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Keiding, B. Speed, P. Lele and J. Crowley, S. Pronzato, H. Wynn, and A. Basford and J. Gilks, S. Aitchison, J. Bailer and W. Philip J. Boland University College Dublin Ireland. This book contains information obtained from authentic and highly regarded sources.
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This book covers many of the diverse methods in applied probability and statistics for students aspiring to careers in insurance, actuarial science and finance.
It should also serve as a valuable text and reference for the insurance analyst who commonly uses probabilistic and statistical techniques in prac- tice. The reader will build on an existing basic knowledge of probability and statistics and establish a solid and thorough understanding of these methods, but it should be pointed out that the emphasis here is on the wide variety of practical situations in insurance and actuarial science where these techniques may be used.
In particular, applications to many areas of general insurance, including models for losses and collective risk, reserving and experience rat- ing, credibility estimation, and measures of security for risk are emphasized. The text also provides relevant and basic introductions to generalized linear models, decision-making and game theory. There are eight chapters on a variety of topics in the book.
Although there are obvious links between many of the chapters, some of them may be studied quite independently of the others. Chapter 1 stands on its own, but at the same time provides a good introduction to claims reserving via the deterministic chain ladder technique and related methods. Chapters 2, 3 and 4 are closely linked, studying loss distributions, risk models in a fixed period of time, and then a more stochastic approach studying surplus processes and the concept of ruin.
Chapter 5 provides a comprehensive introduction to the concept of credibility, where collateral and sample information are brought together to provide reasonable methods of estimation. The Bayesian approach to statistics plays a key role in the establishment of these methods. The final three chapters are quite independent of the previous chapters, but provide solid introductions to methods that any insurance analyst or actuary should know.
Experience rating via no claim discount schemes for motor insurance in Chapter 6 provides an interesting application of Markov chain methods. Chapter 7 introduces the powerful techniques of generalized linear models, while Chapter 8 includes a basic introduction to decision and game theory.
There are many worked examples and problems in each of the chapters, with a particular emphasis being placed on those of a more numerical and practical nature. Solutions to selected problems are given in an appendix. There are also appendices on probability distributions, Bayesian statistics and basic tools in probability and statistics.
Readers of the text are encouraged in checking examples and doing problems to make use of the very versatile and free statistical software package R. The material for this book has emerged from lecture notes prepared for various courses in actuarial statistics given at University College Dublin The National University of Ireland — Dublin over the past 15 years, both at the upper undergraduate and first year postgraduate level.
The Department of Statistics at Trinity College Dublin kindly provided me with accommodation during a sabbatical year used to prepare this material. I also wish to acknowledge encouragement from the Society of Actuaries in Ireland, which has been supportive of both this venture and our program in Actuarial Science at UCD since its inception in Patrick Grealy in particular provided very useful advice and examples on the topic of run-off triangles and reserving.
I have been fortunate to have had many excellent students in both statistics and actuarial science over the years, and I thank them for the assistance and inspiration they have given me both in general and in preparing this text.
Finally, I wish to thank my family and many friends who along the path to completing this book have been a constant source of support and encouragement. In spite of the stochastic nature of most of this book, the first chapter is rather deterministic in nature, and deals with Claims Reserving and Pricing with Run-off Triangles. Methods for dealing with past and future inflation in estimating reserves for future claims are considered.
The average cost per claim method is a popular tool which takes account of the numbers of claims as well as the amounts. The Bornhuetter—Ferguson method uses additional information such as expected loss ratios losses relative to premiums together with the chain ladder technique to estimate necessary reserves. Delay triangles of claims experience can also be useful in pricing new business.
Modeling the size of a claim or loss is of crucial importance for an insurer. In the chapter on Loss Distributions, we study many of the classic probabil- ity distributions used to model losses in insurance and finance, such as the exponential, gamma, Weibull, lognormal and Pareto.
Particular attention is paid to studying the right tail of the distribution, since it is important to not underestimate the size and frequency of large losses. Given a data set of claims, there is often a natural desire to fit a probability distribution with reasonably tractable mathematical properties to such a data set.
Exploratory data analysis can be very useful in searching for a good fit, including basic descriptive statistics such as the mean, median, mode, standard deviation, skewness, kurtosis and various quantiles and plots. Often one may find that a mixture of various distributions may be appropriate to model losses due to the varying characteristics of both the policies and policyholders. We also consider the impact of inflation, deductibles, excesses and reinsurance arrangements on the amount of a loss a company is liable for.
Following on from a study of probability distributions for losses and claims, the chapter on Risk Theory investigates various models for the risk consisting of the total or aggregate amount of claims S payable by a company over a relatively short and fixed period of time. Emphasis is placed on two types of models for the aggregate claims S. In the collective risk model for S, claims are aggregated as they are reported during the time period under consider- ation, while in the individual risk model there is a term for each individual.
Extensive statistical properties of these models are established including the useful recursion formula of Panjer for the exact distribution of S as well as methods of approximating the distribution of S. The models can inform an- alysts about decisions regarding expected profits, premium loadings, reserves necessary to ensure with high probability profitability, and the impact of reinsurance and deductibles.
The chapter on Ruin Theory follows the treatment of risk but the emphasis is put on monitoring the surplus stochastic process of a portfolio of policies throughout time.
The surplus process takes account of initial reserves, net premium income including, for example, reinsurance payments , and claim payments on a regular basis, and in particular focuses on the possibility of ruin a negative surplus. An emphasis is placed on understanding how one may modify aspects of the process, such as the claim rate, premium loadings, typical claim size and reinsurance arrangements, in order to adjust the security level.
Credibility Theory deals with developing a basis for reviewing and revis- ing premium rates in light of current claims experience data in hand and other possibly relevant information from other sources collateral informa- tion. In the Bayesian approach the collateral information is summarized by prior information and the credibility estimate is determined from the posterior distribution result- ing from incorporating sample current claims information.
If the posterior estimate is to be linear in the sample information, one uses the greatest accu- racy approach to credibility, while if one needs to use the sample information to estimate prior parameters then one uses the Empirical Bayes approach to credibility theory. The chapter on Credibility Theory presents in a unified manner these different approaches to estimating future claims and numbers!
They attempt to create homogeneous groups of policyholders whereby those drivers with bad claims experience pay higher premiums than those who have good records. The theory is that they also reduce the number of small claims, and lead to safer driving because of the penalties associated with making claims.
NCD schemes provide a very interesting application of dis- crete Markov chains, and convergence properties of the limiting distributions for the various discount states give interesting insights into the stability of premium income.
Constructing interpretable models for connecting or linking such responses to variables can often give one much added insight into the complexity of the relationship which may often be hidden in a huge amount of data. For example, in what way is the size of an employer liability claim related to the personal charac- teristics of the employee age, gender, salary and the working environment safety standards, hours of work, promotional prospects?
In Nelder and Wedderburn developed a theory of generalized linear models GLM which unified much of the existing theory of linear modeling, and broadened its scope to a wide class of distributions.
The chapter on Generalized Linear Models begins with a review of normal linear models. How generalized linear models extend the class of general linear models to a class of distributions known as exponential families and the important concept of a link function are discussed. Several examples are given treating estimation of parameters, the concept of deviance, residual analysis and goodness-of-fit.
All around us, and in all aspects of life, decisions continually need to be made. We are often the decision makers, working as individuals or as part of team. The decisions may be of a personal or business nature, and often enough they may be both! The action or strategy which a decision maker ultimately takes will of course depend on the criterion adopted, and in any given situation there may be several possible criteria to consider.
In the chapter on Decision and Game Theory, an introduction to the basic elements of zero-sum two- person games is given. Examples are also given of variable-sum games and the concept of a Nash equilibrium. In the treatment of decision theory we concentrate on the minimax and Bayes criteria for making decisions.
A brief introduction to utility theory gives one an insight into the importance of realizing the existence of value systems which are not strictly monetary in nature. Dedication v. Preface vii. Introduction ix. References Appendix A Basic Probability Distributions However, in other areas of general insurance, there can be considerable delay between the time of a claim-inducing event, and the determination of the actual amount the company will have to pay in settlement.
When an incident leading to a claim occurs, it may not be reported for some time. For example, in employer liabil- ity insurance, the exposure of an employee to a dangerous or toxic substance may not be discovered for a considerable amount of time.
In medical malprac- tice insurance, the impact of an erroneous surgical procedure or mistakenly prescribed drug may not be evident for months, or in some cases years. In other situations, a claim may be reported reasonably soon after an incident, but a considerable amount of time may pass before the actual extent of the damage is determined.
In the case of an accident the incident may be quickly reported, but it may be some time before it is determined actually who is liable and to what extent. In some situations, one might have to wait for the outcome of legal action before damages can be properly ascertained.
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By downloading a book for free from our website, you confirm that you will not use the materials of electronic versions of books for commercial purposes. Statistical and Probabilistic Methods in Actuarial Science covers many of the diverse methods in applied probability and statistics for students aspiring to careers in insurance, actuarial science, and finance. The book builds on students' existing knowledge of probability and statistics by establishing a solid and thorough understanding of these methods. It also emphasizes the wide variety of practical situations in insurance and actuarial science where these techniques may be used. Although some chapters are linked, several can be studied independently from the others.
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Statistical and Probabilistic Methods in Actuarial Science covers many of the diverse methods in applied probability and statistics for students aspiring to careers in insurance, actuarial science, and finance. It also emphasizes the wide variety of practical situations in insurance and actuarial science where these techniques may be used. Although some chapters are linked, several can be studied independently from the others. The first chapter introduces claims reserving via the deterministic chain ladder technique. The next few chapters survey loss distributions, risk models in a fixed period of time, and surplus processes, followed by an examination of credibility theory in which collateral and sample information are brought together to provide reasonable methods of estimation.
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Keiding, B. Speed, P. Lele and J.
Эти слова повергли Сьюзан в еще большее смятение. Шифровальный алгоритм - это просто набор математических формул для преобразования текста в шифр. Математики и программисты каждый день придумывают новые алгоритмы.
Она описала дугу и, когда он отпустил руку, с грохотом закрыла люк. Шифровалка снова превратилась в затихшую черную пещеру. Скорее всего Северная Дакота попал в ловушку. Стратмор опустился на колени и повернул тяжелый винтовой замок.
Никто никогда не называл Джаббу дураком, свиньей - быть может, но дураком -. - Свою женскую интуицию ты ставишь выше ученых степеней и опыта Джаббы в области антивирусного программирования. Она взглянула на него с холодным презрением. Бринкерхофф поднял руки в знак капитуляции. - Извини.
Однако, сделав еще несколько шагов, Стратмор почувствовалчто смотрит в глаза совершенно незнакомой ему женщины. Ее глаза были холодны как лед, а ее обычная мягкость исчезла без следа. Сьюзан стояла прямо и неподвижно, как статуя.
Чатрукьян знал: как только Джабба узнает, что Стратмор обошел фильтры, разразится скандал. Какая разница? - подумал. - Я должен выполнять свои обязанности.
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