A practical solution to the problem of ultimate ruin probability

Abstract

It was the Swiss actuary Chr. Moser who, in lectures at Bern University at the turn of the century, gave the name “self-renewing aggregate” to what Vajda (1947) has called the “unstationary community” of lives, namely where deaths at any epoch are immediately replaced by an equivalent number of births. It was Moser too (1926) who coined the expression “steady state” for the stationary community in which the age distribution at any time follows the life table (King, 1887). With such a distinguished actuarial history, excellently summarized by Saxer (1958, Ch. IV), it behoves every actuary to know at least the definitions and modus operandi of today's so-called renewal (point), or recurrent event, processes.     

An approximation to the limit distribution of the epps-pulley test statistic for normality

Summary The approach of Epps and Pulley (1983) based on the empirical characteristic function is one of the most powerful tools for detecting any departures from normality. We obtain the first four moments of the limiting null distribution of the Epps-Pulley Statistic. Johnson- and Pearson curve fitting yields excellent approximations to simulated quantiles, and by modifying the test statistic the procedure may be carried out easily without the use of extensive tables for all sample sizes. Research done while the author was on leave at the University of Gie?en.     

Karl Pearson and the Origin of Kurtosis

Although the kurtosis index proposed by Karl Pearson in 1905 is introduced in statistical textbooks at all levels, the measure is not easily interpreted and has been a subject of considerable debate. In this study, the theoretical development of kurtosis is surveyed from a historical perspective of Pearson's work on evolution. It surprisingly emerges that there was no emphasis in Pearson's papers on kurtosis as measuring (in part) tail heaviness. However, it is found that Pearson used to frequently adjust the formalisation of kurtosis depending on his changing needs. This complex development partly explains the confusion that would surround kurtosis in subsequent literature. Our conclusion is that most misunderstandings arise from improper use of the kurtosis coefficient outside the Pearson system of frequency curves.     

Karl Pearson—The Scientific Life in a Statistical Age by Theodore M. Porter: A Review

Porter presents an excellent account of the young Karl Pearson and his extraordinarily varied activities. These ranged from the Cambridge Mathematical Tripos Exams to German history and folklore, and included free thought, socialism, the woman's question, and the law. Returning to science, Pearson produced the famous Grammar of Science . He decided on a career in statistics only at age 35. Porter emphasizes Pearson's often acrimonious but largely successful battles to show the wide applicability and importance of statistics in many areas of science and public affairs. Eugenics became a passion for Pearson. Avoiding all formulas Porter fails to give any concrete ideas of even Pearson's most important contributions to statistical theory. We try to sketch these here.     

Budget-constrained optimal reinsurance design under coherent risk measures

ABSTRACT

Reinsurance is a versatile risk management strategy commonly employed by insurers to optimize their risk profile. In this paper, we study an optimal reinsurance design problem minimizing a general law-invariant coherent risk measure of the net risk exposure of a generic insurer, in conjunction with a general law-invariant comonotonic additive convex reinsurance premium principle and a premium budget constraint. Due to its intrinsic generality, this contract design problem encompasses a wide body of optimal reinsurance models commonly encountered in practice. A three-step solution scheme is presented. Firstly, the objective and constraint functions are exhibited in the so-called Kusuoka's integral representations. Secondly, the mini-max theorem for infinite dimensional spaces is applied to interchange the infimum on the space of indemnities and the supremum on the space of probability measures. Thirdly, the recently developed Neyman–Pearson methodology due to Lo (2017a) is adopted to solve the resulting infimum problem. Analytic and transparent expressions for the optimal reinsurance policy are provided, followed by illustrative examples.     

Pearson type VII ratio distribution

In modern decision-making processes, ratios or indices of stochastic variables are commonly used criteria. The decision criterion is, however, frequently presented as deterministic despite the fact that sampling has been the dominant collection method for the data underlying the measure. This paper derives the exact distribution of |X/Y| when X and Y are independent Pearson type VII random variables. We provide an application of this result to the exchange rate data of the six major currencies. Some computer programs for use in the applications are also provided.      

A meta‐analysis of the UTAUT model: Eleven years later

The unified theory of acceptance and use of technology (UTAUT) has been widely used to investigate factors influencing the adoption and use of information systems and technologies (IS/IT). However, studies using UTAUT are not conclusive in terms of statistical significance, direction, and magnitude. Through a meta‐analysis of empirical studies on UTAUT from 2003 to 2013, we determine how parsimonious, accurate, and robust UTAUT is at predicting acceptance and use of technology. A meta‐analysis of 74 publications reveals that performance expectancy, effort expectancy, and social influence explain IS/IT adoption, while behavioural intention is the most often measured dependent variable operationalized as a proxy for system use, supporting the strength of UTAUT as an explanatory model of IS/IT acceptance and use. Copyright © 2016 ASAC. Published by John Wiley & Sons, Ltd.     

基于BP神经网络及Pearson相关系数的风机发电机轴承温升故障诊断研究

发电机轴承温升高是直驱风力发电机频发的故障,易导致风机长时间停机,维修难度大,造成设备重大损失。通过建立基于BP神经网络和Pearson相关系数的发电机轴承温升故障诊断模型,并以湖北荆门某风电场为例,试验结果表明:该方法能够较准确预测风机发电机轴承温升变化,通过将温度预测值与正常工况下的温度值进行误差对比,判断发电机是否处于正常工状,实现发电机潜在故障的预警,为风力发电机的故障诊断提供了一种切实可行的方法。     

MIS‐SPECIFICATION TESTING IN RETROSPECT

The primary objective of this paper is threefold. First, to undertake a retrospective view of Mis‐Specification (M‐S) testing, going back to the early 20th century, with a view to (i) place it in the broader context of modeling and inference and (ii) bring out some of its special features. Second, to call into question several widely used arguments undermining the importance of M‐S testing in favor of relying on weak probabilistic assumptions in conjunction with generic robustness claims and asymptotic inference. Third, to bring out the crucial role of M‐S testing in securing trustworthy inference results. This is achieved by extending/modifying Fisher's statistical framework with a view to draw a clear line between the modeling and the inference facets of statistical induction. The proposed framework untangles the statistical from the substantive (structural) model and focuses on how to secure the adequacy of the statistical model before probing for substantive adequacy. A case is made for using joint M‐S tests based on custom‐built auxiliary regressions with a view to enhance the effectiveness and reliability of probing for potential statistical misspecifications.     

Neyman–Pearson Lemma for Fuzzy Hypotheses Testing with Vague Data

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which both hypotheses and observations are imprecise. This paper tries to develop a new approach for testing fuzzy hypothesis when the available data are fuzzy, too. First, some definitions are provided, such as: fuzzy sample space, fuzzy-valued random sample, and fuzzy-valued random variable. Then, the problem of fuzzy hypothesis testing with vague data is formulated. Finally, we state and prove a generalized Neyman–Pearson Lemma for such problem. The proposed approach is illustrated by some numerical examples.