Recurrences formulae for the semi-invariants of some discontinuous frequency functions of n variables

Abstract

Professor Ragnar Frisch has as is well-known given a recurrence formula for the semi-invariants of the Bernouillian frequency function of one variable. Mr. Paul Quale has later generalised his result to certain frequency functions (Pascal and Poisson laws).     

On the compatibility of frequency constants,and on presumptive laws of error

Abstract

1. It is well known that the frequency distribution resulting from repetitions of an observation may be represented by a collection of rational, integral and symmetrical functions of the observations, the most frequently employed functions of this kind being the moments and the semi-invariants. Following the notation of Thiele ¹ we shall denote the rth moment about the origin by s_r , and the rth semi-invariant by µ^r.     

Bayesian multivariate regime-switching models and the impact of correlation structure misspecification in variable annuity pricing

ABSTRACT

We develop Bayesian multivariate regime-switching models for correlated assets, comparing three different ways to flexibly structure the correlation matrix. After developing the models, we examine their relative characteristics and performance, first in a straightforward asset simulation example, and later applied to a variable annuity product with guarantees. We find that the freedom allowed by the more flexible structures enables these models to more accurately reflect the actual asset dependence structure. We also show that the correlation structures inferred by the most commonly used (and simplest) model will result in significantly larger estimates of the cost of the annuity guarantees.     

On the theory of frequency-distributions

Abstract

The following pages contain a brief sketch to a general theory of freq ueney-distributions for n variables. Ta simplify the exposition we begin with frequency-distributions for one variable or frequency curves all though the results here are not novel.     

Aktuareksamen ved Universitetet i Oslo

Abstract

It is a fact that when one is making a decision concerning the probability distribution of a random variable by means of observing this random variable, one is recommended by the statisticians to consider certain functions of the operating characteristic (O.C.) of the decision function as measures of the reliability of the actual decision made. For instance, the confidence coefficient of an interval estimator will as a rule be regarded as a measure of our confidence in the interval.     

On discontinuous frequency functions of two variables

The theory of discontinuous frequency functions of one variable has been well elaborated, especially due to the important researches of Professor Steffensen ¹ See f. i. Steffensen: Matematisk Iagtagelseslære, p. 32 f. while the theory of discontinuous frequency functions of two and more variables has been almost neglected, in spite of the fact that all statisticians agree as to the importance of this topic.     

The Effect of Investor Sentiment on Feedback Trading and Trading Frequency: Evidence from Taiwan Intraday Data

Abstract

Although extensive literature has suggested that investor sentiment may be one of the most important factors in explaining investor trading frequency and trading strategies, how individual investors are significantly influenced by sentiment remains underexplored. The feature of numerous individual investors in the Taiwan stock market provides an avenue to examine the relationship of investor sentiment to trading frequency and positive-feedback trading according to intraday data. Using a vector autoregression model to measure feedback trading in one-minute intervals, we find that trading frequency appears to increase in periods of rising market, suggesting that investor sentiment–driven trading increases market trading frequency without relying on past experiences to conduct trading behavior.     

Derivation of the Wishart distribution of the second order sample moments by straightforward integration of a multiple integral

1. Introduction.

In his textbook of statistics Kendall classifies the methods of deducing exact sampling distributions into four groups:

• (a) straightforward evaluation of the integral in question by ordinary analytical processes such as a convenient change of variable;

• (b) the use of geometrical terminology;

• (c) the use of characteristic functions; and

• (d) other analytical methods.

     

Numerical inversion of characteristic functions

Abstract

To anyone working with characteristic functions, or with Laplace transforms of a non-negative random variable, the three papers by Harald Bohman (1971, 1974, 1975) are invaluable. Numerical integration over an infinite interval is extraordinarily beset with pitfalls (vide Davis & Rabinowitz, 1975) and the publication of actual results achieved in actuarial fields is of great value.     

On the reduction of frequency curves

1. When a frequency distribution is given, it is often important to know whether the corresponding frequency curve is Gauss' normal curve, and if not, whether it can be reduced to one by an adequate change of variable, i.e. by changing bhe method of measuring the attribute. In this note I will give a simple method by which an arbitrary frequency curve may be reduced.     

A Model for Analyzing the Impact of Selective Lapsation on Mortality

Abstract

This paper presents a model for examining the effect of various relationships between mortality rates and lapse rates on the mortality experience of a cohort of insured lives. The approach is individual rather than the aggregate traditionally used in analyzing selective lapsation. The model assumes that insured lives are healthy at policy issue, but later may move to an impaired state from which the lapse rate is zero. Associated with each insured is an unobservable “risk level” random variable, which reflects the heterogeneity of the insured group. Individual mortality and lapse rates are functions of the risk level. A numerical illustration provides some interesting results obtained by using this model.     

On the correlation of acting probabilities

Abstract

In a paper printed in this journal n:r 4–5 1916, I have treated the problem of correlation in homograde statistics. This led in the case of constant acting probabilities to the discussion of an extension of the theorem of Bernoulli to embrace the occurrences of two attributes.     

Representation of concave distortions and applications

ABSTRACT

A family of concave distortion functions is a set of concave and increasing functions, mapping the unity interval onto itself. Distortion functions play an important role defining coherent risk measures. We prove that any family of distortion functions which fulfils a certain translation equation, can be represented by a distribution function. An application can be found in actuarial science: moment-based premium principles are easy to understand but in general are not monotone and cannot be used to compare the riskiness of different insurance contracts with each other. Our representation theorem makes it possible to compare two insurance risks with each other consistent with a moment-based premium principle by defining an appropriate coherent risk measure.     

Predicting the Frequency and Amount of Health Care Expenditures

Abstract

This article extends the standard two-part model for predicting health care expenditures to the case where multiple events may occur within a one-year period. The first part of the extended model represents the frequency of events, such as the number of inpatient hospital stays or outpatient visits, and the second part models expenditure per event. Both component models also use independent variables that consist of an individual’s demographic and access characteristics, socioeconomic status, health status, health insurance coverage, employment status, and industry classification. The second part of the model also includes a variable representing the number of events to predict the expenditure per event, thus capturing dependencies between the first and second parts. This article introduces closed-form predictors of annual total expenditures and demonstrates how to create simulated predictive distributions for individuals and groups. The data for this study are from the Medical Expenditure Panel Survey (MEPS). MEPS panels 7 and 8 from 2003 were used for estimation; panels 8 and 9 from 2004 were used to validate predictions. This annual expenditures model provided a better fit to the data than standard two-part models. The count variable was significant in predicting outpatient expenditures. The aggregate expenditures model provided better point predictions of held-out total expenditures than competing models, including the standard two-part model. The predictive distribution for aggregate expenditures for small groups is long tailed, with both the variability and skewness decreasing as the group size increases, an important point for programs designed to manage expenditures.     

On the theory of testing serial correlation

Abstract

Several different statistics have been proposed for testing the independence between successive observations from a normal population. In order to choose between the various tests a theory of testing this hypothesis in certain populations is needed. In this paper the problem is studied within the framework of the Neyman-Pearson theory. Certain theorems concerning more general problems of quadratic forms are developed and later applied to the question of testing serial correlation.     

On the Gerber-Shiu Discounted Penalty Function for the Ordinary Renewal Risk Model with Constant Interest

Abstract

In this paper we study the Gerber-Shiu discounted penalty function for the ordinary renewal risk model modified by the constant interest on the surplus. Explicit answers are expressed by an infinite series, and a relational formula for some important joint density functions is derived. Applications of the results to the compound Poisson model are given. Finally, a lower bound and an upper bound for the ultimate ruin probability are derived.     

Utility Functions

Abstract

This article is a self-contained survey of utility functions and some of their applications. Throughout the paper the theory is illustrated by three examples: exponential utility functions, power utility functions of the first kind (such as quadratic utility functions), and power utility functions of the second kind (such as the logarithmic utility function). The postulate of equivalent expected utility can be used to replace a random gain by a fixed amount and to determine a fair premium for claims to be insured, even if the insurer’s wealth without the new contract is a random variable itself. Then n companies (or economic agents) with random wealth are considered. They are interested in exchanging wealth to improve their expected utility. The family of Pareto optimal risk exchanges is characterized by the theorem of Borch. Two specific solutions are proposed. The first, believed to be new, is based on the synergy potential; this is the largest amount that can be withdrawn from the system without hurting any company in terms of expected utility. The second is the economic equilibrium originally proposed by Borch. As by-products, the option-pricing formula of Black-Scholes can be derived and the Esscher method of option pricing can be explained.     

New Salary Functions for Pension Valuations

Abstract

This paper investigates salary functions as used in the valuation of pension plans. Pension actuaries as well as researchers in actuarial science may find many of the ideas in this article useful. The main conclusion of this paper is that salary functions, as derived from the parametric models, yield gains and losses that can be quite small and, in some cases, less variable than nonparametric methods. This paper starts by defining the salary function as an accumulation function based on inflation and merit. Next, we investigate traditional estimation methods in the context of this definition. We then present a parametric age-based model for the salary function and compare it with a parametric service-based model. Finally, we apply real pension plan data to derive age-and service-based salary functions and, through the use of two funding methods, investigate how these salary functions affect salary gains and losses.     

On the connection between tests of significance for correlation coefficients and for differences between means

1. Some questions about the connection between statistical tests of significance for simple and multiple correlation coefficients and for differences between sample means (and between sample means and population means) of variables of one or several dimensions are treated in this paper. The distributions of the random variables that are considered in such tests are given, under certain conditions, by frequency functions of the following types ¹ the recently published treatise “Mathematical Methods of Statistics” by Professor Harald Cramér (Uppsala 1945). : where - ∞ < t < ∞, n≧1; where where 0 < t < ∞, k≧1, n≧k; and where .