A comparison of in-sample forecasting methods

Abstract

In-sample forecasting is a recent continuous modification of well-known forecasting methods based on aggregated data. These aggregated methods are known as age-cohort methods in demography, economics, epidemiology and sociology and as chain ladder in non-life insurance. Data is organized in a two-way table with age and cohort as indices, but without measures of exposure. It has recently been established that such structured forecasting methods based on aggregated data can be interpreted as structured histogram estimators. Continuous in-sample forecasting transfers these classical forecasting models into a modern statistical world including smoothing methodology that is more efficient than smoothing via histograms. All in-sample forecasting estimators are collected and their performance is compared via a finite sample simulation study.All methods are extended via multiplicative bias correction. Asymptotic theory is being developed for the histogram-type method of sieves and for the multiplicatively corrected estimators. The multiplicative bias corrected estimators improve all other known in-sample forecasters in the simulation study. The density projection approach seems to have the best performance with forecasting based on survival densities being the runner-up. not reported) claims is often solved via model (2): For each past claim, one considers the date (cohort i) when the accident had happened and the delay (age k) there was until the claim was reported to the insurer.Hence, cohort and age satisfy i + k − 1 ≤ today; given a certain year-wise aggregation. This information is then used to estimate the number of future claims µ ik , i + k − 1 > today, for accidents in the past, i ≤ today.Under model (2), the parameters α i and β k for each cohort i and age k can be estimated from past data. 50Assuming a maximum delay (usually 7 to 10 years in practice, depending on the business line), the estimates of the parameters can be used to forecast the number of future claims with i + k − 1 > today. More details of this age-cohort-reserving example are given in the recent contribution Harnau and Nielsen (2018) and are also included in the highly-cited overview paper of actuarial reserving (England and Verrall, 2002).Other examples where no significant period effect has been found include among many others cancer stud-55 ies (Leung et al., 2002;Remontet et al., 2003), returns due to education (Duraisamy, 2002), unemployment numbers (Wilke, 2017), mesothelioma mortality (Peto et al., 1995;Martínez-Miranda et al., 2014).Given the importance of age-period-cohort models and age-cohort models, it is surprising that continuous versions have not been considered much in the literature. Continuous modeling avoids inefficient pre-smoothing and is in line with recent trends around big data and the drive of modeling and understanding 60 every individual separately. Modeling every individual separately, possibly with additional covariates, results in the estimation of a large number of parameters. An increase of dimension means that ...

Continuous chain-ladder with paid data

“…This is because the hazard function, due to the bounded support, usually increases heavily at the right boundary whereas the shape of the density is less explosive. A simulation study in Bischofberger et al (2019) confirms this heuristic.…”