The ratemaking process is a key issue in insurance pricing. It consists in pooling together policyholders with similar risk profiles into rating classes and assigning the same premium for policyholders in the same class. In actuarial practice, rating systems are typically not based on all risk factors but rather only some of factors are selected to construct the rating classes. The objective of this study is to investigate the selection of risk factors in order to construct rating classes that exhibit maximum internal homogeneity. For this selection, we adopt the Shapley effects from global sensitivity analysis. While these sensitivity indices are used for model interpretability, we apply them to construct rating classes. We provide a new strategy to estimate them, and we connect them to the intra-class variability and heterogeneity of the rating classes. To verify the appropriateness of our procedure, we introduce a measure of heterogeneity specifically designed to compare rating systems with a different number of classes. Using a well-known car insurance dataset, we show that the rating system constructed with the Shapley effects is the one minimizing this heterogeneity measure.

In this paper, we discuss the estimation of conditional quantiles of aggregate claim amounts for non-life insurance embedding the problem in a quantile regression framework using the neural network approach. As the first step, we consider the quantile regression neural networks (QRNN) procedure to compute quantiles for the insurance ratemaking framework. As the second step, we propose a new quantile regression combined actuarial neural network (Quantile-CANN) combining the traditional quantile regression approach with a QRNN. In both cases, we adopt a two-part model scheme where we fit a logistic regression to estimate the probability of positive claims and the QRNN model or the Quantile-CANN for the positive outcomes. Through a case study based on a health insurance dataset, we highlight the overall better performances of the proposed models with respect to the classical quantile regression one. We then use the estimated quantiles to calculate a loaded premium following the quantile premium principle, showing that the proposed models provide a better risk differentiation.

The prominence of the Euler allocation rule (EAR) is rooted in the fact that it is the only return on risk-adjusted capital (RORAC) compatible capital allocation rule. When the total regulatory capital is set using the value-at-risk (VaR), the EAR becomes – using a statistical term – the quantile-regression (QR) function. Although the cumulative QR function (i.e., an integral of the QR function) has received considerable attention in the literature, a fully developed statistical inference theory for the QR function itself has been elusive. In the present paper, we develop such a theory based on an empirical QR estimator, for which we establish consistency, asymptotic normality, and standard error estimation. This makes the herein developed results readily applicable in practice, thus facilitating decision making within the RORAC paradigm, conditional mean risk sharing, and current regulatory frameworks.

Deep neural network models have substantial advantages over traditional and machine learning methods that make this class of models particularly promising for adoption by actuaries. Nonetheless, several important aspects of these models have not yet been studied in detail in the actuarial literature: the effect of hyperparameter choice on the accuracy and stability of network predictions, methods for producing uncertainty estimates and the design of deep learning models for explainability. To allow actuaries to incorporate deep learning safely into their toolkits, we review these areas in the context of a deep neural network for forecasting mortality rates.

Current approaches to fair valuation in insurance often follow a two-step approach, combining quadratic hedging with application of a risk measure on the residual liability, to obtain a cost-of-capital margin. In such approaches, the preferences represented by the regulatory risk measure are not reflected in the hedging process. We address this issue by an alternative two-step hedging procedure, based on generalised regression arguments, which leads to portfolios that are neutral with respect to a risk measure, such as Value-at-Risk or the expectile. First, a portfolio of traded assets aimed at replicating the liability is determined by local quadratic hedging. Second, the residual liability is hedged using an alternative objective function. The risk margin is then defined as the cost of the capital required to hedge the residual liability. In the case quantile regression is used in the second step, yearly solvency constraints are naturally satisfied; furthermore, the portfolio is a risk minimiser among all hedging portfolios that satisfy such constraints. We present a neural network algorithm for the valuation and hedging of insurance liabilities based on a backward iterations scheme. The algorithm is fairly general and easily applicable, as it only requires simulated paths of risk drivers.

This paper introduces the generalized additive mixed model (GAMM) and the quantile generalized additive mixed model (QGAMM) through reanalyses of bilinguals’ lexical decision data from Dijkstra et al. (2010) and Miwa et al. (2014). We illustrate how regression splines can be used to test for nonlinear effects of cross-language similarity in form as well as for controlling experimental trial effects. We further illustrate the tensor product smooth for a nonlinear interaction between cross-language semantic similarity and word frequency. Finally, we show how the QGAMM helps clarify whether the effect of a particular predictor is constant across distributions of RTs.

Assessing conditional tail risk at very high or low levels is of great interest in numerous applications. Due to data sparsity in high tails, the widely used quantile regression method can suffer from high variability at the tails, especially for heavy-tailed distributions. As an alternative to quantile regression, expectile regression, which relies on the minimization of the asymmetric l2-norm and is more sensitive to the magnitudes of extreme losses than quantile regression, is considered. In this article, we develop a new estimation method for high conditional tail risk by first estimating the intermediate conditional expectiles in regression framework, and then estimating the underlying tail index via weighted combinations of the top order conditional expectiles. The resulting conditional tail index estimators are then used as the basis for extrapolating these intermediate conditional expectiles to high tails based on reasonable assumptions on tail behaviors. Finally, we use these high conditional tail expectiles to estimate alternative risk measures such as the Value at Risk (VaR) and Expected Shortfall (ES), both in high tails. The asymptotic properties of the proposed estimators are investigated. Simulation studies and real data analysis show that the proposed method outperforms alternative approaches.

The aim of the present study was to determine whether the association between body mass index (BMI) and the intake of macronutrients varies along the BMI distribution of German adults. Based on a sample of 9214 men and women aged 18–80 years from the representative cross-sectional German National Nutrition Survey (NVS) II, quantile regression was used to investigate the association between BMI and the intake of macronutrients independent of energy intake and other predictors. In both sexes, BMI was positively associated with the intake of total protein and animal protein over its entire range and negatively associated with vegetable protein. A negative association between BMI and the intake of polysaccharides was found along the entire range of BMI in men. There was a weak negative association between BMI and the intake of total fat and saturated fatty acids observed in normal-weight-range women only. In conclusion, the association between BMI and the intake of macronutrients varies along the BMI range. Animal protein intake is positively associated with BMI independent of energy intake in both sexes whereas only in men an inverse association of polysaccharide intake with BMI was shown.

We aim to determine the correlation between parental rearing, personality traits, and obsessive–compulsive disorder (OCD) in different quantiles. In particular, we created an intermediary effect model in which parental rearing affects OCD through personality traits. All predictors were measured at the time of the survey, comprising parental rearing (paternal rearing and maternal rearing), demographics (grade and gender), and personality traits (neuroticism, extroversion, and psychoticism). These results suggest that (a) paternal emotional warmth was negatively correlated with OCD at the 0.40–0.80 quantile, while maternal emotional warmth was positively correlated with the OCD at the 0.45–0.69 quantile. (b) The correlation between negative parental rearing and OCD ranged from the 0.67 to 0.95 quantile for paternal punishment, 0.14–0.82 quantile for paternal overprotection, 0.05–0.36 and >0.50 quantile for maternal over-intervention and overprotection, and 0.08–0.88 quantile for maternal rejection. (c) Extroversion, neuroticism, and psychoticism were not only associated with OCD in a particular quantile but also mediated between parental rearing (namely parental emotional warmth, paternal punishment, paternal overprotection, maternal rejection, maternal over-intervention, and overprotection) and OCD. These findings provide targets for early interventions of OCD to improve the form of family education and personality traits and warrant validation.

This study explored the association between socio-demographic factors and the body mass index (BMI) of women of reproductive age (15–49 years) in Bangladesh. Data from the 2014 Bangladesh Demographic and Health Survey (BDHS-14) were analysed using Multiple Linear Regression (MLR) and Quantile Regression (QR) analyses. The study sample comprised 15,636 non-pregnant women aged 15–49. The mean BMI of the women was 22.35±4.12 kg/m2. Over half (56.75%) had a BMI in the normal range (18<BMI<25 kg/m2), and 18.50%, 20.00% and 4.75% were underweight (BMI≤18 kg/m2), overweight (25≤BMI<30 kg/m2) and obese (BMI≥30 kg/m2), respectively. The results of the MLR found that age, wealth index, urban/rural place of residence, geographical division, womenʼs educational status, husbandʼs educational status, womenʼs working status and total number of children ever born were significantly (p<0.001) associated with respondents’ mean BMI. The QR results showed different associations between socio-demographic factors and mean BMI, as well as a different conditional distribution of mean BMI. Overall, the results indicated that women with uneducated husbands, with little or no education and from less-affluent households from rural areas tended to be more underweight compared with women in other groups. The inter-relationship between the study womenʼs mean BMI and associated socio-demographic factors was assessed using QR analysis to identify the most vulnerable cohorts of women in Bangladesh.

The main objective of this study was to compare the performance of different ‘nonlinear quantile regression’ models evaluated at the τth quantile (0·25, 0·50, and 0·75) of milk production traits and somatic cell score (SCS) in Iranian Holstein dairy cows. Data were collected by the Animal Breeding Center of Iran from 1991 to 2011, comprising 101 051 monthly milk production traits and SCS records of 13 977 cows in 183 herds. Incomplete gamma (Wood), exponential (Wilmink), Dijkstra and polynomial (Ali & Schaeffer) functions were implemented in the quantile regression. Residual mean square, Akaike information criterion and log-likelihood from different models and quantiles indicated that in the same quantile, the best models were Wilmink for milk yield, Dijkstra for fat percentage and Ali & Schaeffer for protein percentage. Over all models the best model fit occurred at quantile 0·50 for milk yield, fat and protein percentage, whereas, for SCS the 0·25th quantile was best. The best model to describe SCS was Dijkstra at quantiles 0·25 and 0·50, and Ali & Schaeffer at quantile 0·75. Wood function had the worst performance amongst all traits. Quantile regression is specifically appropriate for SCS which has a mixed multimodal distribution.

Objectives: A major challenge in cognitive aging is differentiating preclinical disease-related cognitive decline from changes associated with normal aging. Neuropsychological test authors typically publish single time-point norms, referred to here as unconditional reference values. However, detecting significant change requires longitudinal, or conditional reference values, created by modeling cognition as a function of prior performance. Our objectives were to create, depict, and examine preliminary validity of unconditional and conditional reference values for ages 40–75 years on neuropsychological tests. Method: We used quantile regression to create growth-curve–like models of performance on tests of memory and executive function using participants from the Wisconsin Registry for Alzheimer’s Prevention. Unconditional and conditional models accounted for age, sex, education, and verbal ability/literacy; conditional models also included past performance on and number of prior exposures to the test. Models were then used to estimate individuals’ unconditional and conditional percentile ranks for each test. We examined how low performance on each test (operationalized as <7th percentile) related to consensus-conference–determined cognitive statuses and subjective impairment. Results: Participants with low performance were more likely to receive an abnormal cognitive diagnosis at the current visit (but not later visits). Low performance was also linked to subjective and informant reports of worsening memory function. Conclusions: The percentile-based methods and single-test results described here show potential for detecting troublesome within-person cognitive change. Development of reference values for additional cognitive measures, investigation of alternative thresholds for abnormality (including multi-test criteria), and validation in samples with more clinical endpoints are needed. (JINS, 2019, 25, 1–14)