In-sample hazard forecasting based on survival models with operational time

Bischofberger, Stephan M. Article In-sample hazard forecasting based on survival models with operational time Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Bischofberger, Stephan M. (2020) : In-sample hazard forecasting based on survival models with operational time, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 8, Iss. 1, pp. 1-17, https://doi.org/10.3390/risks8010003 This Version is available at: https://hdl.handle.net/10419/257959 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Bischofberger Cass Business School, University of London, London EC1Y 8TZ, UK; [email protected]ty.ac.uk Received: 27 November 2019; Accepted: 24 December 2019; Published: 3 January 2020   Abstract: We introduce a generalization of the one-dimensional accelerated failure time model allowing the covariate effect to be any positive function of the covariate. This function and the baseline hazard rate are estimated nonparametrically via an iterative algorithm. In an application in non-life reserving, the survival time models the settlement delay of a claim and the covariate effect is often called operational time. The accident date of a claim serves as covariate. The estimated hazard rate is a nonparametric continuous-time alternative to chain-ladder development factors in reserving and is used to forecast outstanding liabilities. Hence, we provide an extension of the chain-ladder framework for claim numbers without the assumption of independence between settlement delay and accident date. Our proposed algorithm is an unsupervised learning approach to reserving that detects operational time in the data and adjusts for it in the estimation process. Advantages of the new estimation method are illustrated in a data set consisting of paid claims from a motor insurance business line on which we forecast the number of outstanding claims. Keywords: accelerated failure time model; chain-ladder method; local linear kernel estimation; non-life reserving; operational time 1. Introduction The parametric accelerated failure time (AFT) model has been well established in medical statistics and other applications (Kalbfleisch and Prentice 2002) for decades. The aim of this paper is to introduce a nonparametric generalization of the one-dimensional AFT model for right-truncated data and apply it to estimate the number of outstanding claims in non-life insurance. Given a covariate X∈Rd and given no failure has occurred until time t , the AFT specifies that the probability of a failure between time t and t+ d t equals θα0(θt) d t with θ=exp(−β0X) for an underlying hazard rate α0 and a deterministic vector β∈Rd . More formally, this model is expressed through the conditional hazard rate α(t|X) = θα0(θt),θ=exp(−β0X). Its interpretation is straightforward, for example, in a medical context where failure time T describes the amount of time for a tumor to reach a critical stage. For each individual i , the value of θi depends on its covariate Xi (the patient’s medical data). A value of θi= 2, for instance, means that the development of the tumor happens twice as fast for a patient and θi= 0.9 means 10% slower development than usual. This is in contrast to the proportional hazard model α(t|X) = θα0(t) , where the interpretation of θ is non-trivial (Cox 1972). For the statistical analysis in the AFT model, one can transform the observed failure times through Ti7→ θiTi (if one knows θi ). The transformed survival time θiTi follows the same distribution for all individuals and is independent of the covariate Xi. Risks 2020,8, 3; doi:10.3390/risks8010003 www.mdpi.com/journal/risks Risks 2020,8, 3 2 of 17 The AFT model has been studied by various authors including Buckley and James (1979); Louis (1981); Miller (1976), and Ritov and Wellner (1988). Comprehensive overviews have been given in Cox and Oakes (1984) and Andersen et al. (1993). The model is still widely used and adapted to new problems in medical research. A recent modification of the AFT model has been introduced in Li and Jin (2018) and recent applications include AIDS research (Fulcher et al. 2017) and cancer research (Cho et al. 2018) among many others. This article focuses on the one-dimensional case d= 1 and provides a nonparametric generalization of the parametric AFT model above assuming θ= 1 /ϕ(X) . We estimate ϕ nonparametrically and impose no structural assumption. In a finance or insurance context, the unknown function ϕ is often called operational time and it can accelerate or slow down the survival time T . However, with our definition, ϕ(x) has the same effect as θ−1 in the AFT model, i.e., the effect is reversed. We can transform observed survival times Ti and covariates Xi via Ti7→ Ti/ϕ(Xi) = e Ti to obtain identically distributed survival times e Ti that are independent of their covariates Xi as in the AFT model. In our application of non-life reserving, X is the accident date of an insurance claim and T is its settlement delay which can be affected by calendar effects, seasonal effects or a trend in the speed of claims finalization over time, e.g., due to new organizational structures in the insurance company, more efficient IT systems, or changes in legislation. The latter trends over time are captured by our operational time function ϕ . We estimate ϕ and the marginal hazard rate of e T . Together, they yield an estimate of the conditional hazard rate of T given X , which contains full information of the distribution of T given X . This hazard rate is used to estimate outstanding claim numbers through extrapolation with a chain-ladder type algorithm. The proposed algorithm in this article detects the effects of operational time and adjusts for them. If there is no operational time present, the algorithm still estimates smoothed chain-ladder development factors for an optimal bandwidth that is selected through cross-validation. The concept of operational time was originally developed for stochastic processes in Feller (1971) . In actuarial research, it was first used for processes of claim numbers in Bühlmann (1970) and for non-life reserving in Reid (1978) and Taylor (1981,1982). Comprehensive summaries about operational time in reserving have been provided in Taylor et al. (2008) and Taylor and McGuire (2016). For an overview of its use in mathematical finance, we refer to Swishchuk (2016). The algorithm in this paper is an alternative to the most widely used algorithm in non-life reserving, the chain-ladder method. The difference is that, in chain-ladder, it is assumed that accident date and settlement delay are independent, and thus chain-ladder does not account for calendar time effects like court rulings, emergence of latent claims, or changes in operational time. The first stochastic model around the chain-ladder method was introduced in Mack (1993). Chain-ladder is still widely used in the insurance industry and as a benchmark for new methods in research as explained in overviews of reserving methods in England and Verrall (2002) and, more recently, Taylor (2019). Based on the idea of chain-ladder, different multiplicative models with independent effects of accident date and settlement delay were introduced in Kremer (1982); Kuang et al. (2009); Renshaw and Verrall (1998), and Verrall (1991). Aside from these publications, the greater part of the research on claims reserving can be summarized into two streams: a Poisson process approach and a two-dimensional kernel estimation approach for truncated data. The first (older and more extensive) stream of research focuses on Poisson process models in Antonio and Plat (2014); Avanzi et al. (2016); Huang et al. (2015); Jewell (1989,1990); Larsen (2007), and Norberg (1993,1999). Extensions that investigate dependent covariates or marked Cox processes include Zhao and Zhou (2010); Zhao et al. (2009), or Badescu et al. (2016), respectively. A semiparametric approach very similar to operational time is given in Crevecoeur et al. (2019) , in which the authors allow time on weekends and public holidays to pass faster in order to make up for less claim reports on these days while ensuring a continuous distribution of reporting delay. The approach in this present paper fits into the second stream of reserving research based on “continuous chain-ladder” (Hiabu et al. (2016); Lee et al. 2015,2017;Martínez-Miranda et al. (2013)). In a broader statistical context, the problem was introduced as “in-sample Risks 2020,8, 3 3 of 17 forecasting” (Mammen et al. 2015) and said papers applied their results to forecasting problems beyond actuarial research. These articles have in common that no distributional assumptions are made and that kernel estimation is performed under the assumption of a structural model for the joint density or conditional hazard rate. In the operational time model, Lee et al. (2017) assume the nonparametric factor θ=ψ(X) , i.e., ψ(X) = ϕ−1(X) , and estimate ψ as well as the two marginal densities of accident year and settlement delay. The latter is the closest approach to this present paper; however, we estimate a conditional hazard rate instead of a multivariate density. The advantage of our approach is that we only estimate two functions ( ϕ and α0 ) instead of three, and then extrapolate claim numbers to estimate the number of outstanding claims. This extrapolation is analogous to the algorithm in the chain-ladder method. Since our estimated conditional hazard rates are similar to chain-ladder development factors (Hiabu 2017), we consider it more natural to extrapolate in a hazard framework than to perform extrapolation with density functions. Therefore, we only forecast the effect of the accident date X and do not estimate its distribution. All mentioned continuous chain-ladder publications including this present paper focus on claim numbers instead of payment amounts. Recently, Bischofberger et al. (2019) have shown how to extend the models and estimators for payment amounts. This extension is also feasible for our approach; however, adding extensive additional technicalities is beyond the scope of this paper. In traditional statistical learning, learning problems are classified as “supervised” and “unsupervised” (Hastie et al. 2008). For supervised learning algorithms, the goal is to predict an outcome measure for a given input. For this purpose, the algorithm trains on paired data consisting of (input, output) and then applies the learned structure to predict an output from a new input. On the other hand, in unsupervised learning, there is no output in the data and the goal of the algorithm is often to find patterns in the data minimizing a loss criterion. Nonparametric kernel estimation is used in both approaches: for nonparametric regression in supervised learning and for kernel density estimation in unsupervised learning (Hastie et al. 2008). The new forecasting procedure in this article can be classified as an unsupervised machine learning technique. Although the goal is to give an estimate of the number of outstanding claims from past data, our algorithm cannot be trained on a data set of input and output (in form of past claims and future claims) and then applied to a new input. The presented algorithm estimates the conditional distribution of settlement delay given the accident date that is specific for the data set it is used on. This estimation involves kernel hazard estimators and the minimization of a loss function. Very recently, following a trend in applied statistics, various other machine learning approaches to claims reserving that do not belong to any of the previous streams have arisen. Soon these articles may constitute a third big stream of research. Useful machine learning techniques for reserving include regression trees (Baudry and Robert 2019;Wüthrich 2018) and neural networks (Kuo 2019) among others. These approaches also take dependence between accident date and delay into account and are thus more flexible than many of the aforementioned models. In contrast to the algorithm in this article, they are all based on supervised learning. A neural network architecture based on classical chain-ladder literature, into which the over-dispersed Poisson reserving model of Renshaw and Verrall (1998) is embedded, has been introduced in Gabrielli et al. (2019). This article is structured as follows. The underlying mathematical model is introduced in Section 2. Section 3explains an algorithm to estimate operational time and the baseline hazard. Section 4illustrates how to estimate outstanding liabilities from an operational time and a baseline hazard estimate. A data-driven bandwidth selection procedure is introduced in Sections 5and 6containing an illustration for a real data set. 2. Model We start with a general mathematical model for hazard rates with operational time but without filtering and afterwards adapt it to observations on a run-off triangle in the context of claims reserving. Risks 2020,8, 3 4 of 17 Since this particular triangular data structure can be expressed as truncated data, a counting process survival model lends itself to our cause. 2.1. General Model Let (T , X) be a two-dimensional random variable on the square S={(t , x): 0 ≤t , x≤ T }) for T ≥ 0. Suppose that Tcan be written as T=e Tϕ(X)(1) for a random variable e T that is independent of X and a function ϕ:[ 0, T]→[ 0, T/e T] . We call ϕ operational time and Equation (1) operational time model. The support of e T is [ 0, e T] for some 0 ≤e T ≤ T . In the sequel, we define quantities for each realization (Ti , Xi) of (T , X) with i= 1, . . . , n . In a counting process framework, we identify the survival time with Ti and treat Xi as a one-dimensional covariate. We first define the counting process setting before linking it to the random variable (Ti , Xi) . Suppose we observe a counting process {Ni(t): 0 ≤t≤ T } with respect to a suitable filtration {Fi t: 0 ≤t≤ T } (Andersen et al. 1993, p. 60). The intensity of Ni at time t is defined as λi(t) = lim h↓0h−1E[Ni((t+h)−)−Ni(t−)| Fi,t−]. To illustrate the effect of operational time on the intensity, we start with a simple model for unfiltered data, denoted by the superscript unfilt . We use the notation with superscripts since we will focus on a specific hazard later on, for which we want to reserve the plain notation λ and N . For illustration, we define the counting process Nunfilt i(t) = I(Ti≤t) with the adapted filtration Funfilt i,t=σ({Nunfilt i(s) , Xi(s) , s≤t}) . The intensity of Nunfilt i given Xi(t) = x satisfies Aalen’s multiplicative model (Aalen 1980) with λunfilt i(t) = αunfilt(t|x)I(t≤Ti), =1 ϕ(x)αunfilt 0t ϕ(x)I(t≤Ti), where αunfilt(t|x) = limh↓0h−1P(T∈[t,t+h)|T≥t,X=x) is the conditional hazard of T given X and αunfilt 0(˜ t) = limh↓0h−1P(e T∈[˜ t , ˜ t+h)|e T≥˜ t) is the marginal hazard of e T . We want to emphasize that the hazard rate αunfilt 0 of e T is in particular not conditioned on X because e T and X are independent. The fact that αunfilt 0 is a function of just one argument is the advantage of assuming the structural Model (1) because one can now easily derive an estimator for αunfilt 0 . For unique identification of ϕ , we choose the normalization ϕ(0) = 1 in the sequel. The advantage of this framework is that we can easily handle certain filtering schemes like right-censoring and left-truncation. If the observations of T are right-censored, we observe (Xi , T∗ i , δi) where T∗ i=min{Ti , C} is the censored value of Ti with respect to some censoring time C and δi=I(Ti<C) is the corresponding censoring variable. Moreover, suppose our observations to be left-truncated. In particular, we assume the special case of left-truncation Xi≤Ti . Hence, we use the counting process Nfilt i(t) = I(Ti≤t)δi with respect to its adapted filtration Ffilt i,t=σ({Nfilt i(s),Xi(s),s≤t})and with intensity λfilt i(t) = αfilt(t|Xi(t))Zfilt i(t), for exposure Zfilt i(t) = I(Xi≤t<T∗ i) . The conditional hazard has the same structure as in the last case, αfilt(t|x) = 1 ϕ(x)αfilt 0t ϕ(x). (2) Risks 2020,8, 3 5 of 17 The model in Equation (2) has been investigated for nonparametric regression in Linton et al. (2011) ; however, their model did not allow for right-truncation in run-off triangles. The next chapter introduces the operational time hazard model for right-truncation, which will be used in the sequel. 2.2. Model on the Run-Off Triangle with Right-Truncation When estimating future claim numbers, reserving departments in the non-life insurance industry work with data of historical claims aggregated in two dimensions: the accident date of the claim and the settlement delay, i.e., the time between accident date and payment to the policy holder. Note that, as in the chain-ladder method, we will not need the number of individuals under risk (the number of underwritten policies) for the estimation of future claim numbers. Therefore, we suppose the data contain only paid claims. We denote by X the underwriting date of the policy and by T the settlement delay. Hence, adopting the notation from above, we follow the settlement delay as survival time and the accident date as covariate on which we will condition. In Model (1) , the operational time function ϕ links a non-observable random delay e T to the observed settlement delay depending on the accident date. The independent delay e T can be seen as pure delay, cleared of all external factors. A value ϕ(x)> 1 implies a larger delay T with the heuristic that “time is running slower” and vice versa. This is best explained on the data set that is used in Section 6. The estimator of ϕ has values smaller than 1 for accident dates after January 2006 (see Section 6). This phenomenon is most likely due to the improved use of technology in the insurance company and has also been observed on the same data set in Lee et al. (2017). Instead of treating this as a special case, we let time run faster in this period and use the same delay throughout the whole range of accident dates. In particular, this does prevent discontinuities in the distribution of the delay. Since time was running faster for accident dates in 2006 and later, their actual delay effect e T , cleared of operational time, is larger (and sometimes even beyond the diagonal in the run-off triangle) in Figure 1. The operational time estimate already has a downwards trend for accident dates in 2004 and 2005; however, values in 2004 and and at the end of 2005 are larger than 1. On these dates, time was running slower in our model which is why the independent delay e Tfor early accidents is slightly shorter than in the original data. To adapt the operational time hazard Model (2) to the needs of our application, we assume pairs of observations (Ti , Xi) , i= 1, . . . , n , on the triangle I={(t , x)∈ S : 0 ≤x+t≤ T } . Hence, we have right-truncated observations of T because it now holds Ti≤ T − Xi . To circumvent this difficulty, we invert time and look at observations (T − Ti , Xi) which are left-truncated in T − Ti (Ware and DeMets 1976), so we can apply Model (2) . Note that our observations (Xi , Ti) only have the same distribution as (X , T) if conditioned on {X+T≤ T } . We do not assume any censoring in the following. As before, we focus on a counting process Ni(t) = I(T − Ti≤t) . The intensity of the time-reversed counting process Niwith respect to its natural filtration now equals λi(t) = α(t|x)Zi(t), where α(t|x) is the conditional hazard of T − T given X=x and Zi(t) = I(t+Xi≤ T , t≤ T − Ti) . In particular, we get α(t|x) = 1 ϕ(x)α0T − T − t ϕ(x), (3) with the marginal hazard α0(z) = limh↓0h−1P(T − e T∈[z , z+h)|T − e T≥z) of T − e T since T − e T and X are independent. We will also refer to α0 as a baseline hazard in the following. The reason for the unintuitive argument of α0 is that operational time is defined for T in “forward time”; however, α0 is the hazard rate in reversed time but cleared of operational time, c.f., Equation (2). Risks 2020,8, 3 6 of 17 012345678910 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 observed settlement delay T accident date X 1 100 200 count (a) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 012345678910 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 estimated underlying settlement delay T ~ accident date X 1 100 200 count (b) Figure 1. Original data and data with unobservable delay e T cleared of operational time. The operational time in ( b ) is estimated in Section 6. Claim counts are aggregated into monthly bins for visualization, and settlement delay is displayed in years. The red line represents the date of data collection and the green points are the date of data collection cleared of operational time effects (with respect to accident date). (a) original data; (b) data cleared of operational time. It can be easily derived that our model coincides with the structured model f(x , t) = f1(x)f2(tψ(x)) on the joint density f considered in Lee et al. (2017) for the choice f1 and f2 being the marginal densities of X , e T , respectively, and ψ(x) = ϕ(x)−1 . The advantage of our approach is that we only estimate two functions ϕ and α0 instead of three because we use the algorithm illustrated in Section 4to estimate the outstanding reserve. Hence, we only forecast the effect of given underwriting data X=x and do not estimate the distribution of X . For full inference on X , the roles of Tand Xhave to be swapped. Note that a multivariate extension of the operational time Model (1) for covariates X∈Rd and ϕ:Rd→R with d> 1 is possible and would result in the same hazard Model (3) with analogous baseline hazard α0 if right-truncation is well-defined (for instance T − Ti≤Xi,1 with Xi,1 being first component of Xi ). However, the estimation of ϕ and α0 explained in the next section would get rather involved including a d-dimensional numerical minimization for the estimation of ϕ. 3. Estimation of Baseline Hazard and Operational Time In this section, we show how to estimate the components ϕ and α0 and then combine the estimators into a structured estimator of the conditional hazard. We want to recall that the whole estimation procedure is done in reversed time T − T instead of T for the reporting delay. Hence, the following estimators are defined for Ni(t) = I(T − Ti≤t) and Zi(t) = I(t+Xi≤ T , t≤ T − Ti) . This technical difficulty is necessary because of the right-truncation described in the last section. However, it does not constitute an issue since, once the components are estimated, we can evaluate all functions at T − t to get the results for t . We also want to remark again that the underwriting date X is always considered in “forward time”. In the following, we see the conditional hazard α(t|x) as a function of two arguments α(t , x) and denote its estimators by ˆ α(t , x) . The unstructured hazard estimator in step 1 is analogously denoted by ˆ α[0](t,x). The proposed estimation procedure is as follows. The necessary expressions (7) and (10) , and the loss criterion (11) will be introduced below: Risks 2020,8, 3 7 of 17 1. Estimate the (unstructured) conditional hazard by ˆ α[0](t,x)through Equation (7). 2. Set ˆ ϕ[0]≡1 and r=1. 3. Estimate ˆ α[r] 0through Equation (10) using ˆ ϕ[r−1]. 4. Estimate ˆ ϕ[r]by minimizing the loss in (11) numerically for every xusing ˆ α[r−1] 0. 5. Repeat steps 3 and 4 for r=2, 3, 4, . . . until the convergence criterion ZT 0ˆ ϕ[r](x)−ˆ ϕ[r−1](x)2dx<10−5 is satisfied in iteration r∗. 6. Set the final conditional hazard estimator to ˆ α(t,x) = 1 ˆ ϕ(x)ˆ α0T − T − t ˆ ϕ(x), (4) for ˆ ϕ=ˆ ϕ[r∗], (5) ˆ α0=ˆ α[r∗] 0. (6) The final estimator in (non-reversed) “forward time” is set to ˆ αf(t,x) = ˆ α(T − t,x). Note that the first conditional estimator ˆ α[0](t , x) in step 1 is unstructured, which means that, in general, it does not satisfy Equation (3) . We also want to remark that the final estimator ˆ αf(t , x) is used to extrapolate claim numbers in the next section. Despite being more intuitive, it does not occur in a well-defined model because of the right-truncation T≤ T − X. All estimators ˆ α[0](t , x) , ˆ α[r] 0 , ˆ ϕ[r] are defined via integrated quadratic loss criteria and the hazard estimators ˆ α[0](t,x),ˆ α[r] 0have closed form representations as local linear kernel estimators. 3.1. Pre-Step: Unstructured Conditional Hazard We start with the unstructured conditional hazard estimator. Let Ui(t) = (t , Xi(t)) and u= (t , x) to simplify the notation. For convenience, we will also write u= (u1 , u2) . For any (t , x) , the local linear kernel hazard estimator ˆ α[0](t,x)is defined as the first component θ0minimizing the loss function L(θ0,θ1) = n ∑ i=1Z"1 εZs+ε sdNi(v)−θ0−θT 1(u−Ui(s))2 −ξ(ε)#Kb(u−Ui(s))Zi(s)ds, for θ0 , θ1∈R . Moreover, we use a two-dimensional kernel K and bandwidth b= (b1 , b2) for b1 , b2> 0 as well as the common notation Kb(u1 , u2) = b−1 1b−1 2K(u1/b1 , u2/b2) . The term ξ(ε) = ε−1Rs+ε s d Ni(s)2 is needed to make the expression well-defined. The loss criterion L results in the closed form solution ˆ α[0](t,x) = b O(t,x) b E(t,x), (7) for occurrence and exposure estimators b O(u1,u2) = 1 n n ∑ i=1Zh1−(u−Ui(s))D(u)−1c1(u)iKb(u−Ui(s))dNi(s), b E(u1,u2) = 1 n n ∑ i=1Zh1−(u−Ui(s))D(u)−1c1(u)iKb(u−Ui(s))Zi(s)ds, (8) Risks 2020,8, 3 8 of 17 where the components of the two-dimensional vector c1are c1j(u) = n−1n ∑ i=1ZKb(u−Ui(s))(uj−Uij(s))Zi(s)ds,j=0, . . . , d, and the entries (djk)j,k=1,2 of the (2×2)-dimensional matrix D(u)are given by djk(u) = n−1n ∑ i=1ZKb(u−Ui(s))(uj−Uij(s))(uk−Uik(s))Zi(s)ds. The closed form solution ˆ α[0] has been derived in Nielsen (1998). This paper focusses on the local linear kernel estimator because of its good performance at boundaries. The simpler and more intuitive (Nadaraya–Watson type) local constant hazard estimator is given in Appendix Aas an alternative. However, on bounded support, it is known to suffer from bias at boundaries (Nielsen 1998;Nielsen and Tanggaard 2001). 3.2. Estimation of Baseline Hazard Given Operational Time Starting with the pilot estimator ˆ ϕ[0]≡ 1, we calculate the first iteration ˆ α[1] 0 and then recursively update ˆ α[r] 0 making use of ˆ ϕ[r−1] . For the r -th iteration, we define ˆ α[r] 0 as the hazard rate α0 minimizing the loss l(α0,ϕ,ˆ α) = ZT 0ZT 0ˆ α(t,x)−1 ϕ(x)α0T − T − t ϕ(x)2 (ˆ α(t,x))−1b E(t,x)w(t,x)dxdt, (9) for operational time ϕ=ˆ ϕ[r−1] and the conditional hazard estimate ˆ α=ˆ α[0] . The loss function reflects the principle of minimizing a chi-square criterion (Berkson 1980) in which a least squares criterion is weighted by an estimate of the inverse of the asymptotic variance of ˆ α(t , x)) , here (ˆ α(t , x))−1b E(t,x) . The function w is a weighting function, which is used to ensure that the resulting hazard estimator is a ratio between an occurrence estimator and an exposure estimator. It will be specified later. The minimization of (9) has the analytic solution ˆ α0(t) = RT 0b E(ˆ ϕ[r−1] ∗(t,x),x)w(ˆ ϕ[r−1] ∗(t,x),x)dx RT 0b E(ˆ ϕ[r−1] ∗(t,x),x)w(ˆ ϕ[r−1] ∗(t,x),x)ˆ ϕ(x)−1ˆ α(ˆ ϕ[r−1] ∗(t,x),x)−1dx , where ˆ ϕ[r−1] ∗(t , x) = T−(T−t)ˆ ϕ[r−1](x) for t∈[ 0, T] . The derivation is analogous to Linton et al. (2011). Now, setting the weighting w(t,x) = ˆ ϕ[r−1](x)ˆ α(t,x)results in ˆ α[r] 0(t) = RT 0b O(ˆ ϕ[r−1] ∗(t,x),x)ˆ ϕ[r−1](x)dx RT 0b E(ˆ ϕ[r−1] ∗(t,x),x)dx . (10) The transformation ϕ∗(t , x) = T − (T − t)ϕ(x) = tϕ(x)+( 1 −ϕ(x))T adds the effect of operational time to occurrence and exposure estimators that were constructed with respect to e T . The function ˆ ϕ[r] ∗ is the estimate of ϕ∗ in the r -th iteration. Hence, we evaluate b O and b E at x but at the value of tthat was corrected with the operational time effect. It is worth pointing out that we do not get two marginal one-dimensional hazard estimator despite X and the cleared delay e T=T/ϕ(X) being independent. 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