that the two treatments are equivalent with respect to their therapeutic effectiveness. Recently researchers have invested much time and effort in describing situations in which TE may be achieved. These have involved direct hypothesis testing procedures and confidence interval techniques. The latter involves determining if such an interval lies within predefined equivalent regions. The authors (Al Bartolucci, Karan Singh) have published previously on this subject when the parameters of interest were from growth curve or survival distributions. In this research endeavor the focus is on the binomial parameters characterizing the response from clinical trials. The prior involves the natural conjugate beta family of distributions. The primary parameter of interest is the difference of the two binomial parameters. Limiting values of the hyperparameters of the conjugate family are used to demonstrate the robustness of the outcomes. Several equivalence regions are utilized to test whether or not equivalence has been achieved and under what conditions the attainment of equivalence may not be established. The procedure has wide application as well in the quality control setting. Quasivertex – Total Graphs With Crossing Numbers B. Basvanagoud Department of Mathematics Karnataka University Post Graduate Centre Belgaum – 590001, INDIA ABSTRACT The quasivertex – total graph Q ( G ) of a graph G as the graph whose point set is the union of the set of points and the set of lines of G in which two points are adjacent if and only if they correspond to two nonadjacent points of G or to two adjacent lines of G or to two adjacent points of G or to a point and a line incident to it in G. In this paper we establish a necessary and sufficient condition for quasivertex – total graph to have crossing number 1. We also prove that the Quasivertex – total graph Q (G) of a connected graph never has crossing number 2. In addition we deduce a necessary and sufficient condition for quasivertex – total graph to have crossing number 1 in terms of forbidden subgraphs . Further we investigate some hamiltonian properties of quasivertex – total graphs. On Concomitant of Order Statistics in Some Specific Bivariate Distributions M. I. Beg [email protected] University of Hyderabad Andhra Pradesh, INDIA S. Balasubramanian Indian Statistical Institute Delhi, INDIA ABSTRACT We find the distribution of the concomitant of the r-th order statistic of one of the components for some specific bivariate distributions e.g., bivariate exponential distribution of Marshall's and Olkin, Morgenstern type bivariate bivariate exponential distribution and Gumbel's bivariate type bivariate exponential distribution and Gumbel's bivariate exponential distribution. Properties that concomitants get from the corresponding order statistics are used to derive a number of results. Recurrence relations between moments of concomitants are also obtained. Finally, we consider the joint distribution of two concomitants and obtain their product moments. General Inference for Stochastic Processes B. R. Bhat University of Botswana ABSTRACT It is well-known that for an infinite sequence of random variables {Xn}, the almost sure (a.s.) limit of likelihood ratios (l.r.) of (X1 , X2 , ..., Xn ) is the l.r. of the sequence. Likelihood ratios of many (separable) stochastic processes are obtained by this method. But this method is applicable, if the limit is finite; conditions ensuring this case(Pq _P0 ) are widely discussed in the literature, (where, the underlying probability measures are Pq and P0 (say)). The measures will be mutually singular (perpendicular), denoted (Pq | P0), if the limit is zero or infinite, almost certainly on R¥ . This method fails, if the limit is zero or infinite on a set with probability strictly between 0 and 1. This set, called the set of singularity of Pq and P0 ,will have probability unity if (Pq |P0) . Otherwise, there will be non-trivial Lebesgue decomposition of Pq into absolutely continuous part and singular part (w.r.t. P0 ). The case of singular measures has been discussed by Grenander (Abstract Inference, (Wiley), 1981). This last important possibility has been ignored by workers on Inference for Stochastic Processes. In this paper, we highlight this non-ergodic case, pointing out its implications. We shall also give simple illustrative examples.