Wai-Sum Chan

APPROXIMATION OF POLYGONAL CURVES WITH MINIMUM NUMBER OF LINE SEGMENTS OR MINIMUM ERROR

We improve the time complexities for solving the polygonal curve approximation problems formulated by Imai and Iri. The time complexity for approximating any polygonal curve of n vertices with minimum number of line segments can be improved from O(n2log n) to O(n2). The time complexity for approximating any polygonal curve with minimum error can also be improved from O(n2log2n) to O(n2log n). We further show that if the curve to be approximated forms part of a convex polygon, the two problems can be solved in O(n) and O(n2) time respectively for both open and closed polygonal curves.

Some results on ruin probabilities in a two-dimensional risk model

Abstract Ruin theory under multi-dimensional risk models is very complex. Even in the two-dimensional case, the problem is challenging. In this paper, we consider a bivariate risk model. Three different types of ruin probabilities are defined. Using some results of one-dimensional risk processes, simple bounds for the two-dimensional ruin probabilities are obtained. Numerical examples and simulation experiments are given to illustrate the tightness of the bounds. A partial integral–differential equation satisfied by the two-dimensional ruin probabilities is derived. Although special cases and examples in this paper provide some exciting results, the problem of ruin probability in a multi-dimensional risk model is still far from solved. We hope that this paper stimulates more research by actuaries in this area.

Asymptotic properties of covariate-adjusted response-adaptive designs

Response-adaptive designs have been extensively studied and used in clinical trials. However, there is a lack of a comprehensive study of response-adaptive designs that include covariates, despite their importance in clinical trials. Because the allocation scheme and the estimation of parameters are affected by both the responses and the covariates, covariate-adjusted response-adaptive (CARA) designs are very complex to formulate. In this paper, we overcome the technical hurdles and lay out a framework for general CARA designs for the allocation of subjects to K (≥ 2) treatments. The asymptotic properties are studied under certain widely satisfied conditions. The proposed CARA designs can be applied to generalized linear models. Two important special cases, the linear model and the logistic regression model, are considered in detail.

An audit of the safety of an acute pain service

We audited and analysed the adverse effects and safety of postoperative pain management on 2509 consecutive patients under care of the Acute Pain Service at a tertiary referral teaching hospital over a 32‐month period. Our standard respiratory monitoring consisted of continuous pulse oximetry, hourly respiratory rate counting, sedation scoring and intermittent arterial blood gas sampling. This protocol was reliable and detected six episodes of bradypnoea, 13 of hypercapnia and 23 of oxygen desaturation occurring in 39 patients (1.8% of all spontaneously breathing patients). Two patients required naloxone injection and none had long‐term sequelae. Hypotension due to epidural bupivacaine 0.0625% and fentanyl 3.3 μg.ml−1 infusion occurred in four patients (1.2%), all with a sensory block higher than T5. They readily responded to fluid infusion and ephedrine (two patients). Postoperative nausea or vomiting occurred in 723 (28.8%) and 380 (15.1%) patients, respectively. Odds ratio analysis showed that the risk factors for postoperative nausea and vomiting were: female gender, gynaecological operations, nongeriatric patients and systemic analgesia. Postoperative nausea and vomiting decreased analgesic efficacy by discouraging the use of patient‐controlled analgesia and was regarded as equally distressing as pain. Other side‐effects included: pruritus in 182 patients; dizziness in 333 and lower limb weakness in 73 (21.2% of patients receiving epidural local anaesthetics). It is concluded that a standard monitoring and management protocol, an experienced nursing team and reliable Acute Pain Service coverage is mandatory for the safe use of modern analgesic techniques.

Methods for Estimating the Case Fatality Ratio for a Novel, Emerging Infectious Disease

Abstract During the course of an epidemic of a potentially fatal disease, it is important that the case fatality ratio be well estimated. The authors propose a novel method for doing so based on the Kaplan-Meier survival procedure, jointly considering two outcomes (death and recovery), and evaluate its performance by using data from the 2003 epidemic of severe acute respiratory syndrome in Hong Kong, Peoples Republic of China. They compare this estimate obtained at various points in the epidemic with the case fatality ratio eventually observed; with two commonly quoted, naïve estimates derived from cumulative incidence and mortality statistics at single time points; and with estimates in which a parametric mixture model is used. They demonstrate the importance of patient characteristics regarding outcome by analyzing subgroups defined by age at admission to the hospital.

Approximation of Polygonal Curves with Minimum Number of Line Segments

We improve the time complexity to solve the polygonal curve approximation problem formulated by Imai and Iri from O(n2 log n) to O(n2). If the curve to be approximated forms part of a convex polygon, we show that the time complexity can be further reduced to O(n).

The Lee-Carter Model for Forecasting Mortality Revisited

Abstract Interrupting phenomena are commonly encountered in time-series data analysis with the study of mortality trends being no exception. Nevertheless, previous demographic forecasts have paid little attention to the existence of such phenomena. In this study we use mortality data from Canada and the United States to perform time-series outlier analysis on the key component of the Lee-Carter model: the mortality index. We begin by employing a systematic outlier detection process to ascertain the timing, magnitude, and persistence of any outliers present in historical trends of the mortality index. We then try to match the identified outliers with important events that could possibly justify the vacillations in human mortality levels. At the same time, we adjust the effect of the outliers for model reestimation. The empirical results indicate that the outlieradjusted model could achieve better fits and more efficient forecasts of variables such as the central rates of death and the life expectancies at birth. Finally, we conclude our study with possible extensions on the valuations of life annuities and the probabilistic distribution of the highest attained age, incorporating the effect of mortality improvement portrayed by the revised model.

Outlier analysis and mortality forecasting: The United Kingdom and Scandinavian countries

The Lee-Carter model and its variants have been extensively employed by actuaries, demographers, and many others to forecast age-specific mortality. In this study, we use mortality data from England and Wales, and four Scandinavian countries to perform time-series outlier analysis of the key component of the Lee-Carter model – the mortality index. We begin by employing a systematic outlier detection process to ascertain the timing, magnitude, and persistence of any outliers present in historical mortality trends. We then try to match the identified outliers with imperative events that could possibly justify the vacillations in human mortality levels. At the same time, we adjust the effect of the outliers for model re-estimation. A new iterative model re-estimation method is proposed to reduce the chance of erroneous model specification. The empirical results indicate that the outlier-adjusted model could achieve more efficient forecasts of variables such as death rates and life expectancies. Finally, we point out that the Lee-Carter forecasts are especially vulnerable to outliers near the forecast origin, and discuss the potential limitations of the application of the Lee-Carter model to mortality forecasting.

The CBD Mortality Indexes: Modeling and Applications

Most extrapolative stochastic mortality models are constructed in a similar manner. Specifically, when they are fitted to historical data, one or more series of time-varying parameters are identified. By extrapolating these parameters to the future, we can obtain a forecast of death probabilities and consequently cash flows arising from life contingent liabilities. In this article, we first argue that, among various time-varying model parameters, those encompassed in the Cairns-Blake-Dowd (CBD) model (also known as Model M5) are most suitably used as indexes to indicate levels of longevity risk at different time points. We then investigate how these indexes can be jointly modeled with a more general class of multivariate time-series models, instead of a simple random walk that takes no account of cross-correlations. Finally, we study the joint prediction region for the mortality indexes. Such a region, as we demonstrate, can serve as a graphical longevity risk metric, allowing practitioners to compare the longevity risk exposures of different portfolios readily.

A note on time series model specification in the presence of outliers

We investigate the usefulness of sample autocorrelations and partial autocorrelations as model specification tools when the observed time series is contaminated by an outlier. The results indicate that the specification power of these statistics could be significantly jeopardized by an additive outlier. On the other hand, an innovational outlier seems to cause no harm to them.