At the beginning of 2004, we draw a sample of size 0.25 million people from the inventory of the health insurer AOK. We followed their health claims until 2013. Our aim is the effect a stroke on the dementia onset probability, for Germans born in the first half of the 20$^{th}$ century. People deceased before 2004 are randomly left-truncated. Filtrations, modelling the missing data, enable to circumvent the unknown number of truncated persons by using a conditional instead of the full likelihood. Dementia onset after 2013 is a conditionally fixed right-censoring event. For each observed health history, Jacod's formula yields the conditional likelihood contribution. Asymptotic normality of the estimated intensities is derived, relative to a sample size definition that includes the truncated people. Yet, the standard error is observable. The claims data reveal that after a stroke, with time measured in years, the intensity of dementia onset increases from 0.02 to 0.07. Using the independence of the two estimated intensities, a 95\%-confidence interval for their difference is [0.050,0.056]. The effect halves, when we extend the analysis to an age-inhomogeneous model, but does not change further when we additionally adjust for multi-morbidity.
In this paper, non-life insurance claims were modelled under the three parameter discrete generalised Pareto distribution. Data from the National Insurance Commission of Ghana on reported and settled claims were considered for the period 2012-2016. The maximum likelihood estimation principle was adopted in fitting the discrete Pareto distribution to the yearly and aggregated data. The estimation involved two steps. Firstly, the $μ$ and $(μ+1)$ frequency method of \citet{Prieto2014} was modified to suit the characteristics of the data under study. Secondly, a bootstrap algorithm was implemented…
A number of models for generating statistical data in various fields of insurance, including life insurance, pensions, and general insurance have been considered. It is shown that the insurance statistics data, as a rule, are truncated and censored, and often multivariate. We propose a non-parametric estimation of the distribution function for multivariate truncated-censored data in the form of a quasi-empirical distribution and a simple iterative algorithm for its construction. To check the accuracy of the proposed evaluation of the distribution function for truncated-censored data, simulatio…
According to the Chinese Health Statistics Yearbook, in 2005, the number of traffic accidents was 187781 with total direct property losses of 103691.7 (10000 Yuan). This research aims to fill the gap in the literature by investigating the extreme claim sizes not only for the entire portfolio. This empirical study investigates the behavior of the upper tail of the claim size by class of policyholders.
Large data sets comprising diagnoses about chronic conditions are becoming increasingly available for research purposes. In Germany, it is planned that aggregated claims data including medical diagnoses from the statutory health insurance with roughly 70 million insurants will be published on a regular basis. Validity of the diagnoses in such big data sets can hardly be assessed. In case the data set comprises prevalence, incidence and mortality, it is possible to estimate the proportion of false positive diagnoses using mathematical relations from the illness-death model. We apply the method …
A stratified sampling plan to audit health insurance claims is offered. The stratification is by dollar amount of the claim. The plan is representative in the sense that with high probability for each stratum, the difference in the average dollar amount of the claim in the sample and the average dollar amount in the population, is ``small.'' Several notions of ``small'' are presented. The plan then yields a relatively small total sample size with the property that the overall average dollar amount in the sample is close to the average dollar amount in the population. Three different estimators…
We propose a new class of claim severity distributions with six parameters, that has the standard two-parameter distributions, the log-normal, the log-Gamma, the Weibull, the Gamma and the Pareto, as special cases. This distribution is much more flexible than its special cases, and therefore more able to to capture important characteristics of claim severity data. Further, we have investigated how increased parameter uncertainty due to a larger number of parameters affects the estimate of the reserve. This is done in a large simulation study, where both the characteristics of the claim size di…
The appropriate estimation of incurred but not reported (IBNR) reserves is traditionally one of the most important task of actuaries working in casualty and property insurance. As certain claims are reported many years after their occurrence, the amount and appropriateness of the reserves has a strong effect on the results of the institution. In recent years, stochastic reserving methods had become increasingly widespread. The topic has a wide actuarial literature, describing development models and evaluation techniques. We only mention the summarizing article \cite{EV2002} and book \cite{MV20…
This paper presents a score-based weighted likelihood estimator (SWLE) for robust estimations of generalized linear model (GLM) for insurance loss data. The SWLE exhibits a limited sensitivity to the outliers, theoretically justifying its robustness against model contaminations. Also, with the specially designed weight function to effectively diminish the contributions of extreme losses to the GLM parameter estimations, most statistical quantities can still be derived analytically, minimizing the computational burden for parameter calibrations. Apart from robust estimations, the SWLE can also …
In actuarial research, a task of particular interest and importance is to predict the loss cost for individual risks so that informative decisions are made in various insurance operations such as underwriting, ratemaking, and capital management. The loss cost is typically viewed to follow a compound distribution where the summation of the severity variables is stopped by the frequency variable. A challenging issue in modeling such outcome is to accommodate the potential dependence between the number of claims and the size of each individual claim. In this article, we introduce a novel regressi…
Random shifting typically appears in credibility models whereas random scaling is often encountered in stochastic models for claim sizes reflecting the time-value property of money. In this article we discuss some aspects of random shifting and random scaling in insurance focusing in particular on credibility models, dependence structure of claim sizes in collective risk models, and extreme value models for the joint dependence of large losses. We show that specifying certain actuarial models using random shifting or scaling has some advantages for both theoretical treatments and practical app…
Exponentiated models have been widely used in modeling various types of data such as survival data and insurance claims data. However, the exponentiated composite distribution models have not been explored yet. In this paper, we introduce an improvement of the one-parameter Inverse Gamma-Pareto composite model by exponentiating the random variable associated with the one-parameter Inverse Gamma-Pareto composite distribution function. The goodness-of-fit of the exponentiated Inverse Gamma-Pareto was assessed using three different insurance data sets. The two-parameter exponentiated Inverse Gamm…
This project works with the risk model developed by Li et al. (2015) and quests modelling, estimating and pricing insurance for risks brought in by innovative technologies, or other emerging or latent risks. The model considers two different risk streams that arise together, however not clearly separated or observed. Specifically, we consider a risk surplus process where premia are adjusted according to past claim frequencies, like in a Bonus-Malus (BM) system, when we consider a classical or historical risk stream and an unforeseeable risk one. These are unknown risks which can be of high unc…
Propensity scores are commonly used to reduce the confounding bias in non-randomized observational studies for estimating the average treatment effect. An important assumption underlying this approach is that all confounders that are associated with both the treatment and the outcome of interest are measured and included in the propensity score model. In the absence of strong prior knowledge about potential confounders, researchers may agnostically want to adjust for a high-dimensional set of pre-treatment variables. As such, variable selection procedure is needed for propensity score estimati…
Tweedie's compound Poisson model is a popular method to model insurance claims with probability mass at zero and nonnegative, highly right-skewed distribution. In particular, it is not uncommon to have extremely unbalanced data with excessively large proportion of zero claims, and even traditional Tweedie model may not be satisfactory for fitting the data. In this paper, we propose a boosting-assisted zero-inflated Tweedie model, called EMTboost, that allows zero probability mass to exceed a traditional model. We makes a nonparametric assumption on its Tweedie model component, that unlike a li…
