bluuebunny/crossref_metadata_2025 huggingface dataset:
Fuzzy Set Theory
Authors: A. Derrig Richard , M. Ostaszewski Krzysztof
Published: September 2014

Traditional actuarial methodologies have been built upon probabilistic models. They are often driven by stringent regulation of the insurance business. Deregulation and global competition of the last two decades have opened the door for new methodologies, among them, being fuzzy methods. The defining property of fuzzy sets is that their elements possibly have partial membership (between 0 and 1), with degree of membership of zero indicating not belonging to the set considered, and degree of membership of one indicating certainty of membership.

This article provides a quick overview of developing fuzzy sets methodologies in actuarial science, and the basic mathematics of fuzzy sets.
Random Variable, Overview
Authors: Asmussen Søren
Published: September 2014

In intuitive terms, a random variable (r.v.)
X
is a number, whose value is determined by random factors (chance). For example, the exchange rate Euro/USD next New Year or the number of car accidents in a certain month and county.

In the car accident example, the month may be in the past or in the future. In the first case,
X
is then completely known, say
X
 = 738, and rather than as a r.v., one speaks of 738 as the
outcome
of the r.v.
X
. The precise interpretation of such issues in terms of the axioms for probability theory is discussed.

A r.v. is usually assumed to be real‐valued,
X
 ∈ Υ, like in the above examples. However, in principle also, other sets of values are possible. The most prominent example is multivariate r.v.s,
X
 = (X
1
,…, X
d
) ∈ Υ
d
(e.g. the set of exchange rates in January,
d
 = 31), which are discussed in detail in this article.

Random variables are one of the most fundamental concepts of probability theory and treated in any elementary or intermediate textbook.
Under‐ and Overdispersion
Authors: Xekalaki Evdokia
Published: September 2014

Very often, in connection with applications, one is faced with data that exhibit a variability, which differs from that anticipated under the hypothesized model. This phenomenon is termedover dispersionif the observed variability exceeds the expected variability orunder dispersionif it is lower than expected.

Such discrepancies between the empirical variance and the nominal variance can be interpreted to be induced by a failure of some of the basic assumptions of the model. These failures can be classified by how they arise, that is, by the mechanism leading to them. In traditional experimental contexts, for example, they are caused by deviations from the hypothesized structure of the population, such as lack of independence between individual item responses, contagion, clustering, and heterogeneity. In observational study contexts, they are the result of the method of ascertainment, which can lead to partial distortion of the observations. In both cases, the observed valuexdoes not represent an observation on the random variableX, but an observation on a random variableYwhose probability distribution (the observed probability distribution) is a distorted version of the probability distribution ofX(original distribution). This can have a variance greater than that anticipated under the original distribution (over dispersion) or, much less frequently in practice, lower than expected (under dispersion).
Mixture of Distributions
Authors: Panjer Harry H.
Published: September 2014

This article gives an overview of some probability distributions that are obtained as mixtures of known distributions.
Thinned Distributions
Authors: Willmot Gordon E.
Published: September 2014

A discussion of closure under thinning is given for various discrete parametric distributions. This issue has important practical applications for loss payment analysis in the presence of a deductible.
Zero‐Modified Frequency Distributions
Authors: Panjer Harry H.
Published: September 2014

This article examines zero modification of basic discrete distributions used as counting distributions describing the number of events such as insurance losses.
Ammeter, Hans (1912‐1986)
Authors: Kupper Josef
Published: September 2014

The article describes the life and the important scientific research of the Swiss actuary Hans Ammeter.
Beard, Robert Eric (1911‐1983)
Authors: Teugels Jozef L.
Published: September 2014

Apart from being one of the authors of the first book on Risk Theory, Beard contributed to other aspects of actuarial science as well.
Graunt, John (1620‐1674)
Authors: Teugels Jozef L.
Published: September 2014

John Graunt can be considered the first scholar who applied statistical thinking to demographic questions. After some information on his life, we deal with his major contribution and its influence on further developments in the field.
History of Actuarial Education
Authors: Forfar David O.
Published: September 2014

This article looks at the history of actuarial education where, from the start, there was a difference in approach between the Anglo‐Saxon world and Continental Europe. In the Anglo‐Saxon world, entry to the actuarial profession depended on passing the examinations set by the profession and it was not necessary to have gone to a university (although exemptions could be granted in certain countries from some of the earlier examinations if sufficient relevant academic training had been undertaken). In Continental Europe, entry to the profession depended, in the main, on having successfully completed the relevant courses at university (or technische hochschule) or having gained sufficient practical ‘on‐the‐job’ experience.
History of Insurance
Authors: Sibbett Trevor
Published: September 2014

From early times, communities have looked for ways to pool resources in order to protect individuals from unexpected losses. One of the most common and long lasting arrangements, originating in the Mediterranean area and dating from before the Christian era, was a loan, with a high rate of interest, secured on the voyage of a ship, the loan being repayable only if the voyage were completed successfully. In English, this became known as bottomry, referring to the bottom of the ship.

There is evidence that the Romans practiced an early form of life insurance and bought as well as sold annuities.

In the year 1234, Pope Gregory IX banned the taking of interest and artificial contracts were developed to supplement bottomry. Following Pope Gregory's ban on bottomry, marine insurance in its basically current form developed in Florence and then in Genoa.

This article looks at insurance where a premium is paid to a third party to cover risk. Arrangements that have an element of insurance, some of which date from Babylonian times, are well covered by Trenerry and others.
Lidstone, George James (1870‐1952)
Authors: Teugels Jozef L.
Published: September 2014

The contributions of Lidstone to life insurance are numerous. Even on the purely mathematical scene, he has left his mark.
Screening Methods
Authors: Malutov Michail , Wynn Henry P. , Zhigljavsky Anatoly
Published: September 2014

Screening is a term applied to the general approach of detecting active or significant factors, elements, units, and so on when there are a large number of inactive factors. The meaning of ‘active’ depends on the context: effective treatments, presence of a disease, fault, etc. The methods are of considerable importance in medical and industrial applications. The theoretical aspects, particularly finding optimal or efficient experimental designs are closely related to search methods and special combinatorial constructions.
Parameter and Model Uncertainty
Authors: Cairns Andrew J.G.
Published: September 2014

Almost every actuarial problem involves one or more sources of risk. The task of an actuary is to assess the following risks:

To derive a distribution for the outcome at some fixed future time or at a series of future dates.

To make decisions or observations based on this risk assessment (e.g. to provide a risk measure, to set a premium rate, to calculate the reserve, to decide on an investment strategy).
Robustness in the Linear Regression M+N105odel
Authors: Hubert Mia , Rousseeuw Peter J. , Aelst Stefan
Published: September 2014

When applying a statistical method in practice, it often occurs that some observations deviate from the usual assumptions. However, many classical methods are sensitive to outliers. The goal of robust statistics is to develop methods that arerobust against the possibility that one or several unannounced outliers may occur anywhere in the data.
Reliability Classifications
Authors: Cai Jun
Published: September 2014

In this article, we consider reliability classifications for life distributions and discuss closure properties of classes of life distributions under important risk models such as convolution, mixing, and compounding. Applications of the reliability classes in insurance are discussed.
Graduation in Insurance
Authors: B. Miller Robert
Published: September 2014

In actuarial science, graduation refers to the adjustment of a set of estimated quantities to make the adjusted quantities conform to a pattern suitable for insurance purposes. We distinguish between two broad categories of approaches to actuarial graduation: those that involve fitting curves defined by functions and those that involve other methods. Curve‐fitting approaches use popular functional forms such as Gompertz, Weibull, Makeham, and splines. Non–curve‐fitting approaches in this article are grouped under the headings Weighted Moving Average Methods, Kernel Methods, Whittaker Methods, and Bayesian Methods.
Reliability Analysis
Authors: Natvig Bent
Published: September 2014

In reliability analysis, one is interested in studying the reliability of a technological system being small or large. As examples of small systems, we have washing machines and cars, whereas tankers, oil rigs, and nuclear power plants are examples of large systems. By the reliability of a system, we will mean the probability that it functions as intended. It might be tacitly assumed that we consider the system only for a specified period of time (for instance, one year) and under specified conditions (for instance, disregarding war actions and sabotage). Furthermore, one has to make clear what is meant by functioning. For oil rigs and nuclear power plants, it might be to avoid serious accidents such as a blowout or a core‐melt.

Reliability analysis is a very helpful tool in risk assessment when determining the insurance premiums for risks, especially of rare events, associated with large systems consisting of both technological and human components. Furthermore, reliability analysis is relevant in risk management of any technological system, the aim now being to say something helpful on how to avoid accidents. This is an area of growing importance representing an enormous challenge for an insurance company.
Extremes
Authors: Beirlant Jan
Published: September 2014

The basic methods for extreme tail estimation are reviewed. More specifically, we discuss the estimation of extreme quantiles and small tail probabilities. In this, the estimation of the extreme value index is an important intermediate step. An actuarial case study is provided.
Central Limit Theorem
Authors: Heyde C.C.
Published: September 2014

TheCentral Limit Theorem(CLT) is generally regarded as a generic name applied to any theorem giving convergence in distribution, especially to the normal law, of normed sums of an increasing number of random variables. Results of this kind hold under far reaching circumstances, and they have, in particular, given the normal distribution a central place in probability theory and statistics.
Change of Measure
Authors: Rolski Tomasz
Published: September 2014

Change of measure is a technique used for deriving saddlepoint approximations, bounds or optimal methods of simulation. It is used in risk theory.
Extreme Value Theory
Authors: Albin J.M.P.
Published: September 2014

In insurance one often has to deal with extremal measurements like large claims. Comparable to central limit theory for sums of random variables, extreme value theory provides asymptotic and approximating results for these largest data values. The case of independent claims can be dealt with using three traditional extremal types. These types also appear in the stationary case when the dependence is not too strong.
Stochastic Orderings: Theory
Authors: Denuit Michel , Müller Alfred
Published: September 2014

We describe the basic ideas of stochastic orders, which formally describe intuitive ideas like ‘being larger’, ‘being more variable’ or ‘being more dependent’ for risks or for portfolio of risks. We give the definitions of the most important stochastic orders like usual stochastic order, convex order, supermodular order and others, and we consider their most important properties and some applications.
Generalized Linear Models: Applications in Insurance
Authors: Kaas Rob
Published: September 2014

A wide class of claims reserving methods can be described as special cases of Generalized Linear Models. This class includes the Chain‐Ladder method, as well as the arithmetic and geometric separation method.
Regression Models for Data Analysis
Authors: W. Frees Edward
Published: September 2014

Regression is an applied statistical model that is widely used in the social and natural sciences. This article reviews different sets of the model assumptions, the model specification process, and what one hopes to learn from the modeling process.
Risk Management: An Interdisciplinary Framework
Authors: Tapiero Charles S.
Published: September 2014

Risk management is shown to be based on both theory and practice. It is shown to be conceptual and technical, blending behavioral psychology, financial economics, and decision‐making under uncertainty into a coherent whole that justifies the selection of risky choices. Its applications are broadly distributed across many areas and fields of interest. The examples treated here have focused on both finance and insurance and on a few problems in industrial management.
Finance
Authors: Brockett Patrick
Published: September 2014

Finance and insurance are very similar in nature: they both (now) consider the value of a financial instrument or an insurance policy or a firm to be the discounted present value of the future cash flows they provide to their holder. Research in the fields of finance and insurance have been converging, with products and valuation techniques from one field applying to and being used in the other field. Risk pooling, risk sharing, and risk transfer are the essence of insurance and the topic of study in modern finance, with both fields interested in valuing contingent claims, either claims due to financial or liability obligations, market changes, foreign exchange rate, or due to interest rate changes or payments due, contingent upon the occurrence of a particular type of insured loss (e.g. fire insurance payments, life insurance payments, automobile insurance payments, etc.). Thus, finance and insurance are more and more linked, and products in each area are less and less distinct.
Operations Research: Programming Issues
Authors: Bäuerle Nicole
Published: September 2014

In this section, we present the most important optimization problems arising in deterministic operations research and give a concise survey of common solution techniques. More precisely, we consider linear programming, combinatorial optimization, integer programming, nonlinear programming, and dynamic programming. Typical problems in these areas are also mentioned.
Kalman Filter: Overview
Authors: Barnett Glen , Zehnwirth Ben
Published: September 2014

The Kalman Filter is an efficient method of estimation for a state space model. Best linear unbiased estimates of the mean and variance of the unknown state are updated recursively as new data is added. It is used in many areas. In actuarial work it has been mainly applied to credibility and reserving applications.
Mortality Laws
Authors: Forfar David O.
Published: September 2014

Since the time of De Moivre in 1725, various mathematical formulae have been suggested to represent the pattern of mortality rates with age. Some of the formulae have applied at adult ages, some at very young ages, and some over the whole span of life. Certain formulae have been used in the graduation of mortality data. The article gives a survey of the mathematical laws used and examines how mortality might look in the foreseeable future if it goes on improving.
Early Mortality Tables
Authors: Forfar David O.
Published: September 2014

The article begins by looking at what Ulpian's factors tell us about mortality in ancient Rome. The article then looks at the main mortality tables of the seventeenth, eighteenth, and nineteenth centuries.
Accident Insurance
Authors: Jean Jaime
Published: September 2014

This article examines the different types of accident insurance found in the global insurance marketplace, placing emphasis on the benefits provided and presenting a general pricing model.
Aquaculture Insurance
Authors: Theodorou John A. , Tzovenis Ioannis P.
Published: September 2014

Modern marine aquaculture emerged a few decades ago, triggered by the wild catch decline. Global production more than doubled between 1987 and 1997 and the prospects for future growth especially in Europe are very promising.

During the last thirty years, the sector was a high‐risk activity as production technology was – and still is – developing with many experimental farms and a semiacademic character of most hatcheries. Reasonably, nonmanageable risks could be transferred to third parties by means of insurance, and as insurance coverage was important for the banks to finance the new sector, aquaculture drew the attention of insurance underwriters to focus on the venture's technology, history, and experience, financial strength, presentation, and technical qualities. However, aquaculture insurance treads on thin ice as there are few independent consultants and fewer loss adjusters with appropriate experience; protocols are variable, often propriety rendering it difficult to ascertain the claims, or discern mismanagement. Therefore, definition of risks and appropriate counteractions need to be established.
Commercial Multi‐Peril Insurance
Authors: Lynch James
Published: September 2014

Commercial multiple peril insurance is the most popular commercial package policy in US property and casualty insurance. It typically combines several property and general liability coverages under a single policy, creating a convenience in coverage for the insured. It often is sold at a discount, mainly reflecting better experience among risks that buy the package. It is rated by applying factors known as package modifications at the end of the rating process. This permits each coverage in the package to be analyzed in combination with results from the monoline coverage.
Employment Practices Liability Insurance
Authors: Barto Morris
Published: September 2014

Employment Practices Liability Insurance (EPLI) is gaining in popularity both within and outside the US. This policy responds to claims by employees against employers for wrongful employment practices. Such claims include allegations of wrongful refusal to hire, wrongful termination, sexual harassment, and discrimination based on age, race, ethnicity, or disability. Such claims also include claims for retaliation against workers for participating in legally protected activity. EPLI does not cover bodily injury or property damage, but responds for employees' economic losses as well as mental anguish. The coverage is almost always written on a claims‐made basis with expenses eroding limits. EPLI business tends to be highly segmented depending on the limits of coverage, the size of the self‐insured retention or deductible, the size and sophistication of the policyholders, and whether the policyholders are public or private entities.
Financial Insurance
Authors: Polanskyj Peter
Published: September 2014

Putting aside any distinction between the concepts of financial insurance and financial guaranty insurance, one of the most fundamental definitions available for financial guaranty insurance is an insurance contract that guarantees a cash (or cash equivalent) payment from a financial obligation, or a stream of such payments, at specified points of time.
Fire Insurance
Authors: Cassidy Paul
Published: September 2014

Fire policies cover owners of commercial or industrial property against material damage caused by fire and extraneous perils. Fire policies cover a wide range of insured properties from small rural sheds to billion dollar high‐rise buildings. Losses against these policies are dominated by large individual fire losses and other large losses caused by natural peril and manmade events that generate a large number of simultaneous claims.
Marine Insurance
Authors: Seltmann Astrid
Published: September 2014

Shipping is a business with long‐standing traditions as well as great risks at stake. But although marine insurance is the oldest form of insurance, it seems historically to have been relatively little touched by actuaries. This article will give an overview of marine insurance and its characteristics, the markets, and actuarial aspects. It will also suggest some reasons for the historically low actuarial involvement as well as point out the increasingly interesting potential, not the least due to recent developments in shipping and the general trend towards more sophisticated technical risk analysis and loss prevention measures in the industry in general.
P&I Clubs
Authors: Fry Martin
Published: September 2014

P&I (short for Protection and Indemnity) clubs provide a mutual not‐for‐profit indemnity pool (
see
Pooling in Insurance, Pooling of Employee Benefits) for owners or operators of ships. P&I insurance primarily covers owners or operators of ships for liability to third parties, although the clubs have responded to members' needs and to legislative changes with a range of other policies. The clubs operate by ‘calls’ on members. These calls are an allocation of anticipated claims costs, with adjustments made either by further calls or by refunds.
Property Insurance‐Personal
Authors: A. Ward Kimberley
Published: September 2014

Personal Property insurance in the United States is written on a number of insurance forms. By far, the most common ones are home owners package forms and private automobile physical damage coverage parts. Other forms include farm owners packages, dwelling forms, personal inland marine, and watercraft/recreational vehicle coverages.
Risk Classification, Pricing Aspects
Authors: Feldblum Sholom
Published: September 2014

The cost of insurance coverage depends on the attributes of the insured: males have higher mortality than females and young male drivers have higher accident rates than young female drivers. Insurance premiums are based on expected losses and vary similarly. Differentiation among risks by expected cost is termed risk classification.

Risk classification uses the characteristics of a group to set rates for its members. Certain forms of classification are prohibited in many jurisdictions, such as classification by race or ethnic group; other types of classification, such as applicant age for life insurance or type of firm for workers' compensation insurance, are commonly accepted as the proper basis for insurance rates.

A good risk classification system has several attributes, not all of which are essential. Much debate focuses on the importance of each attribute; actuaries and economists emphasize the equity and homogeneity attributes, whereas other groups may emphasize more the intuition, incentives, or social acceptability attributes.
Background Risk
Authors: Gollier Christian
Published: September 2014

The presence of a background risk is believed to reduce the consumer's willingness to accept other independent risks. Under expected utility, this is true only under some restrictions on the utility function.
Equilibrium Theory
Authors: Citanna Alessandro
Published: September 2014

The idea of equilibrium was imported from the physical sciences into economics already in the nineteenth century [63]. Economists focused on the study of equilibrium because of the idea that this represents a rest point of an underlying dynamic system. A rest point is interpreted as a state of the economic system in which the dynamic forces governing its evolution have come to a static configuration. In economics, this static configuration has been assumed to mean essentially two things: (a) that individual economic behavior is socially consistent, that is, consumption, production, and exchange plans are feasible and can be carried out; and (b) that plans are optimal in the sense of satisfying the objectives of the individual decision makers (consumers/workers, managers, traders, and so on). The study of equilibrium was first developed in the context of markets in which each individual believes to have a negligible impact on aggregate variables such as the commodity or asset prices. These markets are called perfectly competitive, and the theory of equilibrium for perfectly competitive markets – known asWalrasian equilibrium– achieved a formal, general, and rigorous characterization in the 1950s, especially with the work of Arrow and Debreu and McKenzie. The work of Debreu and Smale established local uniqueness of equilibria, paving the way for comparative statics analysis for the Walrasian equilibrium. An alternative view would consider equilibria for an economic system in which each decision‐maker's actions have a nonnegligible impact on everyone else, and on the aggregate as well. The notion of equilibrium for these economies had been developed by Nash.
Financial Economics
Authors: Aase Knut K.
Published: September 2014

We consider a one period model of efficient risk sharing, when the set of possible sharing rules are constrained to be linear. This can be interpreted as a model of a market for common stocks. Here we study the properties of a competitive equilibrium in an incomplete market. The lack of Pareto optimality is then the typical case.

Since the marketed subspace
M
is a closed, linear subspace of
L
2
, we employ Hilbert space techniques to find the first order conditions. We also discuss conditions for the existence of an equilibrium in an incomplete market.

We conclude with a discussion of the different features of idiosyncratic risks in insurance, and risks in financial markets, where a common ground is suggested.
Noncooperative Game Theory
Authors: Thijssen Jacco
Published: September 2014

Noncooperative game theory is, roughly, the mathematical theory of conflict. In this article its main ideas, concepts, and results are presented. Special focus is on the distinction between games of complete and incomplete information. Furthermore, attention is paid to some applications in actuarial sciences, namely adverse selection and moral hazard.
Optimal Risk Sharing
Authors: Aase Knut K.
Published: September 2014

Optimal risk sharing is considered from the perspective of the risk sharing model introduced by Karl Borch in the late 50's. First we introduce, in a modern setting, the main concepts from this theory. These we apply on the risk sharing problem between an insurer and an insurance customer. We motivate the development through a simple example, illustrating some fine points of this theory. In order to explain deductibles, we introduce transaction costs that are paid out ex post. Moral hazard provides another example.
Asset Management
Authors: Sherris Michael
Published: September 2014

Asset management is integral to the actuarial management of any financial security system, although the earliest practical developments in actuarial science took place mostly in life insurance. In more recent years, there has been an increased integration of modern financial theories into actuarial science and the adoption of actuarial techniques in areas of asset management such as credit risk and operational risk.

This article will give a brief overview of the historical development of asset management in actuarial science, then briefly outline major asset management strategies used, as well as key models that have been developed, and the issues in implementing asset management strategies.
Derivative Securities
Authors: Cuthbertson Keith
Published: September 2014

There are three main types of derivative securities, namely futures, options, and swaps. Derivative securities are assets whose value depends on the value of some other (underlying) asset and their value is
derived
from the value of this underlying asset. A key feature of futures and options is that the contract calls for
deferred delivery
of the underlying asset (e.g. AT&T shares), whereas spot assets are for
immediate delivery
(although in practice, there is usually a delay of a few days). Derivative securities can be used by hedgers, speculators, and arbitrageurs.

Derivatives often receive a ‘bad press’ partly because there have been some quite spectacular derivatives losses. Perhaps the most famous are the losses of Nick Leeson who worked for Barings Bank in Singapore and who lost $1.4 bn when trading futures and options on the Nikkei‐225, the Japanese stock index. This led to Barings going bankrupt. More recently in 1998, Long Term Capital Management (LTCM), a hedge fund that levered its trades using derivatives had to be rescued by a consortium of banks under the imprimatur of the Federal Reserve Board. This was all the more galling since Myron Scholes and Robert Merton, two academics who received the Nobel Prize for their work on derivatives were key players in the LTCM debacle. The theory of derivatives is a bit like nuclear physics. The derivative products that they have spawned can be used for good but they can also be dangerous, if used incorrectly.
Hedging and Risk Management
Authors: Cairns Andrew J.G.
Published: September 2014

This article summarizes some of the techniques that are used to hedge derivative contracts and manage the risks that arise within a portfolio of derivatives. For simplicity of exposition, the article concentrates here on equity derivatives.

The most straightforward form of hedging is a buy‐and‐hold strategy. This involves finding the optimal mix of cash and equity that will, for example, minimize the variance of the hedging error at the maturity date of the derivative. However, this method can only reduce risk by a small amount. To improve on it, one needs to consider dynamic portfolio strategies, particularly ones where the optimal mix of cash and equity depends on the state of the market at given times in the future.
Interest‐Rate Modeling
Authors: Cairns Andrew J.G.
Published: September 2014

This article describes some of the main developments in interest‐rate modeling since Black & Scholes' and Merton's original articles on the pricing of equity derivatives. In particular, the article focues on continuous‐time, arbitrage‐free models for the fullterm structure of interest rates. Other models that model a limited number of key interest rates or which operate in discrete time (e.g. the Wilkie model) will be considered elsewhere.

In this article we will describe the basic principles of arbitrage‐free pricing and cover various frameworks for modeling: short‐rate models (e.g. Vasicek, Cox–Ingersoll–Ross, Hull–White); the Heath–Jarrow–Morton approach for modeling the forward‐rate curve; pricing using state‐price deflators including the Flesaker–Hughston/Potential approach; and the Markov‐functional approach.

The article works through various approaches and models in a historical sequence. Partly, this is for history's sake, but, more importantly, the older models are simpler and easier to understand. This will allow us to build up gradually to the more up‐to‐date, but more complex, modeling techniques.
Value‐at‐Risk
Authors: Dowd Kevin
Published: September 2014

The Value at Risk (VaR) on a portfolio is the maximum loss we might expect over a given period, at a given level of confidence. It is therefore defined over two parameters – the period concerned, usually known as the holding period, and the confidence level – whose values are arbitrarily chosen. The VaR is a relatively recent risk measure whose roots go back to Baumol, who suggested a risk measure equal to μ –
k
σ, where μ and σ are the mean and standard deviation of the distribution concerned, and
k
is a subjective parameter that reflects the user's attitude to risk. The term
value at risk
only came into widespread use much later. The notion itself lay dormant for nearly 30 years, and then shot to prominence in the early to mid 1990s as the centerpiece of the risk measurement models developed by leading securities firms. The most notable of these, Morgan's Risk‐Metrics model, was made public in October 1994, and gave rise to a widespread debate on the merits, and otherwise, of VaR. Since then, there have been major improvements in the methods used to estimate VaR, and VaR methodologies have been applied to other forms of financial risk besides the market risks for which it was first developed.
W
ilkie Investment Model
Authors: Hardy Mary R.
Published: September 2014

The Wilkie stochastic investment model, developed by Professor Wilkie, is described fully in two papers; the original version was published in 1986 and the model was reviewed, updated, and extended in 1995. In addition, the latter paper includes much detail on the process of fitting the model and estimating the parameters. It is an excellent exposition, both comprehensive and readable. It is highly recommended for any reader who wishes to implement the Wilkie model for themselves, or to develop and fit their own model.

The Wilkie model is commonly used to simulate the joint distribution of inflation rates, bond yields, and returns on equities. The 1995 paper also extends the model to incorporate wage inflation, property yields, and exchange rates. The model has proved to be an invaluable tool for actuaries, particularly in the context of measuring and managing financial risk. In this article, we will describe fully the inflation, equity, and bond processes of the model. Before doing so, it is worth considering the historical circumstances that led to the development of the model.
Experience‐Rating
Authors: Neuhaus Walther
Published: September 2014

Experience‐rating occurs when the premium charged for an insurance policy is explicitly linked to the previous claim record of the policy or the policyholder.

The most common form of experience‐rating involves two components:

– A tariff premium that is calculated on the basis of the policy's known risk characteristics. Depending on the line of business, known risk characteristics could involve sums insured, type and usage of building or vehicle, the occupation of the insured workers, and so on.

– An adjustment that is calculated on the basis of the policy's previous claim record.

Because it makes such obvious sense to consider a policyholder's claim record before deciding on his or her renewal premium, every pricing actuary and underwriter is experience‐rating most of the time – although not always in a formula‐driven way.

Actuaries have studied formula‐driven ways of experience‐rating under two main headings: Credibility theory and Bonus–malus systems. The formalized use of credibility methods probably originated in workers' compensation insurance schemes in the United States about 100 years ago. Bonus–malus Systems (BMS) are an integral feature of motor vehicle insurance in many developed countries.

This article provides a brief outline of different methods of experience‐rating, discusses properties that an experience‐rating scheme should have and outlines some practical issues.
Burning Cost
Authors: J. Mata Ana
Published: September 2014

The burning cost is an experience‐rating method used in reinsurance to estimate the expected loss cost in a layer. The burning cost ‘translates’ the past losses of a portfolio of policies into the prospective reinsurance period by adjusting claims for inflation and other trends.
Pooling of Employee Benefits
Authors: Ott Martin
Published: September 2014

This article explains the basic concept of multinational pooling of employee benefits and the various forms of risk sharing. It highlights the impact of risk retention on the result after pooling, as well as other aspects that are crucial for the profitability of pooled business.
Reinsurance Pricing
Authors: Schmitter Hans
Published: September 2014

The technical premium is the sum of the risk premium, the fluctuation loading, and the loading for management expenses. In practice, excess‐loss risk premiums are estimated using experience‐rating or exposure rating. Stop‐loss risk premiums are usually estimated by means of rather simple mathematical models. More accurate methods for stop‐loss calculations are applied for determining sliding‐scale premium rates.
Retention and Reinsurance Programmes
Authors: de Lourdes Centeno Maria
Published: September 2014

In this article, we give an overview of the literature on the optimal forms of reinsurance as well as on results on how to fix retentions.
Berkson,Joseph
Authors: O'Fallon W. Michael
Published: September 2014

Born in 1899 in Brooklyn, NY; died in 1982 in Rochester, MN. Developed Berkson's fallacy (or Berkson's bias) pointing out that studies of hospitalized patients have a likely selection bias that may invalidate results. Among other health problems, he applied his methods to links between smoking and lung cancer
Cornfield, Jerome
Authors: Greenhouse Samuel W. , Greenhouse Joel B.
Published: September 2014

Born in 1912 in New York City; died in 1979 in Herndon, VA. Best known for helping develop Cornfield's inequality linking the observed risk ratio to the prevalence of the omitted variable in smoking and nonsmoking groups.
Greenhouse, Samuel W
Authors: Greenhouse Joel B. , Lachin John M.
Published: September 2014

Samuel W. Greenhouse helped pioneer the use of statistical methods in epidemiological research and was influential in the early development of the theory and practice of clinical trials.

As one of the founding statisticians at the National Institutes of Health, he helped create the current infrastructure that supports biostatistical research and service at the NIH.
Halperin,Max
Authors: Greenhouse Samuel W.
Published: September 2014

Apart from two periods working in the private sector, Halperin (1917–1988) had a career at the National Institutes of Health and later at George Washington University. He contributed to a wide range of statistical theory, including multivariate analysis, regression, interval estimation, distribution‐free tests, and multiple comparisons. He helped to lay the foundations of multicenter clinical trials, writing on planning, data and safety monitoring, and stochastic curtailment.
H
ill,
A
ustin
B
radford
Authors: Doll R.
Published: September 2014

Born 1897; died 1991. A. Bradford Hill introduced the principle of randomization in clinical trials. He was influential in establishing the link between smoking and lung cancer, and as a result of this research drew up guidelines for reaching a positive conclusion about causality. These criteria have come to be widely used in epidemiology and, on occasion, the law.
H
otelling,
H
arold
Authors: Gehan Edmund A.
Published: September 2014

Harold Hotelling was born on September 29, 1895, in Fulda, Minnesota and died on December 26, 1973, in Chapel Hill, North Carolina. He was responsible for much original theoretical work in both statistics and mathematical economics and did much to advance the teaching of statistics at US universities, including Columbia and the University of North Carolina. Hotelling's undergraduate degree was a B.A. in journalism from the University of Washington in 1919, but his mathematical talent was recognized and encouraged by Eric T. Bell. Hotelling received an M.S. degree in Mathematics at the University of Washington in 1921 and a Doctorate of Philosophy at Princeton University in 1924, with a dissertation in the field of topology.
Kempthorne,Oscar
Authors: David Herbert A. , Pollak Edward
Published: September 2014

Oscar Kempthorne (1919–2000) made important contributions to statistics, especially inference and experimental design, and to statistical genetics. He was born and educated in England, but worked at Ames, Iowa, from 1947 to 1989.
M
antel,
N
athan
Authors: Hale Lauren , Hale Benjamin
Published: September 2014

Nathan Mantel (1919–2002) was most noted for his contribution to the Mantel–Haenszel procedure and its extensions, which allows for combining data from different sources, and so can be used in a variety of contexts: retrospective studies, prospective studies, and laboratory experiments, including those with litter‐matched samples. The technique was also used to develop the logrank test.
S
nedecor,
G
eorge
W
addel
Authors: Cox D. F.
Published: September 2014

G.W. Snedecor (1881–1974) worked at Iowa State University from 1913 to 1958, directing the new Statistical Laboratory from 1933 to 1947. He introduced a course in statistics in 1915 and, with H.A. Wallace, wrote a book on correlation and machine calculation in 1925. He promoted the methods of R.A. Fisher and in 1937 wrote his influential book on statistical methods, the fifth and subsequent editions of which were coauthored with W.G. Cochran.
Categorical Data Analysis
Authors: Imrey Peter B. , Koch Gary G.
Published: September 2014

Categorical data consist of observations counted and subclassified into one among a set of disjoint, exhaustive groups. The groups may be defined on the basis of a single or a combination of qualitative characteristics and/or quantitative measurements. When such categorizations are to be modeled as probabilistic “responses” or “outcomes”, statistical analysis must be based upon discrete distributions and requires extensive generalization of the classical notions of regression and analysis of variance commonly used for continuous observations. This overview article reviews statistical inference for categorical data, from the basic issues in analyses of 2 × 2 contingency tables through generalized linear mixed models, with emphasis on approaches that have increasingly unified our concepts of categorical and continuous data modeling.
Categorizing Continuous Variables
Authors: Altman Douglas G.
Published: September 2014

Continuous variables are often converted to categorical variables by grouping values. For descriptive purposes, this can be useful but problems can arise when this is done as part of a data analysis. This article discusses relevant issues with categorization of explanatory variables in regression. Effects of categorization are discussed, such as loss of information, power, and efficiency. This is considered first for categorizing into two groups and then for categorizing into several groups. Rather than group according to observed values with several variables, one might group according to a weighted sum of the variables–a prognostic index.
Chi‐Square, Partition of
Authors: Gilula Z. , Haberman Shelby J.
Published: September 2014

A chi‐square statistic suitable for testing a primary hypothesis can be partitioned into components such that each component gives a test for a corresponding secondary hypothesis. Some partitionings are exact and some are approximate. The theory is based on the Fisher–Cochran theorem about decomposing quadratic functions of normal variables. The history of this technique is surveyed. Applications are described for contingency tables, with the main focus on the two‐way table and likelihood‐ratio statistics. Brief mention is also made of partitioning into non‐chi‐square components, such as a decomposition that forms the basis of correspondence analysis.
Exact Inference for Categorical Data
Authors: Corcoran Christopher D. , Senchaudhuri Pralay , Mehta Cyrus R. , Patel Nitin R.
Published: September 2014

Commonly used methods for analyzing categorical data rely on asymptotic theory and may be misleading for small data sets. ExactPvalues and confidence limits are available based on the permutation distributions of appropriate test statistics. Details are given here for a wide range of settings with contingency tables, which may be stratified or involve clustering, including also logistic regression. Various software packages are available.
Multinomial Distribution: Overview
Authors: Zelterman Daniel
Published: September 2014

The multinomial distribution is the generalization of the binomial to more than two dichotomous categorical outcomes. This multivariate discrete distribution describes the joint behavior of frequencies of each of
k
possible outcomes following
n
independent trials. The
k
individual frequencies follow a binomial distribution and are mutually negatively correlated. The probabilities of each of the
k
outcomes are usually estimated using loglinear models or Bayesian methods. Goodness‐of‐fit and exact tests assess the adequacy of the model to explain the empirical data. These topics are illustrated with an example from genetics.
Ordered Categorical Data
Authors: Becker Mark P.
Published: September 2014

Many categorical variables have a natural ordering of the categories. Special methods are available for such data that are more powerful and more parsimonious than methods that ignore the ordering. This article discusses ways of modeling association for ordered categorical data and gives examples of them. These include loglinear models, related association models for two‐way tables that have row and column effects, and logit regression models for an ordinal response variable and a set of explanatory variables. The best‐known logit model, called a proportional odds model, uses the same effect parameters for each set of logits of cumulative probabilities for the ordered categorical response variable.
Poisson Regression: Biostatistical Applications
Authors: H. Seeber Gilg U.
Published: September 2014

Poisson regression models are regression models that assume a Poisson distribution for a count response variable. Usually they have loglinear form, modeling the log of the mean in terms of the explanatory variables. Such models are special cases of generalized linear models. A common application is in multiplicative modeling of rates, for which there are connections with Poisson process models and proportional hazards models. This article introduces these types of models, shows how to fit them using maximum likelihood, and discusses checking of model fit.
Power Divergence Methods
Authors: Imrey Peter B.
Published: September 2014

Power divergences are measures of discrepancy between two distributions, with roots in information theory. For two multinomials with common support, the power divergence of order λ is based on the weighted sum ∑ipj(pj/qj)λof exponentiated ratios of the probabilitiespj,qjrespectively assigned to thejthpossible outcome. Note that these measures are directional rather than symmetric. The limiting logarithmic forms as λ → 0 or −1 are also considered power divergences. Asymptotically chi‐square goodness‐of‐fit statistics for count data are generally multiples of power divergences, and hence, general theoretical results and computational approaches for the power‐divergence family apply to these statistics and to estimators that minimize them. In particular, comparisons within this framework of tests and estimators based on the Pearson, Neyman, Freeman–Tukey, likelihood ratio, and discrimination information chi‐square statistics have shed considerable light on their absolute and relative performance in small sample settings, sparse contingency tables, and in the presence of outlying or zero counts. The “Cressie–Read” statistic, for which λ = 2/3, has been suggested as a possible replacement for the classical test statistics due to its good performance from several perspectives. This article defines the power divergence class of goodness‐of‐fit statistics and associated estimators, summarizes some basic comparative results, and briefly surveys the increasing use of power divergence in various statistical estimation settings.
Scores
Authors: Podgor Marvin J. , Gastwirth Joseph L.
Published: September 2014

This article discusses the choice of scores when conducting analyses with ordered categorical variables. Types of scores discussed include rank scores and scores that assume an underlying distribution. Examples are provided, as well as discussion of related topics and recent developments.