Background: The Beta distribution is useful for fitting variables that measure a probability or a relative frequency. Methods: We propose a Sarmanov distribution with Beta marginals specified as generalised linear models. We analyse its theoretical properties and its dependence limits. Results: We use a real motor insurance sample of drivers and analyse the percentage of kilometres driven above the posted speed limit and the percentage of kilometres driven at night, together with some additional covariates. We fit a Beta model for the marginals of the bivariate Sarmanov distribution. Conclusions: We find negative dependence in the high quantiles indicating that excess speed and night-time driving are not uniformly correlated.
Quantile regression provides a way to estimate a driver’s risk of a traffic accident by means of predicting the percentile of observed distance driven above the legal speed limits over a one year time interval, conditional on some given characteristics such as total distance driven, age, gender, percent of urban zone driving and night time driving. This study proposes an approximation of quantile regression coefficients by interpolating only a few quantile levels, which can be chosen carefully from the unconditional empirical distribution function of the response. Choosing the levels before interpolation improves accuracy. This approximation method is convenient for real-time implementation of risky driving identification and provides a fast approximate calculation of a risk score. We illustrate our results with data on 9614 drivers observed over one year.
Marketers are faced with the daunting challenge of identifying insights anddelivering the right combination of online and offline tactics to engageconsumers at various stages along the consumer journey. In this paper, weinvestigate the effects of retargeting in a multichannel environment. Using athree-stage modeling approach, we find retargeting is an effective advertisingactivity to influence purchase incidence, but only when combined with otherspecific marketing activities. While catalogs and emails have positivesynergies with retargeting, website visits and retargeting have a negativesynergy on a consumer’s decision to make a purchase. One possibleexplanation to the negative synergistic effect is that consumers may findretargeting obtrusive when browsing online, whereas it may serve as awelcome reminder when, combined with emails or catalogs. Rather thannudging consumers along the consumer journey some combinations ofadvertising activities may actually deter customers from engaging with a firm.
Given a risk level or tolerance, quantile regression is a predictive model that fits the corresponding percentile of the continuous response variable. Given a fixed percentage value, we identify the effect of each predictor variable in the cumulative distribution up to that level of the dependent variable. In this article, we show how this methodology can be used in motor insurance data analysis and we propose an extension of quantile regression inspired by the need to predict the expectation of the conditional tail. To this end, specific R routines have been developed and a resampling procedure has been implemented to approximate standard errors. The main conclusion is that this type of models allows us to analyze which factors affect accident risk and can be used to mitigate or to evaluate risk in the insurance field. Keywords: predictive modelling, value-at-risk, tail value at-risk, optimization, resampling Resumen Dado un nivel o tolerancia de riesgo, la regresión cuantílica es un modelo predictivo que ajusta el correspondiente percentil de la variable respuesta continua. Fijado un determinado valor porcentual, se identifica el efecto de cada variable predictora en la distribución acumulada hasta ese nivel de la variable dependiente. En este artículo mostramos cómo puede utilizarse esta metodología en el análisis de datos en el seguro de automóvil y proponemos una extensión de la regresión cuantílica inspirada en la necesidad de predecir la esperanza de la cola condicional. Para ello se han desarrollado rutinas específicas en R y se ha implementado un procedimiento de remuestreo para la aproximación de los errores estándar. La principal conclusión es que este tipo de modelos permite analizar qué factores inciden en el riesgo de accidente y pueden ser utilizados para mitigarlo o para valorarlo en el ámbito asegurador. Palabras clave: modelización predictiva, valor en riesgo, valor en riesgo de la cola, optimización, remuestreo
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