Bootstrap Consistency for the Mack Bootstrap

Mack's distribution-free chain ladder reserving model belongs to the most popular approaches in non-life insurance mathematics. Proposed to determine the first two moments of the reserve, it does not allow to identify the whole distribution of the reserve. For this purpose, Mack's model is usually equipped with a tailor-made bootstrap procedure. Although widely used in practice to estimate the reserve risk, no theoretical bootstrap consistency results exist that justify this approach. To fill this gap in the literature, we adopt the framework proposed by Steinmetz and Jentsch (2022) to derive asymptotic theory in Mack's model. By splitting the reserve into two parts corresponding to process and estimation uncertainty, this enables - for the first time - a rigorous investigation also of the validity of the Mack bootstrap. We prove that the (conditional) distribution of the asymptotically dominating process uncertainty part is correctly mimicked by Mack's bootstrap if the parametric family of distributions of the individual development factors is correctly specified. Otherwise, this is not the case. In contrast, the (conditional) distribution of the estimation uncertainty part is generally not correctly captured by Mack's bootstrap. To tackle this, we propose an alternative Mack-type bootstrap, which is designed to capture also the distribution of the estimation uncertainty part. We illustrate our findings by simulations and show that the newly proposed alternative Mack bootstrap performs superior to the Mack bootstrap.


Introduction
In a non-life insurance business an insurer needs to build up a reserve to be able to meet future obligations arising from incurred claims. The actual sizes of the claims are unknown at the time the reserves have to be built, since the claims are incurred, but either not been reported yet or they have been reported, but not settled yet. This process of forecasting of outstanding claims is called reserving. An accurate estimation of the outstanding claims is crucial for pricing future policies and for the assessment of the solvency of the insurer. A popular and widely used technique in practice to forecast future claims is the Chain Ladder Model (CLM), which provides an algorithm to predict future claims. In this respect, the most popular model is the recursive model proposed by Mack (1993), which extends the CLM by allowing also the calculation of the standard deviation of the reserve.
Alternatively, frameworks based on general linear models (GLMs) considered e.g. in Renshaw and Verrall (1998) make use of over-dispersed Poisson and Log-normal distributions for modeling mean and variance of the reserve. However, such parametric assumptions are often restrictive and the knowledge of the first two moments of the reserve is not satisfactory for actuaries to draw sufficient conclusions about the reserve risk and the solvency of the insurance company.
The reserve risk is defined as the risk that the economic-valued reserve does not suffice to pay for all outstanding claims, which inevitably requires the knowledge of high quantiles of the reserve. For this purpose, England and Verrall (2006) proposed the Mack bootstrap which equips Mack's model with a tailor-made bootstrap procedure. Alternative bootstrap procedures for GLM-based setups have been addressed also in England and Verrall (1999;2006), England (2002 and Pinheiro et al. (2003). Without providing any consistency results, Björkwall et al. (2009) review these bootstrap techniques and suggest alternative non-parametric and parametric bootstrap procedures. Similarly, Björkwall et al. (2010) suggest bootstrap techniques for the separation method, that takes calendar year effects into account. In recent years, bootstrapbased approaches have been favored by many actuaries, because such methods usually produce plausible distributions in practice. However, as demonstrated by Gibson et al. (2007) and Bruce et al. (2008), Mack's model and GLM-type models in combination with the bootstrap do not produce satisfactory results in certain situations. In this regard, refined approaches have been proposed to improve the finite sample performance. For example, Verdonck and Debruyne (2011) investigate the influence of outliers for the parameter estimation in the GLM framework and calculate its leverage on the CLM. Hartl (2010) propose to use deviance residuals instead of Pearson residuals for the GLM framework. Tee et al. (2017) provide an extensive case study for bootstrapping the GLM using a (over-dispersed) Poisson model, the Gamma model and the Log-normal model in combination with different residual types. Peremans et al. (2017) propose a robust bootstrap procedure in a GLM setting based on M-estimators using influence functions. Peters et al. (2010) compare the Mack bootstrap with a Bayesian bootstrap.
Nevertheless, already for the original Mack bootstrap method, the existing literature lacks a deeper and mathematically rigorous understanding. For this purpose, it is desirable to provide a suitable theoretical framework to be able to justify the application of the Mack bootstrap.
Only recently, Steinmetz and Jentsch (2022) proposed a suitable theoretical (stochastic and asymptotic) framework, which allows the derivation of conditional and unconditional asymptotic theory for the reserve in Mack's model. They split the reserve (centered around its best estimate) into two parts, that carry the process uncertainty and the estimation uncertainty, respectively. This allows to derive unconditional limiting distributions for both parts of the reserve, and when conditioning on the latest observed cumulative claims As risk reserving is generally a prediction task, these conditional limiting distributions serve well as benchmarks for the corresponding Mack bootstrap distributions, when addressing the question of bootstrap consistency. While the conditional limiting distribution of the estimation uncertainty part turns out to be Gaussian under mild regularity conditions and when properly inflated, the conditional limiting distribution of the process uncertainty part will be generally non-Gaussian. Considering both parts jointly, the process uncertainty part dominates asymptotically, which leads to a non-Gaussian limiting distribution of the reserve in total.
In this paper, we adopt the theoretical framework introduced in Steinmetz and Jentsch (2022) to investigate the long-standing question of Mack bootstrap consistency. Our contributions are twofold. First, we derive bootstrap asymptotic theory for both parts of the (centered) Mack bootstrap reserve corresponding to process uncertainty and estimation uncertainty, respectively.
We prove that the (conditional) bootstrap distribution of the asymptotically dominating process uncertainty part is correctly mimicked if the parametric family of distributions of the Mack bootstrap individual development factors is correctly specified. Otherwise, this will be generally not the case. In contrast, the (conditional) distribution of the estimation uncertainty part is generally not correctly captured. Second, inspired from our asymptotic findings, we propose an alternative Mack-type bootstrap, which is designed to capture also the distribution of the estimation uncertainty part.
The paper is organized as follows. Section 2 introduces the required notation and assumptions for the CLM, discusses parameter estimation in Mack's model, and provides the asymptotic and stochastic framework of Steinmetz and Jentsch (2022). In Section 3, we discuss the Mack bootstrap approach as proposed by England and Verrall (2006). In Section 4, we summarize the (conditional) asymptotic results from Steinmetz and Jentsch (2022) for the process uncertainty and estimation uncertainty terms in Section 4.1, which will serve as benchmarks for the Mack bootstrap results. Then, in Section 4.2, we derive bootstrap asymptotic theory for both parts of the (centered) Mack bootstrap reserve corresponding to process uncertainty and estimation uncertainty, respectively. Based on these results, we propose an alternative Mack-type bootstrap in Section 5 and derive its asymptotic properties in Section 6. We illustrate our findings in simulations in Section 7 and show that the newly proposed alternative Mack-type bootstrap performs superior to the original Mack bootstrap in finite samples. Section 8 concludes. All proofs, auxiliary results and additional simulations are deferred to the appendix.

The Chain Ladder Model
Reserves are the major part of the balance sheet for non-life insurance companies such that their accurate prediction is crucial. For this purpose, insurers summarize all observed claims Development Year j 0 1 2 · · · I − 1 I Accident Year i 0 C0,0 C0,1 C 0,I−1 C 0,I 1 C1,0 · · · C 1,I−1 C 1,I · · · · · · · · · · · · · · · · · · I − 1 C I−1,0 C I−1,1 · · · I C I,0 C I,1 C I,I Table 1. Observed upper loss triangle D I (upper-left triangle; white and orange) with accident years (vertical axis), development years (horizontal axis), diagonal Q I (orange), and unobserved lower loss triangle D c I (lower-right triangle; green). of a business line in a loss triangle (upper-left triangle in Table 1). Its entries, the cumulative amount of claims C i,j , are sorted by their years of accident i (vertical axis) and their years of occurrence j (horizontal axis), where i, j = 0, . . . , I with i + j ≤ I. Hence, the (observed) loss triangle contains all cumulative claims C i,j that have already been observed up to calendar year I. It constitutes the available data basis and is denoted by The total aggregated amount of claims of the same calendar year k with k = 0, . . . , I are lying on the same diagonal (from lower-left to upper-right corner) of the loss triangle. We denote these diagonals by Q k = {C k−i,i |i = 0, . . . , k}. In this setup, I is the current calendar year corresponding to the most recent accident year and development period such that the diagonal Q I (orange diagonal in Table 1) summarizes the latest cumulative claim amounts collected in year I.
noting that R 0,I = C 0,I − C 0,I = 0 by construction. Hence, for each accident year i and being in calendar year I, to get an estimate of R i,I , we have to predict the unobserved ultimate claim C i,I . Starting from C i,I−i , this is done by predicting sequentially all future, yet (at time I) unobserved claims {C i,j |j = I − i + 1, . . . , I}. By doing this for all i = 0, . . . , I, the whole unobserved lower loss triangle D c I has to be predicted, and by summing-up all predictions for R i,I , we get a prediction also for R I . However, to make the CLM setup above accessible for the derivation of asymptotic theory for predictive inference, Steinmetz and Jentsch (2022) introduced a suitable stochastic and asymptotic framework for Mack's model, which is adopted here as well and will be described in the following.

Asymptotic framework for reserve prediction.
With the loss triangle D I at hand, an asymptotic analysis conditional on the diagonal Q I , which contains the most up-to-date information in the loss triangle, is of much interest for insurers. However, for this purpose, we will not rely on a seemingly "natural" asymptotic frameworkbased on I → ∞, where increasing I means adding new diagonals Q I+h = {C I−i,i |i = 0, . . . , I + h}, h ≥ 1 to the loss triangle D I (see Table 2, upper panel). Instead, as common in predictive inference (see e.g. Paparoditis and Shang (2021)), we employ a different asymptotic framework throughout this paper. That is, we keep the latest cumulative claims in D I , that is, Q I , fixed and let D I grow by adding new rows of cumulative claims {C −h,i |i = 0, . . . , I + h}, h ≥ 1 (see Table 2, lower panel). Nevertheless, both versions of differently growing loss triangles displayed in Table 2 are equal in distribution. In what follows, all asymptotic results are derived under the framework that a sequence of (upper) loss triangles D I,n = {C i,j |i = −n, . . . , I, j = 0, . . . , I + n, −n ≤ i + j ≤ I} , n ∈ N 0 = {0, 1, 2, . . .}, denote the corresponding diagonals. Note that D I,0 = D I , Q I,0 = Q I and that D I,n (and Q I,n ) is obtained by sequentially adding n rows of lengths I + 2, I + 3, . . . , I + n + 1, respectively, on top to D I (see Table 2, lower panel). As before, for all n ∈ N 0 , we augment the (observed) upper loss triangle D I,n by an unobserved lower triangle D c I,n = {C i,j |i = −n, . . . , I, j = 0, . . . , I + n, i + j > I} that contains all future claims that have not been observed (yet) up to time I. Further, according to (2.2), the aggregated total amount of the reserve is denoted by where R i,I+n = C i,I+n − C i,I−i , n ∈ N 0 and R −n,I+n = C −n,I+n − C −n,I+n = 0 by construction.
While we keep I and n fixed in the expositions of the remainder of this section and of Section 3, we let n → ∞ to derive the limiting distribution of the reserve in Section 4.
By adopting the notion of the asymptotic framework described in Section 2.1, the conditions of Mack's Model originally proposed in Mack (1993) can be summarized as follows.
Assumption 2.1 (Mack's Model). For any n ∈ N 0 , let C I,n = (C i,j , i = −n, . . . , I, j = 0, . . . , I + n) denote random variables on some probability space (Ω, A, P ) and suppose the following holds: (i) There exist so-called development factors f 0 , . . . , f I+n−1 such that . . , I, j = 0, . . . , I + n − 1. (2.6) (ii) There exist variance parameters σ 2 0 , . . . , σ 2 I+n−1 such that V ar(C i,j+1 |C i,j ) = σ 2 j C i,j , i = −n, . . . , I, j = 0, . . . , I + n − 1. (2.7) (iii) The cumulative claims are stochastically independent over the accident years i = −n, . . . , I, that is, the cumulative claim matrix C I,n consists of independent rows C i,• = (C i,0 , . . . , C i,I+n ), i = −n, . . . , I. For any n ∈ N 0 , based on the available data D I,n , all development factors f j and variance parameters σ 2 j for j = 0, . . . , I + n − 1 are unknown and have to be estimated from D I,n . The development factors f 0 , . . . , f I+n−1 can be (consistently) estimated by f 0,n , . . . , f I+n−1,n , where According to Mack (1993), these estimators are unbiased, i.e. E( f j,n ) = f j , and pairwise uncorrelated, i.e. Cov( f j,n , f k,n ) = 0 for all j = k. By plugging-in the f j,n 's, the best estimate of the ultimate claim C i,I+n (point predictor) of the ultimate claim C i,I+n is calculated by Consequently, given C i,I−i , the best estimate R i,I+n of the reserve R i,I+n is given by . . , I, and the best estimate R I,n of the total reserve R I,n defined in (2.5) computes to (2.10) noting that R −n,I+n = 0 due to I+n−1 j=I+n f j,n := 1. Furthermore, Mack (1993) proposed to estimate the variance parameters σ 2 0 , . . . , σ 2 I+n−1 by , j = 0, . . . , I + n − 2, (2.11) which are unbiased estimators, i.e. E( σ 2 j,n ) = σ 2 j , and by setting σ 2 I+n−1,n = 0. Of particular interest is in the distribution of the difference of the stochastic (unobserved) reserve R I,n and its best estimate R I,n (based on the observed data D I,n ), which is denoted as the predictive root of the reserve in the following. That is, by combining (2.5) and (2.10), it computes to While a common approach to approximate an unknown (finite sample) distributions is the derivation of asymptotic theory, Mack's conditions summarized in Assumption 2.1 are not (yet) sufficient to establish limiting distributions for the predictive root of the reserve R I,n − R I,n .
2.3. A fully-described stochastic framework of Mack's Model.
Following Steinmetz and Jentsch (2022, Section 2.2), to establish a theoretical framework sufficient to derive asymptotic theory for parameter estimators f j,n and σ 2 j,n , which finally also enables the derivation of the limiting distributions of the predictive root of the reserve R I,n − R I,n , we introduce Assumptions 2.2, 2.3, and 2.5 on the stochastic mechanism that generates the cumulative claim matrix C I,n , and Assumption 2.4 on the sequences of development factors and of variance parameters. They resemble the Assumptions 2.2, 2.3 and 3.3 as well as Assumption 4.1 in Steinmetz and Jentsch (2022), respectively. This framework will also allow to rigorously investigate consistency properties of the Mack bootstrap in Section 4.
The first assumption addresses the initial claims, i.e. the first column of C I,n (and of D I,n ).
Assumption 2.2 (Initial claims). Let the initial claims (C I−n,0 , n ∈ N 0 ) be independent and The independence of the initial claims is a direct consequence of Assumption 2.1 (iii). In addition, Assumption 2.2 imposes an identical distribution for the initial claims. In practice, the condition on the support [1, ∞) of C i,0 is not restrictive and can be relaxed to C i,0 being bounded away from zero.
In view of E(C i,j+1 |C i,j ) in (2.6), suppose that the cumulative claims C i,j+1 are recursively defined by . . , I, j = 0, . . . , I + n − 1, (2.13) where the individual development factors F i,j are assumed to fulfill the following condition.
for some ≥ 0 such that F i,j and F k,l are independent given (C i,j , C k,l ) for all (i, j) = (k, l) with (2.14) Note that Mack's original model setup in Assumption 2.1 is implied by Assumptions 2.2 and 2.3 together. Also note that the stochastic mechanism determined by (2.13) and Assumption 2.3 are assumed for the whole cumulative claim matrix C I,n . However, recall that only those C i,j in C I,n are observed that are contained in the upper loss triangle D I,n . Hence, by using the multiplicative relationship in (2.13), we have also perfect knowledge of F i,j , i = −n, . . . , I − 1, According to Lemma 2.4 in Steinmetz and Jentsch (2022), Assumptions 2.2 and 2.3 allow to derive formulas for the (unconditional) means and variances of C i,j , i = −n, . . . , I, j = 0, . . . , I + n leading to where µ 0 and τ 2 0 are defined in Assumption 2.2. Together with Assumption 2.4 below, according to Lemma 4.2 in Steinmetz and Jentsch (2022), both sequences (µ j , j ∈ N 0 ) and (τ 2 j , j ∈ N 0 ) are non-negative, monotonically non-decreasing, and converging with µ j → µ ∞ and τ 2 Assumption 2.4 (Development Factors and Variance Parameters). Letting n → ∞ in the setup of Assumptions 2.2 and 2.3 leads to The conditions imposed on the sequences of development factors (f j , j ∈ N 0 ) and variance parameters (σ 2 j , j ∈ N 0 ) in Assumption 2.4 are rather mild. In practice, each claim has a finite, but possibly unknown horizon until it is finally settled, which varies by the insurance lines. Altogether, as done in Steinmetz and Jentsch (2022, Section 3.1), this setup allows to derive central limit theorems (CLTs) for (smooth functions of) the parameter estimators f j,n for n → ∞.
According to Steinmetz and Jentsch (2022, Section 3.2), the following additional assumption has to be imposed to derive a CLT also for σ 2 j,n . Although the distributional properties of σ 2 j,n do not show asymptotically in the distribution of the reserve, √ I + n-consistency of σ 2 j,n as obtained in Steinmetz and Jentsch (2022, Theorem 3.5) is required for establishing the bootstrap asymptotic theory in Section 4.
Assumption 2.5 (Higher-order conditional moments of individual development factors). For all i ∈ Z, i ≤ I, j ∈ N 0 , suppose that conditional on C i,j , the third and fourth (central) moments exist and are finite, respectively.
Using (2.13), conditional on Q I,n , the reserve R I,n can be written as Hence, by plugging-in (2.9) and (2.16), the predictive root of the reserve from (2.12) becomes (2.17) where we flipped the index i to I − i in the last step.

Mack's Bootstrap Scheme
The Mack bootstrap, introduced by England and Verrall (2006), equips Mack's Model with a resampling procedure to estimate the whole distribution of the (predicted) reserve. It is very popular and widely used in practice as it describes a rather simple to implement algorithm to estimate the reserve risk by estimating high quantiles of the reserve distribution.
As proposed by England and Verrall (2006), to mimic the distribution of the predictive root of the reserve R I,n − R I,n , the Mack bootstrap constructs a certain bootstrap version R * I,n − R I,n of it. On the one hand, this bootstrap predictive root relies on the same best estimate of the reserve and centers R * I,n also around R I,n . On the other hand, it constructs a certain double-bootstrap version of the reserve R I,n , that is R * I,n , by combining two complementing (non-parametric and parametric) bootstrap approaches for resampling the individual development factors in the upper triangle and in the lower triangle: (i) First, a non-parametric residual-based bootstrap (see Step 4 below) is applied to construct bootstrap individual development factors F * i,j , j = 0, . . . , I +n−1, i = −n, . . . , I − j − 1, that is, for the upper triangle, in order to get bootstrap development factor estimators f * j,n , j = 0, . . . , I + n − 1. (ii) Second, the bootstrap development factor estimators f * j,n from (i) together with a parametric bootstrap (see Step 5 below) are used to construct also bootstrap individual development factors F * i,j , i = −n, . . . , I, j = 0, . . . , I + n − 1 and i + j ≥ I, that is for the lower triangle. For this purpose, a parametric family of (conditional) bootstrap distributions has to be chosen.
Finally, as we are dealing with a prediction problem when estimating the reserve risk, the limiting properties of the predictive root of the reserve conditional on the latest observed cumulative claims are relevant and have to be mimicked by a suitable resampling procedure. For this purpose, the Mack bootstrap is employed to estimate the conditional distribution of R I,n − R I,n given Q I,n by the conditional bootstrap distribution of R * I,n − R I,n given Q * I,n = Q I,n and D I,n .

Mack's Bootstrap Algorithm.
With the upper triangle D I,n at hand, Mack's bootstrap algorithm is defined as follows: Step 1. Estimate the development factors f j and the variance parameters σ 2 j from D I,n by computing f j,n and σ 2 j,n for j = 0, . . . , I +n−1 as defined in (2.8) and (2.11), respectively.
Step 4. Define the bootstrap individual development factors Step 5. Choose a parametric family for the (conditional) bootstrap distributions of F * i,j given C * i,j , D I,n and F * I,n such that F * i,j > 0 a.s. with where E * (·) := E * (·|D I,n ), V ar * (·) := V ar * (·|D I,n ), etc. denote the Mack bootstrap mean, variance, etc., respectively, that is, conditional on the data D I,n . Then, given Q * I,n = Q I,n , generate the bootstrap ultimate claims C * i,I+n and the reserves R * i,I+n = C * i,I+n − C * i,I−i for i = −n, . . . , I using the recursion Step 6. Compute the bootstrap total reserve R * I,n = I+n i=0 R * I−i,I+n and its bootstrap predictive root Step 7. Repeat Steps 3 -6 above B times, where B is large, to get bootstrap predictive roots (R * I,n − R I,n ) (b) , b = 1, . . . , B, and denote by q * (α) the α-quantile of their empirical distribution.
1 Note that σ 2 I+n−1,n = 0 by construction such that r−n,I+n−1 is excluded in (3.1) such that (at most) (I + n + 1)(I + n)/2 − 1 = ((I + n + 1)(I + n) − 2)/2 residuals can be computed. If σ 2 j,n = 0 holds also for other j, the corresponding residuals are excluded in (3.1) as well and the formulas for r and s have to be adjusted accordingly. In the following, for notational convenience, we assume that only σ 2 I+n−1,n = 0 and σ 2 j,n > 0 holds for all j = 0, . . . , I + n − 2 and all n ∈ N0.

Remark 3.1 (On Mack's bootstrap proposal).
(i) While the Mack bootstrap predictive root of the reserve R * I,n − R I,n uses the same best estimate R I,n for centering (as in R I,n − R I,n ), it relies on a certain type of doublebootstrap version R * I,n of the total reserve R I,n , which employs f * j,n instead of just f j,n , but uses σ 2 j,n . However, although E * (F * i,j |C * i,j , D I,n , F * I,n ) = f * j,n holds, we still have E * (F * i,j |C * i,j ) = f j,n for the lower triangle individual development factors. In contrast, for the variances, we have V ar (ii) Due to the fixed-design bootstrap in Step 4, which does not generate bootstrap cumulative claims C * i,j (and consequently no bootstrap upper loss triangle D * I,n ), but only F * i,j 's, the bootstrap development factor estimators f * j,n and f * k,n defined in (3.3) are independent for j = k conditional on D I,n . This is on contrast to the development factor estimators f j,n and f k,n , which are asymptotically independent for j = k, but only uncorrelated in finite samples such that E( f 2 j,n f 2 k,n ) < 0 for j = k. (iii) The non-parametric bootstrap used to construct the f * j,n 's in Step 4 uses residuals, but according to Assumption 2.2 and 2.3, there are no errors in Mack's model that are approximated by these residuals. In fact, each (possibly parametric) bootstrap proposal that successfully mimics the first and second conditional moments of C i,j+1 given C i,j will correctly mimic the limiting distribution of the f j,n 's.
(iv) In view of the discussion above, a fully parametric implementation that uses the same parametric family from Step 5 also in Step 4 to get bootstrap development factors F * i,j 's can be used.
(v) A fully non-parametric approach that uses the non-parametric bootstrap from Step 4 also in Step 5 is thinkable, but would suffer from issues arising from potentially negative F * i,j 's leading to a reduced finite sample performance.

Asymptotic Theory for the Mack Bootstrap
Although the Mack bootstrap as proposed by England and Verrall (2006) and described in Section 3 is widely used in practice for reserve risk estimation, limiting results that confirm its consistency are still missing in the literature. In this section, based on the asymptotic and stochastic framework described in Section 2, we derive asymptotic theory for the Mack bootstrap, which enables a rigorous investigation of its consistency properties.
The Mack bootstrap is designed to mimic the distribution of the predictive root of the reserve R I,n − R I,n conditional on Q I,n based on the bootstrap distribution of the corresponding Mack bootstrap predictive root of the reserve R * I,n − R I,n conditional on Q * I,n = Q I,n and D I,n . Hence, a closer inspection of both expressions is advisable. Picking-up the representation of the predictive root of the reserve R I,n − R I,n in (2.17), it can be decomposed into two additive parts that account for the prediction error and the estimation error, respectively. Precisely, by subtracting and adding I+n i=0 C I−i,i I+n−1 j=i f j , we get where (R I,n − R I,n ) 1 represents the process uncertainty (that carries the process variance) and (R I,n − R I,n ) 2 the estimation uncertainty (that carries the estimation variance).
Similarly, for the Mack bootstrap predictive root of the reserve R * I,n − R I,n from (3.5), by subtracting and adding I+n i=0 C * where (R * I,n − R I,n ) 1 and (R * I,n − R I,n ) 2 are the Mack bootstrap versions of (R I,n − R I,n ) 1 and (R I,n − R I,n ) 2 , respectively.
As main interest is in the distribution of the predictive root of the reserve R I,n − R I,n conditional on Q I,n , in view of the decompositions (4.1) and (4.2), it is instructive to first consider separately the (limiting) distributions of (R I,n − R I,n ) 1 and (R I,n − R I,n ) 2 conditional on Q I,n , respectively. They will serve as valuable benchmark distributions for the investigation of consistency properties of the Mack bootstrap in Section 4.2. Such asymptotic results have been established in Steinmetz and Jentsch (2022, Section 4). We will briefly summarize the relevant conditional limiting distributions below in Section 4.1.

4.1.
Conditional asymptotics for the predictive root of the reserve.
In the following, we review the conditional asymptotic results established in Steinmetz and Jentsch (2022, Theorems 4.3, 4.10, 4.12, and Corollary 4.13) separately for the process uncertainty term (R I,n − R I,n ) 1 in Section 4.1.1, for the estimation uncertainty term (R I,n − R I,n ) 2 in Section 4.1.2, as well as jointly for R I,n − R I,n in Section 4.1.3, respectively. 4.1.1. Conditional asymptotics for reserve prediction: process uncertainty.
Based on Theorem 4.3 from Steinmetz and Jentsch (2022), the following theorem provides the limiting distribution of the process uncertainty term (R I,n − R I,n ) 1 conditional on Q I,n .
Theorem 4.1 (Asymptotics for (R I,n − R I,n ) 1 conditional on Q I,n ). Suppose Assumptions 2.2, 2.3 and 2.4 hold. Then, as n → ∞, conditionally on Q I,n , (R I,n − R I,n ) 1 converges in L 2 -sense to the non-degenerate random variable (R I,∞ − R I,∞ ) 1 . That is, we have The (conditional) L 2 -convergence result in Theorem 4.1 immediately implies also (conditional) convergence in distribution. That is, for n → ∞, we have Moreover, according to Theorem 4.1 (see also the discussion in (Steinmetz and Jentsch 2022, Remark 4.4)), the conditional limiting distribution G 1 |Q I,∞ will be typically non-Gaussian and depending on the (conditional) distribution of the individual development factors F i,j |C i,j . 4.1.2. Conditional asymptotics for reserve prediction: estimation uncertainty.
In comparison to the conditional limiting result for (R I,n − R I,n ) 1 displayed in Theorem 4.1, the derivation of asymptotic results for (R I,n − R I,n ) 2 is rather different and also much more cumbersome. In particular, to obtain non-degenerate limiting distributions, we have to inflate (R I,n − R I,n ) 2 by √ I + n + 1 and the obtained (Gaussian) distribution relies on CLTs for (smooth functions of) development factor estimators f j,n established in (Steinmetz and Jentsch 2022, Section 3 and Appendix C). For the derivation of asymptotic theory, conditional on Q I,n , it is instructive to further decompose (R I,n − R I,n ) 2 to get where (R I,n − R I,n ) (1) 2 is measurable wrt Q I,n and f j,n (Q I,n ) := µ In addition to the condition on the support and the variance parameters in Assumption 4.2, a regularity condition for the backward conditional distribution of cumulative claim C i,j given fulfilled such that, for all K ∈ N 0 , k ≥ 0 and j, j 1 , j 2 ∈ {0, . . . , K}, j 1 ≤ j 2 , we have While Mack's model is designed to generate loss triangles in a rather simple forward way according to the recursion (2.13), which allows to easily calculate forward conditional means 2 is measurable with respect to Q I,n , Assumptions 4.2 and 4.3 allow to prove asymptotic normality of √ I + n + 1(R I,n − R I,n ) (2) 2 conditional on Q I,n .
According to Theorem 4.4(ii), in contrast to G 1 |Q I,∞ in Theorem 4.1, the limiting distribution G (2) 2 |Q I,∞ is Gaussian. Together with Theorem 4.4(i), conditional on Q I,n , the estimation uncertain term

4.1.3.
Conditional asymptotics for the whole predictive root of the reserve.
By combining the results derived for (R I,n − R I,n ) 1 and (R I,n − R I,n ) 2 in Theorems 4.1 and 4.4, respectively, joint asymptotic results for R I,n − R I,n conditional on Q I,n can also be established. According to Theorem 4.4, (R I,n − R I,n ) 2 requires an inflation factor √ I + n + 1 to get convergence to a non-degenerate limiting distribution. As this is not the case for (R I,n − R I,n ) 1 in Theorem 4.1, the process uncertainty term (R I,n − R I,n ) 1 asymptotically dominates the predictive root of the reserve R I,n − R I,n .
Hence, we can conclude that asymptotic normality of the (predictive root of the) reserve does generally not hold, which casts the common practice to use a normal approximation for the reserve in Mack's model into doubt. Moreover, the shape of G 1 |Q I,∞ does depend on the true (conditional) distribution family of the individual development factors F i,j |C i,j .

4.2.
Conditional bootstrap asymptotics for the Mack bootstrap predictive root of the reserve.
In view of the decomposition R I,n − R I,n = (R I,n − R I,n ) 1 + (R I,n − R I,n ) 2 in (4.1) and the conditional limiting distributions of (R I,n − R I,n ) 1 and (R I,n − R I,n ) 2 gathered in Section 4.1, it is instructive to consider the corresponding Mack bootstrap quantities (R * I,n − R I,n ) 1 and (R * I,n − R I,n ) 2 from (4.2) and check whether they are correctly mimicking such limiting distributions. While (R I,n − R I,n ) 1 and (R I,n − R I,n ) 2 are analyzed conditional on Q I,n , the bootstrap quantities (R * I,n − R I,n ) 1 and (R * I,n − R I,n ) 2 have to be considered conditional on Q * I,n = Q I,n , but also on D I,n .
For the derivation of bootstrap asymptotics, we have to impose additional smoothness properties of the parametric family of (conditional) distributions of the individual development factors to assure that consistent estimation of development factors and variance parameters implies also consistent estimation of the whole distribution.
Assumption 4.6 (Parametric family of (conditional) distributions of F i,j ). The (conditional)

belongs to a parametric family of distributions H
such that: As the limiting distribution derived in Theorem 4.1 is generally non-Gaussian and depends on the (conditional) distribution (family) of the individual development factors, we require also that the bootstrap individual development factors F * i,j , i = −n, . . . , I, j = I − i, . . . , I + n − 1, that is, for the lower triangle, follow the true parametric family of (conditional) distributions as the F i,j 's according to Assumption 4.6.
Step 5 of the Mack Bootstrap in Section 3.1 belongs to the true parametric family of (conditional) distributions H used to generate F i,j |C i,j according to Assumption 4.6. That is, we have Together with the setup of Theorem 4.1, the Assumptions 4.6 and 4.7 allow to prove the following result.
Theorem 4.8 (Bootstrap asymptotics for (R * I,n − R I,n ) 1 conditional on Q * I,n = Q I,n and D I,n ). Suppose Assumptions 2.2, 2.3, 2.4, 2.5, 4.6, and 4.7 hold. Then, as n → ∞, conditionally on Q * I,n = Q I,n and D I,n , (R * I,n − R I,n ) 1 converges in distribution to G 1 |Q I,∞ in probability, which is the (limiting) distribution of (R I,∞ − R I,∞ ) 1 |Q I,∞ according to (4.6) described in Theorem 4.1. Moreover, for all n ∈ N 0 , it holds E * ((R * I,n − R I,n ) 1 |Q * I,n = Q I,n ) = 0 and, for n → ∞, we have in probability, where L * (·) denotes a bootstrap distribution conditional on D I,n , and d 2 is the Mallows metric, that is defined for two distributions G and H as where the infimum is taken over all joint distributions of (X, Y ) with marginals X ∼ G and Y ∼ H.

Conditional bootstrap asymptotics for reserve prediction: estimation uncertainty.
In view of the decomposition (R I,n − R I,n ) 2 = (R I,n − R I,n ) (4.7), for the derivation of corresponding bootstrap asymptotic theory, it is seemingly instructive to further decompose also its bootstrap counterpart (R * I,n − R I,n ) 2 in the same way conditional on Q * I,n = Q I,n and D I,n . That is, by taking into account the specific definition of f * j,n in (3.3), we get where we used that C i,j is measurable with respect to D I,n and that F * i,j is stochastically independent of the condition C * i,I−i = C i,I−i given D I,n . This is because the Mack bootstrap relies on a fixed-design approach based on the C i,j 's instead of recursively generating C * i,j to get a whole bootstrap loss triangle D * I,n . Altogether, using E * (F * i,j ) = f j,n , we get 2 . Hence, in comparison to (R I,n − R I,n ) 2 , which was decomposed into two parts (R I,n − R I,n ) (1) 2 and (R I,n − R I,n ) (2) 2 , such an analogous decomposition of (R * I,n − R I,n ) 2 does not exist. However, for n → ∞, it remains to check the limiting properties of (R * I,n − R I,n ) 2 in the following. In contrast to the derivation of the conditional limiting result obtained in Theorem 4.4(ii), which relies on conditional CLTs for the development factor estimators f j,n as stated in (Steinmetz and Jentsch 2022, Appendix C), the derivation of the limiting properties of (R * I,n − R I,n ) 2 rely on unconditional bootstrap CLTs for the Mack bootstrap development factor estimators f * j,n , that is, without conditioning on Q * I,n = Q I,n . For this purpose, to prove asymptotic normality for the f * j,n 's by justifying a Lyapunov condition, we have to impose additional regularity conditions on the estimators for the development factors and variance parameters.
Assumption 4.9 (Uniform boundedness condition). Suppose that the development factor estimators f j,n , j = 0, . . . , I + n − 1 and the variance parameter estimators σ 2 j,n , j = 0, . . . , This allows for the following asymptotic result for the Mack bootstrap estimation uncertainty part.
Consequently, as n → ∞, we have because the limiting normal distribution of √ I + n + 1(R * I,n − R I,n ) 2 conditional on Q * I,n = Q I,n and D I,n deviates in its (zero) mean and variance Ξ(Q I,∞ ) from that of √ I + n + 1(R I,n − R I,n ) 2 conditional on Q I,n , which has mean Q I,∞ , Y ∞ and variance Ξ(Q I,∞ ) according to Theorem 4.4.

4.2.3.
Conditional bootstrap asymptotics for the whole predictive root of the reserve.
As in Section 4.1.3, combining the results for (R * I,n − R I,n ) 1 and (R * I,n − R I,n ) 2 from Theorems 4.8 and 4.10, respectively, joint asymptotics for R * I,n − R I,n conditional on Q * I,n = Q I,n and D I,n can be obtained.
Theorem 4.11 (Bootstrap asymptotics for R * I,n − R I,n conditional on Q * I,n = Q I,n and D I,n ). Suppose the assumptions of Theorems 4.8 and 4.10 hold. Then, conditional on Q * I,n = Q I,n and D I,n , (R * I,n − R I,n ) 1 and (R * I,n − R I,n ) 2 are uncorrelated, and R * I,n − R I,n |(Q * I,n = Q I,n , D I,n ) converges in distribution to G 1 |Q I,∞ . That is, we have in probability.
As already observed in Theorem 4.4 for the estimation uncertainty term (R I,n − R I,n ) 2 , its Mack bootstrap version requires also an inflation factor √ I + n + 1 to establish convergence towards a non-degenerate limiting distribution. As this is not the case for the process uncertainty term (R I,n − R I,n ) 1 in Theorem 4.1 and its bootstrap version in Theorem 4.8, the process uncertainty terms will asymptotically dominate the predictive roots R I,n − R I,n and R * I,n − R I,n . Hence, although the limiting bootstrap distribution of √ I + n + 1(R * I,n − R I,n ) 2 conditional on Q * I,n = Q I,n and D I,n in Theorem 4.10 does not correctly mimic the corresponding limiting behavior of √ I + n + 1(R I,n − R I,n ) 2 conditional on Q I,n in Theorem 4.4, the whole bootstrap predictive root R * I,n − R I,n still mimics the limiting distribution of the predictive root R I,n − R I,n correctly.
Hence, in view of the concepts of asymptotic validity and asymptotic pertinence of a bootstrap prediction approach discussed in Pan and Politis (2016), the Mack bootstrap can be regarded as asymptotically valid, but not as asymptotically pertinent under the stated conditions. Remark 4.12 (On the asymptotic results for the Mack bootstrap).
(i) A closer inspection of the decompositions in (4.1) and (4.2) reveals some inconsistencies: -While a term based on products of f j 's is added to and subtracted from R I,n − R I,n to get (R I,n − R I,n ) 1 and (R I,n − R I,n ) 2 , a term using products of f * j,n 's instead of f j,n 's, which would be the natural choice, is added to and subtracted from R * I,n − R I,n to get (R * I,n − R I,n ) 1 and (R * I,n − R I,n ) 2 . -Consequently, while (R I,n − R I,n ) 1 relies on products of F i,j 's centered around products of f j 's, its Mack bootstrap version (R * I,n − R I,n ) 1 relies on products of F * i,j 's, which are not (naturally) centered around products of f j,n 's, but around products of f * j,n 's. -Moreover, while (R I,n − R I,n ) 2 relies on differences between products of development parameters f j and products of their estimators f j,n , its Mack bootstrap version (R * I,n − R I,n ) 2 relies on differences between products of bootstrap development factor estimators f * j,n and products of estimators f j,n . Hence, the sign of (R * I,n − R I,n ) 2 is flipped in comparison to (R I,n − R I,n ) 2 . This may have a negative effect in finite samples, but as the limiting conditional distribution is Gaussian and hence symmetric, this will not be an issue asymptotically.
-According to the latter observation, also the terms (R * I,n − R I,n ) (1) 2 and (R * I,n − R I,n ) (2) 2 in the seemingly natural decomposition of the bootstrap estimation uncertainty term in (4.13) are switched in comparison to (R I,n − R I,n ) (1) 2 and (R I,n − R I,n ) (2) 2 .
(ii) The bootstrap consistency result for the Mack bootstrap process uncertainty part conditional on Q I,n in Theorem 4.8 requires the correct choice of the true family of (conditional) distributions of the F i,j 's also for the F * i,j 's in Step 5 of Section 3.1. Otherwise, only the first and second moments of the conditional distribution will be correctly mimicked asymptotically, but not necessarily the whole distribution.
(iii) The uniform boundedness conditions in Assumption 4.9 are required to establish a Lyapunov condition for bootstrap CLTs for (smooth functions of ) f * j,n because the Mack bootstrap draws bootstrap errors r * i,j from residuals computed from all columns in D I,n . (iv) The bootstrap inconsistency result for the Mack bootstrap estimation uncertainty part conditional on Q * I,n = Q I,n and D I,n in Theorem 4.10 is because the bootstrap approach in Step 4 is not taking the condition Q * I,n = Q I,n into account. Hence, the (always larger!) variance-covariance matrix Σ K, f j shows in the conditional limiting distribution instead of Σ (2) K, f j obtained in Theorem 4.4. Moreover, a decomposition of (R * I,n − R I,n ) 2 resembling the decomposition of (R I,n − R I,n ) 2 in (4.7) does not exist. (v) The requirement of a bootstrap procedure to not only mimic the asymptotically dominating part of the (conditional) predictive distribution that captures the prediction (i.e. process) uncertainty (asymptotic validity), but also the asymptotically negligible part capturing the uncertainty due to model parameter estimation is closely related to the concept coined asymptotic pertinence in Pan and Politis (2016) for time series prediction, which is also discussed by Beutner et al. (2021) from a slightly different perspective.
The discussion above motivates an alternative notion of a Mack-type bootstrap to be introduced in the following section that is designed to eliminate the raised issues. In particular, it should respect the conditioning on Q * I,n = Q I,n and it should generate a whole bootstrap loss triangle D * I,n in a backward manner starting from the diagonal Q * I,n = Q I,n . See e.g. Paparoditis and Shang (2021) for bootstrap predictive inference in a functional time series setup.

An alternative Mack-type Bootstrap Scheme
According to the findings and the discussion in Section 4, the original Mack bootstrap proposal is not capable of mimicking the conditional distribution of the estimation uncertainty part correctly. Although it is asymptotically dominated by the process uncertainty part, it is generally desirable to construct a Mack-type bootstrap that addresses this issue to enable a better finite sample performance.
For this purpose, we propose an alternative Mack-type bootstrap in this section to mimic the distribution of the predictive root of the reserve R I,n − R I,n using an alternative bootstrap predictive root of the reserve R + I,n − R + I,n to be defined below. To distinguish it from the original Mack bootstrap proposal in Section 3, we denote all related bootstrap quantities and operations with a "+" instead of a " * ". This novel approach deviates from the original Mack bootstrap scheme from Section 3 in several ways: (i) First, given the loss triangle D I,n and conditional on Q + I,n = Q I,n , where Q + I,n = {C + I−i,i |i = 0, . . . , I + n}, a recursive backward bootstrap approach is employed to generate a whole bootstrap upper triangle (ii) Second, the development factor estimators f j,n computed from D I,n are used for a parametric bootstrap to construct bootstrap individual development factors F + i,j , i = −n, . . . , I, j = 0, . . . , I + n − 1 and i + j ≥ I, that is for the lower triangle, which also allows to construct R + I,n . (iii) Third, for the construction of the bootstrap predictive root of the reserve R + I,n − R + I,n , the bootstrap reserve R + I,n is not centered around its best estimate R I,n , but around a suitable bootstrap version R + I,n .
Analogous to the original Mack bootstrap, the alternative Mack bootstrap is employed to estimate the conditional distribution of R I,n − R I,n given Q I,n by the conditional bootstrap distribution of R + I,n − R + I,n given Q + I,n = Q I,n and D I,n .

An alternative Mack-type Bootstrap Algorithm.
With the upper triangle D I,n at hand, the alternative Mack-type bootstrap algorithm is defined as follows: Step 1. Estimate the development factors f j and the variance parameters σ 2 j from the data by computing f j,n and σ 2 j,n for j = 0, . . . , I +n−1 as defined in (2.8) and (2.11), respectively.
Step 2. Choose a parametric family for the (conditional) bootstrap distributions of the backward Then, given Q + I,n = Q I,n , generate backwards a bootstrap loss triangle D + I,n using the recursion Step 3. Compute bootstrap development factor estimators f + j,n for j = 0, . . . , I + n − 1, which are defined as f j,n in (2.8), but are calculated from the bootstrap loss triangle D + I,n . That is, we compute Step 4. Choose a parametric family for the (conditional) bootstrap distributions of F + i,j given C + i,j and D I,n such that F + i,j > 0 a.s. with Then, given Q + I,n = Q I,n , generate the bootstrap ultimate claims C + i,I+n and the reserves . . , I using the recursion Step 5. Compute the bootstrap total reserve R + I,n = I i=−n R + i,I+n and the alternative Mack bootstrap predictive root of the reserve where the centering term R + I,n is a bootstrap version of the best estimate R I,n , that is defined by Step 6. Repeat Steps 2 -5 above B times, where B is large, to get (R + I,n − R + I,n ) (b) , b = 1, . . . , B bootstrap predictive roots, and denote by q + (α) the α-quantile of their empirical distribution.
Remark 5.1 (On the alternative Mack-type bootstrap).
(i) In comparison to the Mack bootstrap from Section 3.1, the bootstrap reserve is not a double-bootstrap quantity anymore, the centering is based on a bootstrap version of the best estimate, and the bootstrap for the upper loss triangle is backwards starting in the diagonal.
(ii) The conditional distribution for the G + i,j |C + i,j+1 can be chosen in different ways. For instance, this can be done non-parametrically similar to Steps 2 -4 in Section 3.1 or using the parametric family of distributions used in Step 5 in Section 3.1. However, it is crucial to mimic sufficiently well the first and second backward conditional moments, that is, E(C ij |C i,j+1 ) and V ar(C i,j |C i,j+1 ), respectively.

Asymptotic Theory for the alternative Mack Bootstrap
By adopting the general strategy of Section 4 to investigate the consistency properties of the original Mack bootstrap, the alternative Mack predictive root of the reserve R + I,n − R + I,n can be decomposed also into a prediction error part and an estimation error part, respectively. That is, by adding and subtracting I+n i=0 C + I+n−1 j=i f j,n , we get As in Section 4.2 for the Mack bootstrap, we have to check whether, conditional on Q + I,n = Q I,n and D I,n , the alternative Mack bootstrap quantities (R + I,n − R + I,n ) 1 and (R + I,n − R + I,n ) 2 in (4.2) are correctly mimicking the limiting distributions of (R I,n − R I,n ) 1 and (R I,n − R I,n ) 2 given Q I,n , respectively. 6.1.1. Conditional bootstrap asymptotics for reserve prediction: process uncertainty.
The process uncertainty part (R + I,n − R + I,n ) 1 of the alternative Mack bootstrap differs from the (R * I,n − R I,n ) 1 as the F + i,j 's in (R + I,n − R + I,n ) 1 use f j,n instead of f * j,n and as I+n−1 j=i F + I−i,j is centered around I+n−1 j=i f j,n instead of I+n−1 j=i f * j,n accordingly. However, by using very similar arguments, we get the same limiting result also for the process uncertainty part (R + I,n − R + I,n ) 1 of the alternative Mack bootstrap.

Conditional bootstrap asymptotics for reserve prediction: estimation uncertainty.
In view of the decomposition of (R I,n − R I,n ) 2 in (4.7), conditional on Q + I,n = Q I,n and D I,n , its alternative Mack bootstrap counterpart (R + I,n − R + I,n ) 2 can be also decomposed further. That is, we have 2 , 2 is measurable with respect to Q + I,n = Q I,n and D I,n and f + j,n (Q I,n ) := µ . In comparison to Theorem 4.10, the uniform boundedness condition in Assumption 4.9 can be dropped, but the derivation of (conditional) bootstrap asymptotic theory and consistency results for (R + I,n − R + I,n ) 2 requires additional assumptions on the backward individual development factors G + i,j from Step 3 in Section 5.1. Precisely, it has to be guaranteed that the backward conditional mean E(C i,j |C i,j+1 ) and the backward conditional variance V ar(C i,j |C i,j+1 ) are consistently mimicked by their alternative Mack bootstrap counterparts E + (C + i,j |C + i,j+1 ) and V ar + (C + i,j |C + i,j+1 ), respectively, such that the corresponding limiting distributions obtained in Steinmetz and Jentsch (2022, Theorem C.1) are correctly mimicked. (i) For each fixed K ∈ N 0 , let f + K,n (Q I,n ) = (f + 0,n (Q I,n ), f + 1,n (Q I,n ), . . . , f + K,n (Q I,n )) and define f K,n = ( f 0,n , f 1,n , . . . , f K,n ) . Then, conditional on Q + I,n = Q I,n , we have of inflation factors and the variance-covariance matrix Σ where the variance-covariance matrix Σ K,f is defined in Theorem C.1 in Steinmetz and Jentsch (2022).
In concordance to the derivation of the conditional limiting result obtained in Theorem 4.4(ii), which relies on conditional CLTs for the development factor estimators f j,n given in (Steinmetz and Jentsch 2022, Appendix C), the conditional bootstrap CLTs in Assumption 6.2 allow to state the following theorem, which provides the limiting distribution of the alternative Mack bootstrap estimation uncertainty term (R + I,n − R + I,n ) 2 conditional on Q + I,n = Q I,n and D I,n . Precisely, while (R + I,n − R + I,n ) 2 is measurable with respect to D I,n , Assumption 6.2 allows to establish asymptotic normality of √ I + n + 1(R + I,n − R + I,n ) 2 conditional on Q + I,n = Q I,n and D I,n . Theorem 6.3 (Bootstrap asymptotics for (R + I,n − R + I,n ) 2 conditional on Q + I,n = Q I,n and D I,n ). Suppose Assumptions 2.2, 2.3, 2.5, 2.4, 4.2 and 6.2 hold. Then, as n → ∞, the following holds: (i) Conditional on Q + I,n = Q I,n , 2 converges in distribution to the non-degenerate limiting distribution G (1) 2 . That is, we have √ I + n + 1(R + I,n − R + I,n ) Steinmetz and Jentsch (2022). Here, the sequences Q I,∞ and Y 2 converges in distribution to G 2 |Q I,∞ in probability, where G 2 |Q I,∞ ∼ N (0, Ξ(Q I,∞ ))|Q I,∞ is the (conditional) limiting distribution obtained in Theorem 4.4(ii).

6.1.3.
Conditional bootstrap asymptotics for the whole predictive root of the reserve.
As in Sections 4.1.3 and 4.2.3, combining the results for (R + I,n − R * I,n ) 1 and (R + I,n − R * I,n ) 2 from Theorems 6.1 and 6.3, we get joint asymptotics for R + I,n − R I,n conditional on Q + I,n = Q I,n and D I,n .
Theorem 6.4 (Bootstrap asymptotics for R + I,n − R + I,n conditional on Q + I,n = Q I,n and D I,n ). Suppose the assumptions of Theorems 6.1 and 6.3 hold. Then, conditional on Q + I,n = Q I,n and D I,n , (R + I,n − R + I,n ) 1 and (R + I,n − R + I,n ) 2 are stochastically independent, and R + I,n − R + I,n converges in distribution to G 1 |Q I,∞ in probability. That is, we have According to the discussion below Theorem 4.11 and in view of the concepts of asymptotic validity and asymptotic pertinence of bootstrap predictive inference in Pan and Politis (2016), the alternative Mack bootstrap can be regarded as asymptotically valid and asymptotically pertinent under the stated conditions.
Remark 6.5 (Backward vs. forward bootstrapping). While a backward bootstrap appears to be natural in time series setups addressed in Pan and Politis (2016), they also propagate a simpler forward bootstrap to capture the estimation uncertainty in bootstrap prediction. Asymptotically, in their setup, both approaches are indeed equivalent due to the intrinsic stationarity assumption.
However, in Mack's Model setup considered here, this is not the case and the (fixed-design) forward bootstrap of England and Verrall (2006) does not correctly capture the conditional limiting distribution of the estimation uncertainty part.

Simulation Study
In this section, we compare the original Mack bootstrap from Section 3 and the alternative In all scenarios, the development factors and the variance parameters fulfill f j > 1 and σ 2 j > 0 with f j and σ 2 j decreasing to 1 and 0, respectively. Precisely, we use exponentially decreasing sequences (f j ) j∈N 0 and (σ 2 j ) j∈N 0 with f j = 1 + e −1−0.2j and σ 2 j = 509, 518 · e −1−0.7j . We distinguish between two Setups a) and b), where the parameter are exactly the same in both cases, but the first column C  which deviates from R + I,n in (5.3) as it relies on f * j,n in (3.3), but is based on (parametrically generated) 3) instead of f + j,n defined in (5.1). This choice of the centering term still resembles the decomposition in (6.1), that shares the (sign) properties of (4.1), which is not the case for (4.2). For all bootstraps, whenever a parametric distribution is used to generate the upper bootstrap loss triangle, we choose the same parametric distribution family used already for the lower triangle (to generate R * I,n and R + I,n ). However, as we do not know the correct parametric family of distributions of the F i,j 's, we make use of all three distribution families in (i)-(iii) for all three bootstrap approaches, respectively, to also investigate the effect of a misspecified parametric family of distributions to generate R * I,n and R + I,n .
In the Appendix E, we provide also simulation results that compare the distribution of the first (i.e. the process uncertainty) parts of the bootstrap predictive roots (R * I,n − R I,n ) 1 and (R + I,n − R + I,n ) 1 conditional on Q * I,n = Q I,n or Q + I,n = Q I,n and D I,n , respectively, with the distribution of (R I,n − R I,n ) 1 conditional on Q I,n . Note that this distribution is straightforward to simulate. As expected, in view of Theorems 4.8 and 6.1, we find no differences in the performances of both construction principles.

Simulation results.
First, we consider the bootstrap variances of the bootstrap predictive roots of the reserves obtained for the three Mack-type bootstraps under study. For both Setups a) and b), we find that the alternative Mack-type bootstrap variance is always 1-5 percentage points smaller than the bootstrap variances obtained for the other two approaches, which do not differ much.  Table 3. While the percentages increase for growing n, for all bootstraps, the alternative Mack-type bootstrap consistently achieves percentages that are higher by 1-3 percentage points in comparison to to the two other bootstraps, which turn out to be quite similar throughout.
We consider also the average over all simulations of the squared mean of the deviation of the bootstrap distribution given Q * I,n = Q I,n or Q + I,n = Q I,n , respectively, and D I,n and its true distribution given Q I,n . Therefore, we calculate the mean squared error of each simulation for b = 1, . . . , 10, 000 and then consider the root of the overall mean of the mean squared error

Conclusion
In this paper, we adopt the stochastic and asymptotic framework that was proposed by  Mack bootstrap asymptotics for parameter estimators.
The following theorem is the Mack bootstrap version of the (unconditional!) Theorem 3.1 in Steinmetz and Jentsch (2022) adapted to the asymptotic framework of Section 2.1.
As the unconditional limiting distributions obtained in Theorem A.1 above and in Theorem 3.1 in Steinmetz and Jentsch (2022) coincide, the Mack bootstrap is unconditionally, that is without conditioning on Q * I,n = Q I,n , consistent for an arbitrary, but fixed number of estimators of development factors. That is, for each fixed K ∈ N 0 , we have where f K = (f 0 , f 1 , . . . , f K ) and d K denotes the Kolmogorov distance between two probability distributions.
The following direct corollary is the Mack bootstrap version of Corollary 3.2 in Steinmetz and Jentsch (2022) adapted to the asymptotic framework of Section 2.1.
Corollary A.2 (Asymptotic normality for products of f * j,n 's conditional on D I,n ). Suppose the assumptions of Theorem A.1 hold. Then, as n → ∞, the following holds: (i) For each fixed K ∈ N 0 and i = 0, . . . , K, we have (ii) For each fixed K ∈ N 0 , we have also joint convergence, that is, as derived in the proof of Corollary 3.2 in Steinmetz and Jentsch (2022).
A.1. Proof of Theorem A.1. By construction of the Mack bootstrap estimators f * j,n , j = 0, . . . , I + n − 1 according to (3.5, for each fixed K ∈ N 0 , the K + 1 estimators f * 0,n , f * 1,n , . . . , f * K,n are independent conditional on D I,n . Hence, it is actually sufficient to prove part (i). For any fixed j and from (2.10) and (3.5), using C i,j+1 = C i,j F i,j , we get immediately Noting that, for all j, (Z i,n , i = −n, . . . , I − j − 1, n ∈ N 0 ) forms a triangular array of random variables that are independent conditional on D I,n , we can make use of a (conditional) Lyapunov CLT to prove asymptotic normality. First, for the bootstrap mean, using measurability of all C i,j 's and of f j,n in Z i,n with respect to D I,n , we get Further, by the construction of Mack's bootstrap, for any fixed j and i = −n, . . . , leading to E * (Z * i,n ) = 0. Second, for the bootstrap variance, we get and, from the particular construction of Mack's bootstrap leading to E * (r * ij ) = 0 and E * (r * 2 ij ) = 1, we obtain and, altogether, Letting n → ∞, making use of Assumption 2.5, we get σ 2 j,n → σ 2 j by Theorem 3.5 in Steinmetz and Jentsch (2022), as well as by a WLLN using that, for all j, (C k,j , k ∈ Z, k ≤ I − j − 1) are iid by Assumption 2.1(iii) with (finite) mean µ j and variance τ 2 j according to (2.19) and (2.20), respectively.
Finally, it remains to prove a Lyapunov condition to complete the proof. Choosing δ = 2 for the Lyapunov condition, for any j, it is sufficient to show that Due to measurability of all C i,j 's with respect to D I,n , we get Further, as For this purpose, we have to compute E * ((F * i,j − f j,n ) 4 ) next. By plugging-in for F * i,j , we get Further, as In the following, suppose for convenience that r t,s = r t,s . However, the arguments for r t,s including re-centering (and re-scaling) are essentially the same, but tedious and lengthy. In this case, by plugging-in for r t,s , we get E * r * 4 i,j = 2 (I + n + 1)(I + n) − 2 2 Note that we implicitly assume that only σ 2 I+n−1,n is estimated as zero; see Section 3.1.
Hence, we can bound E * (r * 4 i,j ) above by O P (1) 2 (I + n + 1)(I + n) − 2 Finally, the term in brackets on the last right-hand side is a sum consisting of non-negative summands, which is also O P (1) as its expectation is bounded because the κ Nevertheless, the asymptotic theory for (R * I,n − R I,n ) 1 conditional on Q * I,n = Q I,n and D I,n is not straightforward as it is composed of sums and products consisting asymptotically of infinitely many summands and factors. Hence, we decompose (R * I,n − R I,n ) 1 by truncating these sums and products to be able to apply Proposition 6.3.9 in Brockwell and Davis (1991). For this purpose, let K ∈ N 0 be fixed and suppose I, n ∈ N 0 are large enough such that K < I + n − 1. We begin with showing part a). The parametric family of (conditional) distributions used to generate the F i,j |C i,j and F * i,j |C * i,j is continuous with respect to C i,j , f j , σ 2 j and C * i,j , f * j,n , σ 2 j,n , respectively, by Assumption 4.6. Hence, as f j,n − f j = O P ((I + n − 1) −1/2 ), f * j,n − f j,n = O P * ((I + n − 1) −1/2 ) and σ 2 j,n − σ 2 j = O P ((I + n − 1) −1/2 ) holds for all fixed j ∈ N 0 , we can conclude that, for all fixed K ∈ N 0 and as n → ∞, that

Then, we have
in probability, which proves A * 1,K,I,n |(Q * I,n = Q I,n , D I,n ) d → G 1,K |Q I,∞ . For part b), by letting also K → ∞, we get immediately which proves G 1,K |Q I,∞ d → G 1 |Q I,∞ . Before we prove part c), let us also consider mean and variance of A * 1,K,I,n (conditional on Q * I,n = Q I,n and D I,n ). For the mean, using measurability of C I−i,i with respect to D I,n and the law of iterated expectations, we have E * (A * 1,K,I,n |Q * I,n = Q I,n ) = E * (E * (A * 1,K,I,n |Q * I,n = Q I,n , F * I,n )|Q * I,n = Q I,n ) using similar arguments as used to show E( K j=i F I−i,j ) = K j=i f j . Similarly, using the law of total variance and (B.6), we get for the variance V ar * (A * 1,K,I,n |Q * I,n = Q I,n ) =E * V ar * (A * 1,K,I,n |Q * I,n = Q I,n , F * I,n )|Q * I,n = Q I,n + V ar * E * (A * 1,K,I,n |Q * I,n = Q I,n , F * I,n )|Q * I,n = Q I,n due to the fact that f * k,n 's are independent of the condition Q * I,n = Q I,n and because of V ar * (A * 1,K,I,n |Q * I,n = Q I,n , F * I,n ) = as n → ∞ for all K fixed, because f j,n − f j = O P ((I + n − 1) −1/2 ) and σ 2 j,n − σ 2 j = O P ((I + n − 1) −1/2 ) for all j ∈ N 0 . Finally, letting K → ∞, we get which equals V ar((R I,∞ − R I,∞ ) 1 |Q I,∞ ). Hence, it remains to show part c) to complete the proof. We begin with showing part c) for A * 2,K,I,n . By similar arguments used above, for the mean, we have E * (A * 2,K,I,n |Q * I,n = Q I,n ) = 0 due to E * (A * 2,K,I,n |Q * I,n = Q I,n , F * I,n ) = 0 and, for the variance, we have V ar * (A * 2,K,I,n |Q * I,n = Q I,n ) = E * ((A * 2,K,I,n ) 2 |Q * I,n = Q I,n ) = E * (E * ((A * 2,K,I,n ) 2 |Q * I,n = Q I,n , F * I,n )|Q * I,n = Q I,n ). For the inner expectation, using stochastic independence over accident years leading to stochastic independent summands of A * 2,K,I,n (conditional on Q * I,n = Q I,n , D I,n and F * I,n ), we get E * ((A * 2,K,I,n ) 2 |Q * I,n = Q I,n , F * I,n ) For the term corresponding to the first term in brackets on the last right-hand side, we get (B.12) Using linearity of expectations, for the first expectation on the last right-hand side of (B.12), Similarly, for the second expectation in (B.12), we get and for the third one, we have Altogether, for all K < I + n − 1, this leads to E * ((A * 2,K,I,n ) 2 |Q * I,n = Q I,n , F * I,n ) Plugging-in and making use of the fact that the f * j,n 's are stochastically independent conditional on D I,n and Q * I,n = Q I,n , this leads to V ar * (A * 2,K,I,n |Q * I,n = Q I,n ) obtained by re-arranging terms and due to E * ( f * j,n |Q * I,n = Q I,n ) = E * ( f * j,n ) = f j,n and E * f * 2 j,n |Q * I,n = Q I,n = E * f * 2 j,n = f 2 j,n + σ 2 j,n I−j−1 p=−n C p,j for all j. Next, to argue that V ar * (A * 2,K,I,n |Q * I,n = Q I,n ) ≥ 0 vanishes in probability for K → ∞ and n → ∞ afterwards, it suffices to show that its unconditional expectation is bounded for K → ∞ and that its bound converges to zero as n → ∞. We get E V ar * (A * 2,K,I,n |Q * I,n = Q I,n ) Now, let us consider the three terms on the last right-hand side separately. Using I−h−1 p=−n C p,h ≥ (I + n − h) h ≥ h , the first one can be bounded by Next, using the law of iterated expectations and and, similarly, for the second term, we obtain Together, this term becomes which, using similar arguments as above, can be bounded by Now, letting n → ∞, we get the following upper bound by Assumptions 2.4 and 4.2. Now, letting also K → ∞, the term K i=0 K k=i σ 2 k also remains bounded due to Finally, as f l → 1 and σ 2 Similarly, using the same arguments, the second term in the representation of E(V ar * (A * 2,K,I,n |Q * I,n = Q I,n )) above can be bounded by which, for n → ∞, can be bounded by which vanishes for K → ∞.
Finally, for the third term in the representation of E(V ar * (A * 2,K,I,n |Q * I,n = Q I,n )), we get While, for n → ∞, the second factor 1 − I+n−1 l=K+1 f l can be bounded by 1 − ∞ l=K+1 f l , which converges to 0 due to ∞ l=K+1 f l → 1 for K → ∞, the first factor above can be bounded by which, for n → ∞, can be bounded further by which is bounded as K j=0 (j + 1)σ 2 j → ∞ j=0 (j + 1)σ 2 j < ∞ for K → ∞. This completes the first part of c) for term A * 2,K,I,n . Continuing with A * 3,K,I,n to prove also the second part of c), we have E * (A * 3,K,I,n |Q * I,n = Q I,n ) = 0 using the law of iterated expectations. By using similar calculations as for A * 2,K,I,n , we get Hence, to show that V ar * (A * 3,K,I,n ) ≥ 0 vanishes in probability, we prove that E(V ar * (A * 3,K,I,n |Q * I,n = Q I,n )) is bounded for n → ∞ and that its bound converges to zero as K → ∞. By plugging-in and using similar arguments as above for A * 2,K,I,n , we get E(V ar * (A * 3,K,I,n |Q * I,n = Q I,n )) ≤ behavior of (R * I,n − R I,n ) 1 and to the proof of Theorem 4.7 in Steinmetz and Jentsch (2022) for the (unconditional!) limiting behavior of (R I,n − R I,n ) 2 , we decompose (R * I,n − R I,n ) 2 by truncating the sums and products to be able to apply Proposition 6.3.9 in Brockwell and Davis (1991). For this purpose, let K ∈ N 0 be fixed and suppose I, n ∈ N 0 are large enough such that K < I + n − 1. Then, after inflating (R * I,n − R I,n ) 2 with √ I + n + 1, we get √ I + n + 1 R * I,n − R I,n We begin with part a). That is, for each fixed K ∈ N 0 , we consider B * 1,K,I,n = f m (B.14) for i 1 , i 2 ∈ N 0 . Moreover, as Q I,∞ and Y ∞ are stochastically independent, conditional on Q I,∞ , due to ∞ j=0 (j + 1) 2 σ 2 j < ∞ by Assumption 2.4. We continue with showing part c) for B * 2,K,I,n . Using similar arguments as above, we have to consider Using the unbiasedness of f * j,n conditional on Q * I,n = Q I,n , D I,n for f j,n , that is, for all j, and the independence of the f * j,n 's conditional on Q * I,n = Q I,n and D I,n , we have E * (B * 2,K,I,n |Q * I,n = Q I,n ) = 0 by construction. Hence, it remains to show that V ar * (B * 2,K,I,n |Q * I,n = Q I,n ) is bounded in probability for n → ∞ and its bound vanishes for K → ∞ afterwards. Now, to compute the bootstrap variance V ar * (B * 2,K,I,n |Q * I,n = Q I,n ), for any fixed K ∈ N 0 and n ∈ N 0 large enough such that K < I + n − 1, we get V ar * (B * 2,K,I,n |Q * I,n = Q I,n ) using that, conditional on D I,n , the f * j,n 's are independent of the condition Q * I,n = Q I,n . To calculate the covariance on the last right-hand side, for i 1 ≤ i 2 , first, we consider the mixed moment since f * j,n and f * k,n are independent for j = k and j, k ∈ {0, . . . , I + n − 1} conditional on D I,n . Similarly, we have By rearranging the terms in brackets on the last right-hand side above, it becomes Now, following the same steps as in the proof of Theorem 4.7 in Steinmetz and Jentsch (2022), we can compute the unconditional expectation of the above. Using C i,j > j , E( f c,n |B I,n (c)) = f c , E( σ 2 c,n |B I,n (c)) = σ 2 c as well as for all c ∈ {0, . . . , I + n − 1}, where B I,n (k) = {C i,j |i = −n, . . . , I, j = 0, . . . , k, i + j ≤ I + n}, we can argue that V ar * (B * 2,K,I,n |Q * I,n = Q I,n ) ≥ 0 vanishes in probability for n → ∞ and K → ∞ afterwards, by showing that its unconditional expectation is bounded for n → ∞ and that its bound converges to zero as K → ∞.
we can bound also E(V ar * (B * 2,K,I,n |Q * I,n = Q I,n )) from above. Precisely, putting everything together, we get E(V ar * (B * 2,K,I,n |Q * I,n = Q I,n )) and the leading term of the last right-hand side becomes which can be bounded further by Now, considering the four terms in brackets separately, for the first one, we can argue that it vanishes asymptotically due to is converging and, consequently, also bounded, and due to ∞ j=0 (j + 1) 2 σ 2 j j < ∞ by Assumption 4.2. Similarly, using that ∞ j=K+1 f j → 1 and ∞ j=K+1 (f 2 j + 2σ 2 j j ) → 1 for K → ∞, we can also show that the other three terms vanish asymptotically. This completes the first part of c) for B * 2,K,I,n . Similarly, for showing part c) for B * 3,K,I,n , we have to consider B * 3,K,I,n |(Q * I,n = Q I,n , D I,n ) = By the same arguments as used above for B * 2,K,I,n , we get E * (B * 3,K,I,n |Q * I,n = Q I,n ) = 0 and for any fixed K ∈ N 0 and n ∈ N 0 large enough such that K < I + n − 1, we have V ar * (B * 3,K,I,n |Q * I,n = Q I,n ) =(I + n + 1) To calculate the covariance on the last right-hand side, for i 1 ≤ i 2 , we consider the mixed moment which is just the first term of the mixed moment of the covariance calculated for B * 2,K,I,n . By using similar calculations to get E * ( f * 2 c,n ) (for B * 2,K,I,n ), we obtain Noting that all involved summands and factors are non-negative, taking expectations of the last right-hand side and using the law of iterative expectations and C i,j > j , E( f c,n |B I,n (c)) = f c , E( σ 2 c,n |B I,n (c)) = σ 2 c as well as (B.15), we get such that the leading term of E(V ar * (B * 3,K,I,n |Q * I,n = Q I,n )) becomes 2 I+n−1 For the triple sum on the last right-hand side, we get ≤const.
Further, the sequence (µ i , i ∈ N 0 ) shares the properties of (C I−i,i , i ∈ N 0 ) in a deterministic sense such that j−K l=1 lµ l+K ≤ const.l 2 . Consequently, we have because the inner conditional expectation on the last right-hand side is zero.
We obtain Corollary D.2 (Asymptotic normality for products of f j,n 's conditionally on Q I,n ; Corollary C.2(ii,iv) in Steinmetz and Jentsch (2022)). Suppose the assumptions of Theorem D.1 hold.
Then, as n→∞, the following holds: (i) For each fixed K∈N 0 , unconditionally, we have also joint convergence, that is, (ii) For each fixed K∈N 0 , conditionally on Q I,n , we have also joint convergence, that is, 6 × 10 −9 −2 × 10 +8 −1 × 10 +8 0 1 × 10 +8 2 × 10 +8 Reserve Density Figure 1. Boxplots of skewness and kurtosis as well as five arbitrarily selected density plots for the simulated Mack type bootstrap conditional distribution of (R * I,n − R I,n ) 1 given Q * I,n =Q I,n and D I,n for n=10 and I=10 for the setup of a), where F * i,j follows a (conditional) gamma (top), log-normal (center) and truncated normal distribution (bottom). development factors is skewed, the chosen distribution for F * i,j and F + i,j , respectively, for the lower triangle should be skewed. For example, if we choose a gamma distribution for F * i,j instead of a log-normal distribution as true distributional family of F * i,j , the percentage to fail to reject the null hypothesis is higher compared to if we choose a truncated normal distribution, e.g., for n=40, we get that 69% out of M =500 fail to reject the null hypothesis assuming a gamma distribution compared to 31% using a truncated normal distribution (cf. Table 6). Also, if the true underlying distribution is a truncated normal distribution and we choose a gamma, then 50% out of M =500 fail to reject the null hypothesis or to assume a log-normal distribution, then 39% and if we choose the true underlying distribution, then 84% for n=40 (cf. Table 6).
For setup a) the effect of choosing the wrong distribution is not as high as for b). We can explain this with the property of the gamma and the log-normal distribution. Both tend to 'lose' their skewness and excess of kurtosis for  Figure 2. Boxplots of skewness and kurtosis as well as five arbitrarily selected density plots for the simulated Mack type bootstrap conditional distribution of (R * I,n − R I,n ) 1 given Q * I,n =Q I,n and D I,n for n=10 and I=10 for the setup of b), where F * i,j follows a (conditional) gamma (top), log-normal (center) and truncated normal distribution (bottom).     chosen distribution gamma log-normal trunc. normal true distribution n oMB aMB iMB oMB aMB iMB oMB aMB iMB 0 9.881 9.874 9.961 9.925 9.850 9.919 9.841 9.822 9.849 10 9.644 9.582 9.680 9.722 9.650 9.742 9.414 9.328 9.442 20 9.479 9.317 9.476 9.459 9.362 9.477 9.254 9. 0 9.830 9.789 9.843 9.894 9.876 9.997 9.881 9.874 9.910 10 9.520 9.513 9.524 9.676 9.583 9.517 9.543 9.474 9.575 20 9.167 9.156 9.234 9.234 9.227 9.234 9.344 9.