Misspecification tests for log-normal and over-dispersed poisson chain-ladder models

Harnau, Jonas Article Misspecification tests for log-normal and over-dispersed poisson chain-ladder models Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Harnau, Jonas (2018) : Misspecification tests for log-normal and over-dispersed poisson chain-ladder models, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 6, Iss. 2, pp. 1-25, https://doi.org/10.3390/risks6020025 This Version is available at: https://hdl.handle.net/10419/195817 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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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Misspecification Tests for Log-Normal and Over-Dispersed Poisson Chain-Ladder Models Jonas Harnau ID Department of Economics, University of Oxford & Oriel College, Oxford OX1 4EW, UK; [email protected] Received: 28 February 2018; Accepted: 21 March 2018; Published: 23 March 2018   Abstract: Despite the widespread use of chain-ladder models, so far no theory was available to test for model specification. The popular over-dispersed Poisson model assumes that the over-dispersion is common across the data. A further assumption is that accident year effects do not vary across development years and vice versa. The log-normal chain-ladder model makes similar assumptions. We show that these assumptions can easily be tested and that similar tests can be used in both models. The tests can be implemented in a spreadsheet. We illustrate the implementation in several empirical applications. While the results for the log-normal model are valid in finite samples, those for the over-dispersed Poisson model are derived for large cell mean asymptotics which hold the number of cells fixed. We show in a simulation study that the finite sample performance is close to the asymptotic performance. Keywords: Bartlett test; F-test; over-dispersed Poisson; log-normal 1. Introduction “Can we trust chain-ladder models?” is a central question in non-life insurance claim reserving. It hinges on the model assumptions: if these are violated the answer would be “no”. For example, the popular over-dispersed Poisson chain-ladder model assumes a fixed variance to mean ratio across the run-off triangle. If this is false then distribution forecasts are bound to fail. Yet, there is no statistical theory available to test for a violation of this assumption. We show that testing for a violation of central assumptions is straightforward in two popular chain-ladder models: over-dispersed Poisson and log-normal. While the over-dispersed Poisson model assumes a fixed variance to mean ratio, the log-normal model imposes a common variance of the log data. Further, both models assume a chain-ladder structure. That is, accident year effects do not vary by development year and vice versa. We show that these assumptions are not only testable, but testable with standard tools that can easily be implemented in a spreadsheet. The over-dispersed Poisson model arguably owes its special status to the ubiquitous chain-ladder technique. Kremer (1985) showed that this deterministic technique so commonly used in claim reserving is replicated by maximum likelihood estimation in a Poisson model. However, integer support and the implicit assumption that the variance equals the mean cannot be reconciled with insurance claim data. This explains the need for the over-dispersed Poisson model which relaxes both of these assumptions. Unlike the Poisson model, the over-dispersed Poisson model is moment-based and does not come equipped with a distributional framework. Despite this shortcoming, distribution forecasts are needed and bootstrapping (England 2002;England and Verrall 1999) is in widespread use. Yet, so far we do not have a statistical theory for the bootstrap in this setting. Recently, Harnau and Nielsen (2017) proposed a distributional framework that incorporates the moment assumptions of the over-dispersed Poisson model. This framework allows for a compelling asymptotic theory that does not require a large array but rather large cell means. The practical Risks 2018,6, 25; doi:10.3390/risks6020025 www.mdpi.com/journal/risks Risks 2018,6, 25 2 of 25 implication is that for a run-off triangle with a large, potentially unknown, number of payments, we can use a fixed sample size Gaussian distribution theory. They derive parameter distributions, tests for model reduction, such as the absence of calendar effects, and closed form distribution forecasts. Their assumptions accommodate, among others, many compound Poisson distributions. In insurance, these have the interpretation that each cell of aggregate incremental claims is the sum of a Poisson number of claims each with a random individual claim amount. The asymptotic theory then does not assume that we have many such cells, but rather that the mean of the Poisson number of claims is large. We stress that while Harnau and Nielsen (2017) largely use terminology from the age-period-cohort literature, the theory immediately applies to the reserving literature by renaming age, period, and cohort effects to development, calendar, and accident effects. Modeling aggregate incremental claims as log-normal rather than over-dispersed Poisson is also common. Kremer (1982) introduced a log-normal model with multiplicative mean structure mirroring the over-dispersed Poisson chain-ladder model. While this model does not replicate the classic chain-ladder technique, it is easily estimated by least squares. Recently, Kuang et al. (2015) derived explicit expression for the estimators in the log-normal model. These have interpretation as a geometric, rather than the classic arithmetic chain ladder. Other contributions for the log-normal model are discussed in the excellent overview of stochastic reserving models by England and Verrall (2002). We are of course not the first to question the validity of the assumptions in these models. Yet, so far the problem was dealt with by specifying more flexible models. For example, Hertig (1983) considers a log-normal model that allows the log data variance to vary by development year. The double-chain-ladder model by Martínez Miranda et al. (2012) has, conditional on the incurred counts, an approximate over-dispersed Poisson structure where the over-dispersion varies by accident year. The “distribution-free” model by Mack (1993) has separate variance parameters for each development year. We note that while this model also replicates the classical chain-ladder point forecasts, it differs from the over-dispersed Poisson model and so far lacks a distributional framework that would allow for a rigorous statistical theory. Thus, while it is a popular model, we do not consider it further in this paper. While using more flexible models seems sensible when assumptions are violated, we should not be too quick to dispose of well-known simple models. Particularly for forecasting, such simpler models may be advantageous. A statistical framework for misspecification testing is thus needed. The tests may corroborate the initial modeling choice of the expert, draw attention to an issue, or confirm the suspicion that the model is not well suited for the task. Whichever scenario the expert encounters, the misspecification tests can help to make an informed choice. The test statistics we propose in this paper are well known in an analysis of variance (ANOVA) context. There, the researcher is usually presented with several samples and wants to test for treatment effects. The data are often assumed to be independent Gaussian. The first step is to test for common variances across samples. This is done with a Bartlett test based on an easily computed likelihood ratio statistic. Then, given common variances, a standard F -test can be used to test for different means between the samples, indicating a treatment effect. The difference to the ANOVA application is that we generally have data for only one sample, often a run-off triangle. We thus reverse engineer the ANOVA situation by splitting the data into several artificial sub-samples. This idea has a long history in the econometric literature. For instance, Chow (1960) proposed a test for structural breaks that involved splitting the sample at the known breakpoint. In the (weak) instrumental variable literature, Angrist and Krueger (1995) proposed a split-sample procedure with the objective to break the bias of the instrumental variable estimator towards the ordinary least squares estimator. Figure 1shows examples of how we could split run-off triangles into sub-samples. Risks 2018,6, 25 3 of 25 i,j1 5 10 1 I1 5 I2 10 ( a ) Two sub-samples i,j1 5 11 1 I1 I2 5 I3 11 (b) Three sub-samples i,j1 5 10 1 I1I3 I2 5 I4 10 ( c ) Four sub-samples Figure 1. Examples for splits of run-off triangles into two ( a ), three ( b ) and four ( c ) sub-samples. Sub-samples are denoted by I` . Accident years i are in the rows, development years j in the columns. In Section 2, we give a precise definition for the conditions that both the data set as well as the artificial sub-samples must meet. We note that while we do not provide guidance on how to chose the sub-sample structure in this paper, the choice does not affect the size of the proposed tests under the null hypothesis. In a log-normal model, taking logs yields Gaussian data such that we can directly apply the Bartlett and F -test from the ANOVA scenario. While the finite sample distribution of the Bartlett test statistic has no closed form, it does not have nuisance parameters and critical values could easily be simulated. However, Bartlett (1937) suggests a χ2 approximation to the exact distribution that allows us to sidestep simulations. For a special case with just two sub-samples, we can also apply an F -test for the hypothesis for common variances of the log data ; while Bartlett and F -tests are not identical, simulations indicate that they give similar information. Next, we show that an F -test for common mean parameters is not only straightforward but also independent of the Bartlett test. These results are collected in Section 3. In the over-dispersed Poisson model, the asymptotic framework by Harnau and Nielsen (2017) catapults us into a finite dimensional Gaussian world. Therefore, the results developed for the log-normal model carry over. We can now asymptotically use a Bartlett test as a test for common over-dispersion across sub-samples. Similarly, an F -test for common mean parameters across sub-samples is asymptotically F-distributed and asymptotically independent of the dispersion parameter tests. We stress again that the asymptotic theory does not require a large triangle but rather large means of the cells in the triangle. As for the log-normal model, we could simulate critical values for the Bartlett test; however a χ2approximation can still be justified. We show all this in Section 4. The Bartlett test is easily implemented and makes an empirical application straightforward. The same is true for an F -test on the means. We illustrate the testing procedure, splitting the data, estimating the sub-models, Bartlett testing for common dispersion parameters, and F -testing for common mean parameters, in Section 5with several empirical applications. We clear up remaining questions about the power of the tests and the performance of approximations in a simulation study based on a run-off triangle. First, it would not take much to simulate critical values of the Bartlett test statistic under the null, rather than to use a χ2 approximation. However, we show in a simulation study that this approximation works so well that simulating critical values seems superfluous. Second, we produce power curves under several alternatives for the test for common variances of the log data in a log-normal model. Third, we find that the asymptotic results for the over-dispersed Poisson model are well approximated in finite samples, at least in our simulations. The simulation study is in Section 6. Finally, we discuss some open questions for future research such as how to choose the sub-sample structure and whether one can select between over-dispersed Poisson and log-normal model. With this, Section 7concludes the paper. Risks 2018,6, 25 4 of 25 2. Data and Sub-Samples Our aim is to test model specification by using statistics that are usually employed to test for common parameters across separate samples. However, we are presented with just a single sample, such as a run-off triangle. Thus, we artificially construct separate samples by splitting the data at hand into sub-samples. Many intuitive splits can be accommodated by the theory, for example, all sub-samples in Figure 1. Here, we define precisely the permissible structures for data and sub-samples, illustrated on an example of a run-off triangle. For the theory in this paper, we assume that data are a generalized trapezoid as defined by Kuang et al. (2008). This flexible format allows for different numbers of accident and development years, and can accommodate missing past and future calendar years. Run-off triangles are a special case with as many accident as development years and only future calendar years missing. For accident year i and development year j , we count calendar years k with an offset so k=i+j− 1. Generalized trapezoids are characterized by the index set I={(i,j):Il≤i≤Iu,Jl≤j≤Ju,Kl≤k≤Ku}, where Il and Iu , Jl and Ju , and Kl and Ku are the smallest and largest accident, development and calendar year indices available, respectively. We denote the number of cells in I by n . The run-off triangle in Table 1, taken from Taylor and Ashe (1983), are a generalized trapezoid with Il=Jl=Kl = 1, Iu=Ju=Ku=10 and n=55. Table 1. Insurance run-off triangle taken from Taylor and Ashe (1983) as an example for a generalized trapezoid. Entries are aggregate incremental paid amounts for claims of accident year i and development year j. Calendar years k=i+j−1 are on the diagonals increasing from the top left. i, j 1 2 3 4 5 6 7 8 9 10 1 357,848 766,940 610,542 482,940 527,326 574,398 146,342 139,950 227,229 67,948 2 352,118 884,021 933,894 1,183,289 445,745 320,996 527,804 266,172 425,046 - 3 290,507 1,001,799 926,219 1,016,654 750,816 146,923 495,992 280,405 - - 4 310,608 1,108,250 776,189 1,562,400 272,482 352,053 206,286 - - - 5 443,160 693,190 991,983 769,488 504,851 470,639 - - - - 6 396,132 937,085 847,498 805,037 705,960 - - - - - 7 440,832 847,631 1,131,398 1,063,269 - - - - - - 8 359,480 1,061,648 1,443,370 - - - - - - - 9 376,686 986,608 - - - - - - - - 10 344,014 - - - - - - - - - We also assume that each sub-sample is a generalized trapezoid. We denote sub-samples by I1 , . . . Im . The sub-samples should be disjoint so Is∩ It=∅ and their union should be the original sample so ∪`I`=I . All sub-samples of the examples in Figure 1are generalized trapezoids. For instance, the sub-sample I2 in Figure 1c is specified by Il 2=Jl 2= 2, Iu 2=Ju 2= 5, Kl 2= 6, Ku 2= 9 and n2=10. The purpose of the generalized trapezoid assumption is to ensure parameter identification later on. We note that this assumption is often more restrictive than needed. Examples for arrays that do not fall into the generalized trapezoid category are arrays with missing cells and disconnected arrays such as the combination of sub-samples I1 and I3 in Figure 1b. However, for many of these arrays identification may still be given and then the theory developed below will still be valid. 3. Log-Normal Model Given data and sub-samples, we can specify a log-normal model, define estimators, and provide the theory for specification testing. The idea is to start with a model that allows parameters to vary across sub-samples and then to test for reductions to a model with common parameters. The latter, most restrictive, model is commonly used in claim reserving. If we reject a reduction to this model, it is Risks 2018,6, 25 5 of 25 likely misspecified. Estimation is done by least squares. The first hypothesis is that log data variances are common across sub-samples; we can test this with a Bartlett test. The second hypothesis is for common linear predictors and can be assessed with an independent F-test. 3.1. Model and Hypotheses The unrestricted model allows both log data means and variances to vary across sub-samples. For this model, we assume that the aggregate incremental claims Yij,` for accident year i , development year j, and sub-sample `are independent log-normal with MLN : log(Yij,`)D =N(µij,`,σ2 `),µij,`=αi,`+βj,`+δ`∀(i,j)∈ I`,`∈ {1, . . . , m}. While we focus on linear predictors µij,` with accident and development year effect, the theory in this paper allows for more general or restrictive linear predictors. For example, we could incorporate calendar year effects as in Zehnwirth (1994)orKuang et al. (2011). The first hypotheses restricts log data variances to be common across sub-samples I` . The remaining assumptions are maintained; thus, linear predictors are still allowed to vary across sub-samples. We write the hypothesis as Hσ2:σ2 `=σ2∀`∈ {1, . . . , m}. The model that arises by imposing this restriction is MLN σ2: log(Yij,`)D =N(µij,`,σ2). The second hypothesis nests the first but also restricts linear predictors to be common across sub-samples. The hypothesis is Hµ,σ2:σ2 `=σ2and µij,`=µij =αi+βj+δ∀`∈ {1, . . . , m}. Under this hypothesis, all parameters are common across sub-samples I` . Thus, we can feasibly drop the sub-script `and write the model under this hypothesis as MLN µ,σ2: log(Yij)D =N(µij,σ2). This is the log-normal geometric chain-ladder model. We can also think about the hypotheses on the original scale. Mean and variance parameters on the log-scale map into median and coefficients of variations on the original scale. Taking the model MLN µ,σ2under Hµ,σ2as an example, log(Yij)D =N(µij,σ2) =⇒Median(Yij) = exp(µij),SD(Yij) E(Yij)=qexp(σ2)−1. Thus, the separation between mean and variance on the log-scale translates to separation between median and coefficient of variation on the original scale. Hence, we can alternatively think of Hσ2 as the hypothesis of common coefficients of variation and of Hµ,σ2 as further imposing common median parameters. 3.2. Estimation We estimate on the log-scale with standard estimators, least squares for log data means and residual sum of squares for log data variances. Since the theory for testing developed below is adapted from a Gaussian framework, estimation on the log-scale is intuitive. Before specifying the estimators, we briefly discuss identification. Risks 2018,6, 25 6 of 25 The identification problem is that µij =αi+βj+δ= (αi+a) + (βj+b) + (δ−a−b) for any a , b . Thus, the levels of accident and development effects are not identified. However, the linear predictors µ are identified (Kuang et al. 2008). These are thus invariant to the identification constraints imposed on the individual effects. Therefore, it does not matter whether we impose ad-hoc constraints such as αIl `=βJl `= 0 or non ad-hoc constraints as suggested by Kuang et al. (2008). We choose to discuss estimation based on the latter, which has the advantage that it allows for straightforward counting of degrees of freedom. By way of example, we apply the identification by Kuang et al. (2008) to a run-off triangle with I={(i , j): 1 ≤i , j , k≤I} . Defining the first difference operator as ∆ , the idea is to re-write µij =µ11 + I ∑ s=2 1(i≤s)∆αs+ I ∑ t=s 1(j≤s)∆βs. Then, µij =x0 ijξ where the design vector xij = ( 1, 1 (i≤2) , . . . , 1 (i≤I) , 1 (j≤2) , . . . , 1 (j≤I) , )0 and the identified parameter vector is ξ= (µ11 , ∆α2 , . . . , ∆αI , ∆β2 , . . . , ∆βI)0 . We denote the number of parameters as p=length(ξ) . The identification method can be extended to generalized trapezoids as well as to linear predictors with calendar year effects. 3.2.1. Estimation in Unrestricted Model MLN For the unrestricted model MLN, we estimate linear predictors as ˆ µLN ij,`=x0 ij,`ˆ ξLN `where ˆ ξLN `= ∑ ij∈I` xij,`x0 ij,`!−1(∑ ij∈I` xij,`log(Yij,`)). With degrees of freedom d f`=n`−p`, we estimate log data variances by ˆ σ2,LN `=RSS` d f` where RSS`=∑ ij∈I` {log(Yij,`)−ˆ µLN ij,`}2. (1) 3.2.2. Estimation with Common Variances in MLN σ2 Imposing the restriction of common log data variances Hσ2 does not require re-estimation as the estimators from MLN can be re-used. The estimators for the linear predictors µij,` are identical to those of MLN. The log data variance in MLN σ2is estimated by ¯ σ2,LN = m ∑ `=1 d f` d f. ˆ σ2,LN `=RSS. d f.(2) where d f.=∑m `=1d f`and RSS.=∑m `=1RSS`. 3.2.3. Estimation with Common Variances and Linear Predictors in MLN µ,σ2 Under the hypothesis HLN µ,σ2 , which imposes common log data mean and variance parameters, both estimators change. We drop the ` -subscript indicating the sub-sample since estimation is done over the full sample I. With that, we write the estimators for the linear predictors in MLN µ,σ2as ˆ µLN ij =x0 ij ˆ ξLN with ˆ ξLN = ∑ ij∈I xijx0 ij!−1(∑ ij∈I xij log(Yij)). We estimate the log data variance σ2under this hypothesis, defining d f =n−p, by Risks 2018,6, 25 7 of 25 ˆ σ2,LN =RSS d f where RSS =∑ ij∈I {log(Yij)−ˆ µLN ij }2. 3.2.4. Remarks Least squares estimation for the identified parameter vector ξ is maximum likelihood estimation in the log-normal model. Kuang et al. (2015) derive a representation of the least squares estimators that is interpretable as a geometric chain-ladder, in contrast to the classic, arithmetic, chain-ladder. For many regression models, there is little difference between scaling the residual sum of squares by the degrees of freedom or the number of observations; the former yields an unbiased estimator for σ2 , the latter the maximum likelihood estimator. However, the scaling does matter here due to the large parameter to observation ratio. By way of example, the Taylor and Ashe (1983) data has n= 55 observations but only d f = 36 degrees of freedom so that ˆ σ2,LN is some 50% larger then the rival estimator RSS/n. This is amplified for the sub-samples. 3.3. Testing for Common Variances We show how to test for common log data variances, that is for Hσ2 in MLN using a Bartlett test. In a special case with two sub-samples, we can use an F-test instead of a Bartlett test. The Bartlett test (Bartlett 1937) was designed to test for common variances across several Gaussian samples. Thus, it is directly applicable to the log sub-samples. We only give a rough overview of the theory; for a more detailed derivation in contemporary terminology see Jørgensen (1993, pp. 94–96). The test rests on the independent χ2 -distribution of ˆ σ2,LN ` in MLN . Rather than deriving a test in the Gaussian model for log(Yij,`) ,Bartlett (1937) considers a joint χ2 model for the variance estimators. In this χ2model, the log-likelihood ratio statistic for the hypothesis Hσ2is LRLN =d f.log(¯ σ2,LN)− m ∑ `=1 d f`log(ˆ σ2,LN `)(3) for ˆ σ2 and ¯ σ2 as defined in (1) and (2) , respectively. Define now the Bartlett distribution Ba(·) such that LRLN D =Ba(d f1 , . . . , d fm) under the hypothesis. Considering LRLN as a function of the estimators so LRLN =LR(ˆ σ2,LN 1, . . . , ˆ σ2,LN m), the Bartlett distribution is characterized by P{Ba(d f1, . . . , d fm)≤y}=ZA(y) m ∏ `=1 dG`(x`)(4) where G`(·) is the χ2 d f` cdf and A(y) = {(x1 , . . . , xm):LR(x1 , . . . , xm)≤y} . Likelihood theory tells us that Ba(d f1 , . . . , d fm) and thus LRLN approaches a χ2 m−1 as min(d f1 , . . . , d fm) goes to infinity. However, Bartlett (1937) goes a step further and suggest to divide LRLN by C=1+1 3(m−1) m ∑ `=1 1 d f` −1 d f.!. Comparing LRLN/C rather than LRLN to a χ2 m−1 substantially improves the quality of the approximation and makes it useful even in rather small samples. That is, under Hσ2, BLN =LRLN C D ≈χ2 m−1. (5) The Bartlett correction factor C improves the order of magnitude of the error term. This idea has been shown to apply generally to likelihood ratio tests; see, for instance, Lawley (1956) and Barndorff-Nielsen and Cox (1984). While using an asymptotic approximation for the Bartlett test is appealing, we could also simulate critical values of the exact distribution. This is feasible because the exact distribution of Risks 2018,6, 25 8 of 25 LRLN , Ba , is free of nuisance parameters. However, if Ba/C is sufficiently close to χ2 m−1 , simulating the critical values may be unnecessary even for rather small degrees of freedom. Looking ahead, we confirm in a simulation study in Section 6.1 that the asymptotic approximation indeed works very well. As an alternative to the Bartlett test, we can test the equality of dispersion parameters across two sub-samples with an F -test that is not equivalent to a Bartlett test. The F -test follows quickly given independence and distribution of the log data variance estimators σ2,LN `in (1). Under Hσ2, FLN σ2=σ2,LN 2 σ2,LN 1 D =Fd f2,d f1(6) so that we can use a (two-sided) F -test to test the hypothesis; see, for example, Snedecor and Cochran (1967, chp. 4.15). We can write LRLN as a function of FLN σ2. With r=d f2/d f1, LRLN =LR(FLN σ2) = d f1log 1+rFLN σ2 1+r!+d f2log (1+ (rFLN σ2)−1 1+r−1). (7) This mapping is not monotone. Intuitively, the Bartlett test is one-sided compared to a two-sided F -test. Thus, we would expect LR to be increasing both for small and large FLN σ2 . We can now find scenarios in which the F -test and the Bartlett test lead to different decisions: for example, with d f1= 1 and d f2= 2 an equal-tailed 5% F -test just about rejects the null for a draw FLN σ2= 0.025, while a 5% Bartlett test does not reject with LR( 0.025 ) = 4.23 and a (simulated) exact critical value of 4.91. This leaves the question which test should be used; we investigate this in Section 6.2. Usually, a drawback of both F and Bartlett test is their sensitivity to departures from Gaussianity of the log data log(Yij,`).Box (1953) goes as far as comparing the Bartlett test to a test for Gaussianity and argues in favor of robust tests, prioritizing robustness over other qualities such as power. However, sensitivity to non-Gaussianity is not necessarily undesirable for an application to insurance claim-reserving since distribution forecasts of the log-normal model would also be invalid if the data is not log-normal. Besides, we find F -test and Bartlett test appealing for their simplicity and because they carry over to over-dispersed Poisson models as we will see later. Thus, we do not consider methods to improve robustness to departures from Gaussianity such as made by Shoemaker (2003) for F-tests. 3.4. Testing for Common Linear Predictors Now that we know how to test for common variances, we turn to testing for common linear predictors. The idea is to test sequentially: first for common variances, then for common linear predictors. We show how to use an F -tests for the latter and prove that this test is independent of Bartlett and F-tests for common variances. Thus, size control is not an issue. If we take the model with common variances MLN σ2 as given, then testing for Hµ,σ2 amounts to testing for common linear predictors. Since standard Gaussian theory applies, FLN µ=(RSS −RSS.)/(d f −d f.) RSS./d f. D =Fd f −d f.,d f. under the hypothesis. Thus, we can use a (one-sided) F -test to test for a reduction from MLN σ2 to MLN µ,σ2 . Unlike the dispersion Bartlett and F -tests, this F -test is equivalent to the corresponding exact Gaussian likelihood ratio test. However, a χ2 approximation to the likelihood ratio test may not work well due to rather few degrees of freedom. Thus, we prefer the F-test since it is easier to implement. A sequential test approach for common variance and common linear predictors is sensible. This is because we can show the tests are independent. We formulate the independence result in a theorem; all proofs are in Appendix A. Theorem 1. In model MLN µ,σ2, the test statistic FLN µis independent of FLN σ2and LRLN. Risks 2018,6, 25 15 of 25 thrice repeated sequential tests with a Bonferroni correction, we would reject if any p-value was below 5%/3 ≈0.017. This is not the case so the model survives this battery of tests as well. 5.3. Log-Normal (Extended) Chain-Ladder As a final empirical application, we look at a run-off triangle first considered by Barnett and Zehnwirth (2000) . We show this data in Table A2. These data are known to be modeled best with a predictor with not just accident and development, but also calendar effects. We look at a model with and without calendar effects. Barnett and Zehnwirth (2000) and also Kuang et al. (2011) consider a log-normal model for this data and we follow them in this choice. As before, we split the data, specify the model, and test for the hypotheses. The results are summarized in Figure 4. i,j1 5 11 1 I1 I2 5 I3 11 (a) Sub-sample structure ˆ σ2,LN `d f`¯ σ2,LN d f.ˆ σ2,LN d f I10.0002 6 I20.0013 6 I30.0015 9 I0.0011 21 0.0068 45 BLN =6.06 pχ2 2=0.05 FLN µ=11.20 pF20,16 =0.00 ˆ σ2,LNe `d f e `¯ σ2,LNe d f e .ˆ σ2,LNe d f e I10.0003 3 I20.0013 5 I30.0016 8 I0.0013 16 0.0014 36 BLNe =2.06 pχ2 2=0.36 FLNe µ=1.13 pF20,16 =0.41 (b) Estimation and test results Figure 4. Log-normal chain-ladder ( LN ) and extended chain-ladder ( LNe ) model for Barnett and Zehnwirth (2000) data. Sub-sample structure shown in (a), estimation and test results in (b). Figure 4a shows the sub-sample structure we choose. Given the apparent need for calendar effects, we aim to maximize power for varying dispersion parameters along the same time dimension and split the run-off triangle, this time with eleven accident and development years, after periods five and eight into three sub-samples. The top of Figure 4b shows estimation and test results for a model without calendar effect. This model is given by MLN : log(Yij,`)D =N(αi,`+βj,`+δ`,σ2 `). A Bartlett test for the hypothesis Hσ2 of common log data variances has a χ2 p-value of (just under) 0.05. We may consider this as evidence against Hσ2 . For comparison with the model with calendar effect considered next, we still compute an F -test for the hypothesis Hµ,σ2 . We point out that this test is not strictly a test for common linear-predictors if we are not comfortable to accept MLN σ2 as a model. The statistic FLN µ=11.20 has a 0.00 p-value so that we reject Hµ,σ2. Thus, MLN µ,σ2is not well specified. At the bottom of Figure 4b we show results for a model with calendar effects γ for calendar years k=i+j−1. The model is MLNe : log(Yij,`)D =N(αi,`+βj,`+γk,`+δ`,σ2 `). The theory for specification tests is not affected by this change and thus still valid. A Bartlett test for Hσ2 in this model yields a χ2 p-value of 0.36 so we may feel comfortable to impose common Risks 2018,6, 25 16 of 25 log data variances and take MLNe σ2 as given. An F -test for common linear predictors leaves us with ap-value of 0.41. Thus, reducing the model to MLNe µ,σ2 seems sensible. Therefore, we cannot reject the specification of the model with calendar effect. If we directly compare the two models, we can see that the calendar effect has a substantial impact on the specification tests. While the model with calendar effect seems to be well specified, the model without this effect raises red flags for both a test for common variances and common linear predictors. The test for common linear predictors is much more strongly affected by dropping the calendar effect than the Bartlett test. This indicates that the shift in log data variances is smaller than that in linear predictors. We look at the shift in linear predictor in two ways. First, we can directly test for for dropping the calendar effects from the well specified MLNe µ,σ2 . A standard F -test for the hypothesis Hγ:γk= 0 ∀k yields a p-value of 0.00, consistent with the rejection of the model without calendar effects MLN µ,σ2 above. Alternatively, we can test for a reduction from MLNe σ2: log(Yij,`)D =N(αi,`+βj,`+γk,`+δ` , σ2) to MLN σ2 , corresponding to the hypothesis Hγk,`:γk,`= 0 ∀k , ` . This reduction allows for breaks in linear predictors between sub-samples. Interestingly, an F -test cannot reject Hγk,` (p-value 0.92). As an intuition, we recall that the chain-ladder predictor without calendar effects can accommodate a constant trend in calendar years, but not deviations from that trend. Thus, allowing for separate sets of linear predictors on the sub-samples implicitly allows for three different calendar trends. While still less flexible than the model with an effect for each calendar year, this seems to be good enough. Note, however, that the Bartlett flags the reduction from MLN to MLN σ2(but not from MLNe to MLNe σ2). Overall, the analysis suggests that calendar effects are needed in this data set for two reasons for this sub-sample structure: to capture the structure of the linear predictors themselves, and, to a lesser extent, to achieve homogeneous variance across the log data. We note that for this data, repeating the tests for different sub-samples structures does affect the results. Indeed, considering sub-samples similar to before, the specification of the log-normal extended chain-ladder model is rejected. Specifically, splitting the data into two sub-samples after the fifth accident year, a Bartlett test yields a p-value of 0.017 and an F -test a p-value of 0.004. Considering four sub-samples with splits after the fifth calendar year, the fifth development year and the sixth accident year, the p-value of the Bartlett test is 0.03 and that of the F -test 0.05. Again controlling the size of the repeated tests with a Bonferroni correction, we would reject the null hypothesis if we can find ap-value below about 0.017. This is the case for the F -test and a knife-edge decision for the Bartlett test in the two sub-sample scenario. Thus, for this data we may want to consider a different model or at least be somewhat more skeptical of its results. 6. Simulations The developed theory begs several questions that we answer in a simulation study. First, we argued that we can sidestep simulating critical values of the Bartlett distribution Ba and instead approximate these by a Bartlett corrected χ2 critical value. We show that this works very well. Second, we compute power curves of Bartlett and F -test for common log data variances under several alternatives in a log-normal model to get a better understanding for the tests’ behavior. Third, we show that an asymptotic approximation in an over-dispersed Poisson model resembles the asymptotic distribution closely, both under the null and the considered alternatives. Finally, we derived above that F -tests for common linear predictors in the over-dispersed Poisson model are asymptotically independent of tests for common over-dispersion. We confirm that the size of the former test seems unaffected by conditioning on the results of the latter, even in finite samples. 6.1. Performance of Bartlett test χ2Approximation The theory tells us that the distribution of Ba/C , which is the exact distribution of the Bartlett statistic BLN in the log-normal model, is close to a χ2 for large degrees of freedom. We show that the approximation works very well for a range of degrees of freedom. Risks 2018,6, 25 17 of 25 We draw realizations from the adjusted Bartlett distribution Ba(d f1 , . . . , d fm)/C as follows. For `= 1, . . . , m , we draw independent χ2 distributed V` with d f` degrees of freedom and compute s`=V`/d f`and ¯ s=∑m `=1d f`/d f.s`. Then, {d f.log(¯ s)−∑m `=1d f`log(s`)}/Cis Ba/Cdistributed. Figure 5a shows the upper 10% probability spectrum of a pp-plot for the adjusted Bartlett distribution Ba(d f1 , . . . , d fm)/C against a χ2 m−1 . We show plots for the tuples ( 26, 6 ) , ( 3, 5, 8 ) , ( 6, 6, 9 ) , and ( 6, 3, 6, 6 ) encountered in the empirical applications above. The plots are based on 10 7 draws for each tuple. The plots seem indistinguishable from the 45-degree line, even though we zoomed in to the upper 10% of the spectrum. 0.9 0.925 0.95 0.975 1 P (2 q ) 0.9 0.925 0.95 0.975 1 P (Ba/ C q ) (26, 6) (3, 5, 8) (6, 6, 9) (6, 3, 6, 6) ( a ) Degrees of freedom from empirical applications 0.9 0.925 0.95 0.975 1 P (2 q ) 0.9 0.925 0.95 0.975 1 P (Ba/ C q ) (13, 3) (1, 2, 4) (3, 3, 4) (3, 1, 3, 3) (b) Half the empirical degrees of freedom Figure 5. pp-plots for the adjusted Bartlett distribution Ba/C against χ2 for varying degrees of freedom. ( a ) and ( b ) show results for degrees of freedom corresponding to the empirical applications and half those degrees of freedom, respectively. Figure 5b is constructed in the same way as Figure 5a, except the degrees of freedom are halved and rounded down. Now, we can see some deviations from the 45-degree line. As expected, we can see convergence to the 45-degree line as the degrees of freedom increase. In Table 2, we take a closer look at the approximation at α= 1%, 5%, 10% critical values cα of a χ2 m−1 specifically. The table shows P(Ba/C>cα) , corresponding to the true size of a Bartlett test in a log-normal model if we use the χ2approximation rather than simulated critical values. Table 2. P(Ba/C>cα) where cα is the χ2α critical value. Results are in %. Degrees of freedom shown as (d f1, . . . , d fm). (13, 3) (1, 2, 4) (3, 3, 4) (3, 1, 3, 3) (26, 6) (3, 5, 8) (6, 6, 9) (6, 3, 6, 6) α=10% 9.94 9.05 9.86 9.20 9.98 9.93 9.97 9.92 α=5% 4.93 4.22 4.85 4.37 4.98 4.92 4.97 4.92 α=1% 0.95 0.69 0.92 0.76 0.99 0.95 0.98 0.96 While we can see some differences for some of the halved critical values, we would argue that the approximation for the degree of freedom tuples from the empirical applications is so good that using it is reasonable and should not affect the modeling decision. 6.2. Rejection Frequencies of Tests for Common Variance in Log-Normal Model As a supplement to the behavior of the tests for common log data variance under the null hypothesis in the log-normal model given in Section 3.3, we now also take a look at power. We simulate the three sub-sample structures from the empirical applications and consider rejection frequencies of the tests used in the corresponding applications. We find that the Bartlett and F -test for common Risks 2018,6, 25 18 of 25 variance have very similar power, at least in this simulation. Further, we see that the power does not necessarily decrease with the number of sub-samples. For the sub-sample structures from the empirical applications (see Figure 1), we simulate MLN : log(Yij,`)D =N(µij,` , σ2 `) . Thus, we simulate for m= 2, 3, 4 sub-samples. Before specifying the parameter values, we point out that the distribution of tests in this model depends only on ratios σ2 s/σ2 t , the degrees of freedom d f`, and the number of sub-samples m. To see this, we first re-write LRLN =d f.log(¯ σ2,LN)− m ∑ `=1 d f`log(ˆ σ2,LN `) = m ∑ `=1 d f`log (d f` d f. ` ∑ s=1 RSSs RSS`!). (12) Now, under MLN , RSS` D =σ2 `χ2 d f` independently. Thus, the distribution of LRLN is invariant to common changes in levels of σ2 ` as well as to µij,` . Therefore, we can normalize the smallest σ2 ` to unity and set µij,`= 0 without loss of generality. The distribution of the F -statistic FLN σ2 shares these properties. For each sub-sample scenario, we consider a range of values for the log data variance ratios σ2 s/σ2 t . For m> 2 sub-samples there is more than one ratio such that we cannot effectively visualize all combinations. We thus consider the following special case. For each sub-sample structure, we the compute the spacing of the estimates from the corresponding empirical application. That is, we order the empirical estimates ˆ σ2 (1)<· · · <ˆ σ2 (m) and compute the m spacing-coefficients x`= (ˆ σ2 `−ˆ σ2 (1))/(ˆ σ2 (m)−ˆ σ2 (1)) . We note that x(1)= 0 and x(m)= 1. The spacings (x1 , . . . , xm) in the empirical examples are ( 1, 0 ) (Verrall et al. 2010), ( 0, 0.76, 1 ) (Barnett and Zehnwirth 2000), and (0.09, 1, 0.58, 0)(Taylor and Ashe 1983). The log data variance for the `-th subset is then σ2 `=σ2 (1)+x`(σ2 (m)−σ2 (1)). (13) To trace out power curves, we vary the largest ratio σ2 (m)/σ2 (1) from one, corresponding to Hσ2: σ2 `=σ2 , to twenty in 0.5 increments. As noted above, we can set σ2 (1)= 1 without loss of generality. For each degree of freedom scenario and for each ratio σ2 (m)/σ2 (1), we draw 106sub-samples. For each draw, we compute the test statistics used in the corresponding empirical application, FLN σ2 as in (6) and BLN as in (5) . We note that for m= 3, we compute only the Bartlett test statistic for the model with calendar effect BLNe to make the plot less cluttered. Thus, the degrees of freedom for ˆ σ2,LN ` in the three scenarios are (26, 6),(3, 5, 8)and (6, 3, 6, 6). We use χ2critical values for the Bartlett tests. Figure 6shows rejection frequencies at 5% critical values. We can see that all tests have the right size under Hσ2 , that is for σ2 (m)/σ2 (1)= 1. The power of two-sided F -test and Bartlett test in the two sub-sample scenario is very similar with a slight advantage for the Bartlett test. Thus, the choice between the two test may mostly depend on taste. Comparing Bartlett tests across scenarios, we see that the power for m= 4 sub-samples is larger than that for m= 3 sub-samples. Thus, fewer sub-samples do not necessarily imply higher power. Intuition comes from the degree of freedom weighting. For m= 3 sub-samples, if we drop the variance with the smallest degree of freedom the larger two variances are relatively homogeneous. Meanwhile, for m= 4 sub-samples there is still plenty of variation left among the largest three variances. Thus, since the test attributes more weight to the better estimates with higher degrees of freedom, the scenario with m= 3 sub-samples is a rather tough case. We indicated the σ2 (m)/σ2 (1) ratios we found in the individual empirical applications by vertical lines. We recall that the spacing of intermediate variances is taken from the empirical applications. Therefore, suppose that the empirical estimates are the truth such that Hσ2 is violated. Then we can read of the power against this scenario directly from the plot. For example, in the application to the Verrall et al. (2010) data, the F -test would have a power of about 35% while the Bartlett test power would be closer to 40%. Risks 2018,6, 25 19 of 25 1 5 10 15 20 2 (m)/2 (1) 0 0.05 0.2 0.4 0.6 0.8 1 Rejection Frequency VNJ BZ TA FLN 2 ( m = 2) BLN ( m = 2) BLNe ( m = 3) BLN ( m = 4) Figure 6. Power curves for log-normal dispersion tests based on sub-sample structures from empirical applications. Empirical maximum to minimum ratios indicated by horizontal lines. BZ is short for Barnett and Zehnwirth (2000), VNJ for Verrall et al. (2010), and TA for Taylor and Ashe (1983). 6.3. Performance of Over-Dispersed Poisson Model Asymptotics The theoretical results for the over-dispersed Poisson model are asymptotic, rather than exact as in the log-normal model. We show that an asymptotic approximation works well. Tests for common over-dispersion have the right size under the null. The power under the alternative in finite samples is close to the asymptotic power. Further, F -tests for common linear predictors conditional on non-rejection of over-dispersion tests are very close to F distributed in finite samples. 6.3.1. Rejection Frequencies of Tests for Common Over-Dispersion We can use the rejection frequencies from the log-normal simulations as a benchmark for those in the over-dispersed Poisson model. To see this we recall that as the overall mean E(Y..)→∞ , the over-dispersion estimator ˆ σ2,ODP ` D →σ2 `χ2 d f`/d f` in the over-dispersed Poisson model MODP . This matches the exact distribution of ˆ σ2,LN ` in MLN . Thus, asymptotically, the distribution of LRODP in MODP and LRLN in MLN are identical for identical ratios σ2 s/σ2 t . The same holds for FODP σ2 and FLN σ2 . We simulate for the same three sub-sample structures as in the log-normal simulations. For the simulation design, we set-up an unrestricted model MODP that satisfies the assumptions in Section 4.1. For the distribution of the cells we choose compound Poisson-gamma so Yij,`=∑Cij,` s=1Xs,` where Cij,` D =Poisson{exp(µij,`)} independent of the i.i.d. gamma distributed Xs,` with scale σ2 `− 1 and shape (σ2 `− 1 )−1 . We note that the parametrization for the linear predictors µij,` and the level of the over-dispersion σ2 ` matters in finite samples. This is in contrast to the log-normal model. The reason is that the finite sample distribution of ˆ σ2,ODP ` in MODP is generally not σ2 `χ2 d f`/d f` . Thus, for each considered scenario, we set the linear predictors µij,` to the estimates ˆ µODP ij,` from the data in the corresponding empirical application. Similarly, we set the smallest over-dispersion σ2 (1)=ˆ σ2,ODP (1). We again vary the ratios σ2 (m)/σ2 (1) from one to twenty, using the exact same spacing x` from (13) in the log-normal simulations so σ2 `=σ2 (1)+x`(σ2 `−σ2 (1)) . The only difference is that now σ2 (1)=ˆ σ2,ODP (1) . Therefore, asymptotically, the power for a common σ2 (m)/σ2 (1) is identical in over-dispersed Poisson and log-normal models. We draw 10 6 sub-samples for each over-dispersion ratio σ2 (m)/σ2 (1) and sub-sample structure. Figure 7a shows the rejection frequencies at 5% critical values for the four test statistics from the empirical applications as in the log-normal model but now computed based on ˆ σ2,ODP `. Risks 2018,6, 25 20 of 25 1 5 10 15 20 2 (m)/2 (1) 0 0.05 0.2 0.4 0.6 0.8 1 Rejection Frequency VNJ BZ TA FODP 2 ( m = 2) BODP ( m = 2) BODPe ( m = 3) BODP ( m = 4) (a) Rejection frequencies 1 5 10 15 20 2 (m)/2 (1) 0.00 0.02 0.04 0.06 0.08 0.10 Asymptotic Rejection Frequency VNJ BZ TA (b) Gap to asymptotic rejection frequencies Figure 7. Power gap for log-normal dispersion tests based on sub-sample structures from empirical applications. Empirical maximum to minimum ratios indicated by horizontal lines. Rejection frequencies shown in (a), gap to asymptotic rejection frequencies in (b). For σ2 (m)/σ2 (1)= 1 we are under the null; we can see that the rejection frequencies are very close to 5% so the tests have the correct size. Under the alternative where σ2 (m)/σ2 (1)> 1, the ordering of the rejection frequencies matches that in the log-normal simulations (Figure 6). Generally, the plot is reassuringly reminiscent of its equivalent in the log-normal simulations. Figure 7b shows the gap to the asymptotic rejection frequencies that arises in the finite sample simulations. The simulation set-up implies that this is the difference between rejection frequencies in log-normal and over-dispersed Poisson simulations. Thus, the plots shows the impact of the asymptotic approximation in the over-dispersed model. Since the difference under the alternative is positive throughout, the power in the over-dispersed model is lower than in the log-normal model. We next interpret the plots under the alternative in turn for the three sub-sample scenarios. For m= 2, the power gap of Bartlett and F -test initially increases with σ2 (m)/σ2 (1) , hitting 10 pp (percentage points)for the F -test at σ2 (m)/σ2 (1)≈ 7.5, before it decreases. The initial increase relates to the asymptotic theory by Harnau and Nielsen (2017) which assumes fixed dispersion parameters. Since we keep σ2 (1) constant, the remaining dispersion parameters grow with the ratio. Thus, we would require larger cell-means to achieve the same asymptotic approximation quality. The later decrease reflects the upper bound of one for the power: even as the asymptotic approximation becomes worse, the difference between dispersion parameters becomes so large that it is easily caught. For m= 4, the power gap is increasing throughout the considered range for σ2 (m)/σ2 (1) . The intuition for the increase again comes from the asymptotic theory. We do not see a decrease since the power is still quite far from unity, staying below 80% even for the largest maximum to minimum ratio. Meanwhile, for m= 3, the power gap is essentially zero so that the finite sample power matches the asymptotic power. The intuition for this follows because the dispersion to mean ratio is small. As a rough indication, dividing the largest considered dispersion 20 ·ˆ σ2,ODP (1) by the mean over all cells n−1∑ij Yij yields 0.8% for the Barnett and Zehnwirth (2000) simulations compared with 70% and 56% for the Verrall et al. (2010) and Taylor and Ashe (1983) simulations, respectively. We again indicate the power at the particular alternative generated by taking the estimates in the empirical applications as true values by vertical lines. Figure 7b shows that for these alternatives, the power for all asymptotic approximations is within 5pp of their asymptotic power. Risks 2018,6, 25 21 of 25 6.3.2. Independence of Test for Common Linear Predictors We move on to evaluate the quality of a finite sample approximation to the asymptotic independence in Lemma 1. Specifically, we consider the finite sample distribution of FODP µ as in (11) given that a tests for common over-dispersion did not reject. Arguably, this is the most interesting case since it matches the natural order of the two specification tests. We simulate under the null Hµ,σ2 , that is for a model with common linear predictors and over-dispersion MODP µ,σ2 . As before, cells Yij are compound Poisson-gamma. We consider three scenarios, setting the parameters to the estimates for MODP µ,σ2 in the three empirical examples. We draws 10 6 triangles per scenario. For each draw, we compute tests based on the sub-sample structure of the corresponding empirical application. We first conduct a Bartlett test for Hσ2 at 5% critical values. If we do not reject Hσ2 based on this test, we keep the triangle, otherwise we throw it out. Since we simulate under the null hypothesis, we thus keep about 95% of the draws. Only for the draws we keep do we compute the F -statistic for common linear predictors FODP µ. Figure 8shows a pp-plot for the FODP µ against Fd f −d f.,d f. for the triangles that survived Bartlett testing. 0.900 0.925 0.950 0.975 1.000 P (F df df ., df . q ) 0.900 0.925 0.950 0.975 1.000 P ( FODP q ) m = 2 (VNJ) m = 3 (BZ) m = 4 (TA) Figure 8. Distribution of FODP µconditional on non-rejection of a 5% Bartlett test. To be able to tell a difference from the 45-degree line, we limit our attention to the upper 10% of the probability spectrum since. This is also the most interesting range for testing. Even in this spectrum, each plot is very close to the 45-degree line. Therefore, under Hµ,σ2 , we can be reassured that an F -test for common linear predictors has the correct size in finite samples even if we apply it only conditionally on non-rejection of a test for common over-dispersion. 6.4. Remark We note that all simulations are for tests that consider the correct sub-sample structure under the alternative. Of course, this does not seem realistic in applications. However, for tests computed on a given sub-sample structure, it appears we would generally be able to choose a true, different, sub-sample structure against which the tests would at best have limited power. For example, say we compute the tests on the two sub-samples with a split after the fifth accident year in Figure 1a while really there are three sub-samples with an additional split after the fifth development year. Then, we could choose parameterizations for the three true true sub-samples to balance out the variation between the two incorrectly chosen sub-samples, thus minimizing power. Therefore, it seems to us that such simulation results would be almost entirely driven by our chosen parametrization and provide little insight beyond that. We believe the real answer to this problem must come from a theory that is Risks 2018,6, 25 22 of 25 agnostic to the sub-sample structure as discussed below. However, we stress again that the size of the tests under the null hypothesis is not affected by the chosen sub-sample structure. 7. Discussion Some questions are left open for future research. For example, it is not clear how to best choose the sub-sample structure and the number of sub-samples. Further, the question arises whether we can somehow select between the over-dispersed Poisson and log-normal model. Finally, a misspecification test for independence of the cells would be a useful addition to the modeling toolkit. So far, we chose the sub-sample structures somewhat arbitrarily if potentially informed by prior knowledge of the data. While the size of the tests under the null is not affected by the sub-sample structure, the power of the tests under the alternative is affected both by the chosen number of sub-samples and their structure. In applications, the expert may consider choosing a range of sub-samples structures and conducting tests for each, adjusting the size based on the number of tests to account for multiple testing as shown in the empirical applications. For future research, it would be useful to derive a theory that is agnostic to the number of sub-samples and their structure while still directly controlling size. It might be fruitful to look for ideas in time-series econometrics which has been concerned with tests for parameter breaks for a long time. In this literature, Chow (1960) had proposed a test for parameter breaks that required knowledge of the breakpoint. By now, there are several test available that are agnostic with respect to the number of breaks, related to the number of sub-samples in our problem, and the position of breaks, akin to the sub-sample structure. Examples include Andrews’ test (Andrews 1993), generalizations of Chow tests (Nielsen and Whitby 2015), and indicator saturation (Hendry 1999). However, these tests are designed for data with a single time-scale and results are generally based on long time-series. In contrast, we are confronted with data with three interlinked time-scales and the arrays are often small with a large number of parameters that is growing with the array size. Thus, the known results do not carry over and a it appears that a new theory is needed. Since we have seen two models in this paper, log-normal and over-dispersed Poisson, a natural question is when we should choose which model. As we have seen, the log-normal model assumes a fixed standard deviation to mean ratio while the over-dispersed Poisson model considers the variance to mean ratio to be fixed. Making use of recent results for generalized log-normal models by Kuang and Nielsen (2018) , a class of models that includes the log-normal but is more general, Harnau (2018) proposes a test to distinguish between (generalized) log-normal and over-dispersed Poisson models based on this discrepancy. Finally, a misspecification test for the assumption that the cells in the array are independent would be useful. This is an assumption that both the log-normal and the over-dispersed Poisson model impose. In contrast, the “distribution free” model by Mack (1993) relaxes this somewhat, assuming independence only across accident years. Acknowledgments: The author thanks two anonymous referees for their helpful comments. Discussions with Bent Nielsen (Department of Economics, University of Oxford & Nuffield College, UK) are gratefully acknowledged. The author was supported by the Economic and Social Research Council, grant number ES/J500112/1, and the European Research Council, grant AdG 694262. Conflicts of Interest: The author declares no conflict of interest. Appendix A Appendix A.1. Proof of Theorem 1 The proof relies on two properties. First, RSS −RSS ., the numerator of FLN µ , reduces to a comparison of least squares fitted log-means ∑m `=1∑ij∈I`(ˆ µLN ij −ˆ µLN ij,`)2 , and is therefore, in the Gaussian framework at hand, independent of the residual sum of squares RSS1, . . . , RSSm. Risks 2018,6, 25 23 of 25 Second, the denominator of FLN µ , the aggregated residual sum of squares RSS. and the relative contributions π1 , . . . , πm for π`=RSS`/RSS. are mutually independent. To see this, we first recall that under the hypothesis RSS1 , . . . , RSSm are independent σ2χ2 . The proof is unaffected by setting σ2= 1. Thus, let X1 , . . . Xm be independent χ2 . Recall that the sum of independent χ2 ’s is χ2 . Let S`=∑` s=1Xs and V`=S`−1/S` . We note that we can map V2 , . . . , Vm to the frequencies X1/Sm , . . . , Xm/Sm . For the special case with m= 2, Johnson et al. (1995, p. 212) note the independence of S2 and V2 . The general case for independence of Sm and V2 , . . . , Vm can be proved by induction. Here, we partially replicate the (originally Danish) argument from Andersson and Jensen (1987, p. 180). We show the induction step from m− 1 to m . Suppose V2 , . . . , Vm−1 , Sm−1 and Xm are independent. Then Sm=Sm−1+Xm and Vm=Sm−1/Sm . Sm and Vm match the setting for the special case with m= 2 from above and are thus independent. Hence, V2 , . . . , Vm and Sm are independent, completing the induction step. Independence of V2, . . . , Vmand Smimplies independence of π1, . . . , πmand RSS., Taken together, RSS −RSS ., RSS. , and π1 , . . . , πm , are mutually independent. Now, we can write the test statistics for the dispersion parameters as functions of the relative contributions π`: LRLN =LR(π1, . . . , π`) = m ∑ `=1 d f`log d f` d f.−log(π`),FLN σ2=d f1 d f2 π2 π1 . Thus, FLN µis independent of FLN σ2and LRLN. Appendix A.2. Proof of Theorem 2 This follows from (8) , independence of d` across ` due to disjoint sub-samples made up of independent Yij, the continuous mapping theorem, and the results discussed in Section 3. Appendix A.3. Proof of Lemma 1 Once we show that D−D. , D. and D1/D. , . . . , Dm/D. are asymptotically mutually independent, the result follows from the proof of Theorem 1since the asymptotic distribution of the deviances in the over-dispersed Poisson model matches the exact distribution of the residual sum of squares in the log-normal model. We can set σ2= 1 without loss of generality. Then, to prove mutual independence, we build on the insight of Harnau and Nielsen (2017) that asymptotics for the over-dispersed Poisson model match standard exponential family asymptotics. Thus, D−D. and D1 , . . . , Dm are asymptotically equivalent to quadratic forms (Johansen 1979, Theorem 7.8) of asymptotically Gaussian projections on orthogonal subspaces (Johansen 1979, Theorem 7.6). Thus, independence and hence Lemma 1follows. Table A1. Insurance run-off triangle taken from Verrall et al. (2010, Table 1) as used in the empirical application in Section 5.1 and the simulations in Section 6. i, j 1 2 3 4 5 6 7 8 9 10 1 451,288 339,519 333,371 144,988 93,243 45,511 25,217 20,406 31,482 1729 2 448,627 512,882 168,467 130,674 56,044 33,397 56,071 26,522 14,346 - 3 693,574 497,737 202,272 120,753 125,046 37,154 27,608 17,864 - - 4 652,043 546,406 244,474 200,896 106,802 106,753 63,688 - - - 5 566,082 503,970 217,838 145,181 165,519 91,313 - - - - 6 606,606 562,543 227,374 153,551 132,743 - - - - - 7 536,976 472,525 154,205 150,564 - - - - - - 8 554,833 590,880 300,964 - - - - - - - 9 537,238 701,111 - - - - - - - - 10 684,944 - - - - - - - - - Risks 2018,6, 25 24 of 25 Table A2. Insurance run-off triangle taken from Barnett and Zehnwirth (2000, Table 3.5) as used in the empirical application in Section 5.1 and the simulations in Section 6. i, j 1 2 3 4 5 6 7 8 9 10 11 1 153,638 188,412 134,534 87,456 60,348 42,404 31,238 21,252 16,622 14,440 12,200 2 178,536 226,412 158,894 104,686 71,448 47,990 35,576 24,818 22,662 18,000 - 3 210,172 259,168 188,388 123,074 83,380 56,086 38,496 33,768 27,400 - - 4 211,448 253,482 183,370 131,040 78,994 60,232 45,568 38,000 - - - 5 219,810 266,304 194,650 120,098 87,582 62,750 51,000 - - - - 6 205,654 252,746 177,506 129,522 96,786 82,400 - - - - - 7 197,716 255,408 194,648 142,328 105,600 - - - - - - 8 239,784 329,242 264,802 190,400 - - - - - - - 9 326,304 471,744 375,400 - - - - - - - - 10 420,778 590,400 - - - - - - - - - 11 496,200 - - - - - - - - - - References Andersson, Steen A., and Søren T. Jensen. 1987. Forelæsningsnoter i Sandsynlighedsregning, 3rd ed. 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