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xxiv Actuarial Modelling of Claim Counts Journal which integrates bonus-malus scales in the framework of Markov chains, Professor Norberg introduces the present work. As always, it has been a pleasure to work with Wendy Hunter and Susan Barclay, Project Editors; Simon Lightfoot, Publishing Assistant; Kathryn Sharples, Commissioning Editor in Statistics and Mathematics; and Kelly Board and Sarah Kiddle, Content Editors, Engineering and Statistics at John Wiley & Sons, Ltd and Sunita Jayachandran at Integra Software Services Pvt. Ltd. Last but not least, we apologize to our families for the time not spent with them during the preparation of this book, and we are very grateful for their understanding. Michel Denuit Xavier Maréchal Sandra Pitrebois Jean-François Walhin Louvain-la-Neuve and Brussels, January 2007.
Notation Here are a few words on the notation and terminology used throughout the book. For the most part, the notation used in this book conforms to what is usual in mathematical statistics as well as nonlife insurance mathematics. The real line − + is denoted as . The half positive real line is + = 0 +. The set of the nonnegative integers is = 0 1 2. The real n-dimensional space is denoted as n , and n is the set of all the n-tuples of nonnegative integers. A point of n is an n-dimensional vector with real coordinates. It is represented by a bold letter x; the ith component of x is x i , i = 1 2n. All the vectors are tacitly assumed to be column vectors, that is, ⎛ ⎞ x 1 ⎜ x = ⎝ ⎟ ⎠ x n A superscript ‘T’ is used to indicate transposition. Hence, x T is a row vector with components x 1 x n . The dimension of x is denoted as dimx. The matrices are denoted by a capital letter in boldface, for instance M; M T denotes the transposition of M. The vector of ones, that is 1 11 T , will be denoted by e. The identity matrix (with entries 1 on the main diagonal and 0 elsewhere) is denoted by I. The determinant of the matrix M is denoted as detM, its inverse by M −1 . We denote as oh a function of h that tends to 0 faster than the identity, that is, such that oh lim h↘0 h = 0 Intuitively, oh is negligible when h becomes sufficiently small. The factorial of the positive integer n is denoted as n! and defined by n!=nn − 11. By convention, 0!=1. The binomial coefficient ( n k) denotes the number of different possible combinations of k items from n different items:
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Page 7 and 8: Contents Foreword Preface Notation
Page 9 and 10: Contents vii 2.4 Overdispersion 79
Page 11 and 12: Contents ix 4.2.3 Trajectory 170 4.
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3 Credibility Models for Claim Coun
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Credibility Models for Claim Counts
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Part III Advances in Experience Rat
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8 Transient Maximum Accuracy Criter
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Bibliography Albrecht, P. (1983a).
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Bibliography 347 Daengdej, J., Luko
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Bibliography 349 Greenwood, M., & Y
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Bibliography 351 Norberg, R. (1976)
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Bibliography 353 Walhin, J.-F., & P
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